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''in the narrow sense''
 
''in the narrow sense''
  
An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591701.png" /> that acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591702.png" />-valued functions (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591703.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591704.png" />) defined on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591705.png" />, according to the formula
+
An operator $  A $
 +
that acts on $  k $-
 +
valued functions ( $  k = \mathbf R $
 +
or $  k = \mathbf C $)  
 +
defined on an open set $  U \subset  \mathbf R  ^ {n} $,  
 +
according to the formula
 +
 
 +
$$ \tag{1 }
 +
A u  =  v  \equiv \
 +
\sum _ {i _ {1} + \dots + i _ {n} \leq  m }
 +
a _ {i _ {1}  \dots i _ {n} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591706.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\frac{\partial  ^ {i _ {1} + \dots + i _ {n }  } u }{\partial  x _ {1} ^ {i _ {1} } \dots \partial  x _ {n} ^ {i _ {n} } }
 +
.
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591707.png" /> are functions with values in the same field, called the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591708.png" />. If the coefficients take values in the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l0591709.png" />-dimensional matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917010.png" />, then the linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917011.png" /> is defined on vector-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917012.png" /> and transforms them into vector-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917013.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917014.png" /> it is called a linear ordinary differential operator, and in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917015.png" /> it is called a linear partial differential operator.
+
Here $  a _ {i _ {1}  \dots i _ {n} } $
 +
are functions with values in the same field, called the coefficients of $  A $.  
 +
If the coefficients take values in the set of $  ( t \times s ) $-
 +
dimensional matrices over $  k $,  
 +
then the linear differential operator $  A $
 +
is defined on vector-valued functions $  u = ( u _ {1} \dots u _ {s} ) $
 +
and transforms them into vector-valued functions $  v = ( v _ {1} \dots v _ {t} ) $.  
 +
In the case $  n = 1 $
 +
it is called a linear ordinary differential operator, and in the case $  n > 1 $
 +
it is called a linear partial differential operator.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917016.png" /> be a [[Differentiable manifold|differentiable manifold]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917018.png" /> be finite-dimensional vector bundles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917019.png" /> (all of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917020.png" />, cf. [[Vector bundle|Vector bundle]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917021.png" /> be the sheaves (cf. [[Sheaf|Sheaf]]) of germs of sections of these bundles of the corresponding smoothness class. A linear differential operator in the wide sense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917022.png" /> is a sheaf mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917023.png" /> satisfying the following condition: Every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917024.png" /> has a coordinate neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917025.png" /> within which the bundles are trivial, while the mapping
+
Let $  X $
 +
be a [[Differentiable manifold|differentiable manifold]] and let $  E $
 +
and $  F $
 +
be finite-dimensional vector bundles on $  X $(
 +
all of class $  C  ^  \infty  $,  
 +
cf. [[Vector bundle|Vector bundle]]). Let $  \widetilde{E}  \rightarrow \widetilde{F}  $
 +
be the sheaves (cf. [[Sheaf|Sheaf]]) of germs of sections of these bundles of the corresponding smoothness class. A linear differential operator in the wide sense $  A: E\rightarrow F $
 +
is a sheaf mapping $  \widetilde{E}  \rightarrow \widetilde{F}  $
 +
satisfying the following condition: Every point $  x \in X $
 +
has a coordinate neighbourhood $  U $
 +
within which the bundles are trivial, while the mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917026.png" /></td> </tr></table>
+
$$
 +
A : \Gamma ( U , E )  \rightarrow  \Gamma ( U , F  ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917027.png" /> is the space of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917028.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917029.png" />, acts according to (1), in which local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917030.png" /> and the trivializations
+
where $  \Gamma ( U , E ) $
 +
is the space of sections of $  E $
 +
over $  U $,  
 +
acts according to (1), in which local coordinates $  x _ {1} \dots x _ {n} $
 +
and the trivializations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917031.png" /></td> </tr></table>
+
$$
 +
E \mid  _ {U}  \cong  U \times k  ^ {s} ,\ \
 +
F  \mid  _ {U}  \cong  U \times k  ^ {t}
 +
$$
  
are used. The smallest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917032.png" /> such that (1) is suitable at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917033.png" /> is called the order of the linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917034.png" />. For example, every non-zero connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917035.png" /> is a linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917036.png" /> of the first order. Another equivalent definition of a linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917037.png" /> is the following: It is a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917038.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917040.png" /> is the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917041.png" />.
+
are used. The smallest number $  m $
 +
such that (1) is suitable at all points $  x \in X $
 +
is called the order of the linear differential operator $  A $.  
 +
For example, every non-zero connection on $  E $
 +
is a linear differential operator $  d : E \rightarrow E \otimes \Omega  ^ {1} ( X) $
 +
of the first order. Another equivalent definition of a linear differential operator $  A : E \rightarrow F $
 +
is the following: It is a linear operator $  A : \Gamma ( X , E ) \rightarrow \Gamma ( X , F  ) $
 +
satisfying the condition $  \supp  Au \subset  \supp  u $,  
 +
where $  \supp  u $
 +
is the support of $  u $.
  
A linear differential operator can be defined on wider function spaces. For example, if a positive metric is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917042.png" /> and a scalar product is defined on the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917044.png" />, then the spaces of square-integrable sections of these bundles are defined. A linear differential operator defined by the local expressions (1) determines a linear unbounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917045.png" />. Under certain weak assumptions the latter may be closed as an operator on Hilbert spaces. This closure is also called a linear differential operator. In a similar way one can construct an operator that acts on Sobolev spaces or on spaces of more general scales.
+
A linear differential operator can be defined on wider function spaces. For example, if a positive metric is defined on $  X $
 +
and a scalar product is defined on the bundles $  E $
 +
and $  F $,  
 +
then the spaces of square-integrable sections of these bundles are defined. A linear differential operator defined by the local expressions (1) determines a linear unbounded operator $  A : L _ {2} ( E) \rightarrow L _ {2} ( F  ) $.  
 +
Under certain weak assumptions the latter may be closed as an operator on Hilbert spaces. This closure is also called a linear differential operator. In a similar way one can construct an operator that acts on Sobolev spaces or on spaces of more general scales.
  
A linear differential operator of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917046.png" /> can be extended to an operator on spaces of generalized sections. Such an extension can be constructed by means of a formally adjoint operator. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917047.png" /> be the bundle dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917048.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917050.png" /> is the trivial one-dimensional bundle) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917051.png" /> be the bundle of differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917052.png" /> of maximal degree. There is defined a bilinear mapping
+
A linear differential operator of class $  C  ^  \infty  $
 +
can be extended to an operator on spaces of generalized sections. Such an extension can be constructed by means of a formally adjoint operator. Let $  E  ^  \prime  $
 +
be the bundle dual to $  E $(
 +
that is, $  E  ^  \prime  = \mathop{\rm Hom} ( E , I ) $,  
 +
where $  I $
 +
is the trivial one-dimensional bundle) and let $  \Omega $
 +
be the bundle of differential forms on $  X $
 +
of maximal degree. There is defined a bilinear mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917053.png" /></td> </tr></table>
+
$$
 +
( \cdot , \cdot ) _ {E} : \Gamma ( X, E) \times \Gamma _ {0} (
 +
X , E  ^  \prime  \otimes \Omega )  \rightarrow  k ,
 +
$$
  
which involves integration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917054.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917055.png" /> is the space of sections with compact support. The formula
+
which involves integration over $  X $.  
 +
Here $  \Gamma _ {0} ( \cdot ) $
 +
is the space of sections with compact support. The formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917056.png" /></td> </tr></table>
+
$$
 +
( {}  ^ {t} A v , u ) _ {E}  = \
 +
( v , A u ) _ {F}  $$
  
 
uniquely defines a linear operator
 
uniquely defines a linear operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917057.png" /></td> </tr></table>
+
$$
 +
{}  ^ {t} A : \Gamma _ {0} ( X , F  ^  \prime  \otimes
 +
\Omega )  \rightarrow  \Gamma _ {0} ( X , E  ^  \prime  \otimes \Omega ) .
 +
$$
 +
 
 +
It is induced by the linear differential operator  $  {}  ^ {t} A : F ^ { \prime } \otimes \Omega \rightarrow E  ^  \prime  \otimes \Omega $
 +
which inside the coordinate neighbourhood  $  U $
 +
has the expression
  
It is induced by the linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917058.png" /> which inside the coordinate neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917059.png" /> has the expression
+
$$
 +
{}  ^ {t} A u  = \
 +
\sum (- 1) ^ {i _ {1} + \dots + i _ {n} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917060.png" /></td> </tr></table>
+
\frac{\partial  ^ {i _ {1} + \dots + i _ {n} }
 +
( {}  ^ {t} a _ {i _ {1}  \dots i _ {n} } u ) }{\partial  x _ {1} ^ {i _ {1} } \dots \partial  x _ {n} ^ {i _ {n} } }
 +
,
 +
$$
  
if the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917061.png" /> is trivialized by the choice of the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917062.png" />. The linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917063.png" /> is said to be formally adjoint with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917064.png" />.
+
if the bundle $  \Omega $
 +
is trivialized by the choice of the section $  d x _ {1} \wedge \dots \wedge d x _ {n} $.  
 +
The linear differential operator $  {}  ^ {t} A $
 +
is said to be formally adjoint with respect to $  A $.
  
In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917065.png" /> convergence is defined according to the following rule: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917066.png" /> if the union of the supports of the sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917067.png" /> belongs to a compact set and if in any coordinate neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917068.png" /> over which there is a trivialization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917069.png" />, the vector-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917070.png" /> converge uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917071.png" /> together with all partial derivatives with respect to local coordinates. The space of all linear functionals is called the space of generalized sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917072.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917073.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917074.png" /> takes convergent sequences to convergent sequences and therefore generates an adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917075.png" />. The latter coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917076.png" /> on the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917077.png" /> and is called the extension of the given linear differential operator to the space of generalized sections. One also considers other extensions of linear differential operators, to spaces of generalized sections of infinite order, to the space of hyperfunctions, etc.
+
In the space $  \Gamma _ {0} ( X , E  ^  \prime  \otimes \Omega ) $
 +
convergence is defined according to the following rule: $  f _ {k} \rightarrow f $
 +
if the union of the supports of the sections $  f _ {k} $
 +
belongs to a compact set and if in any coordinate neighbourhood $  U \subset  X $
 +
over which there is a trivialization of $  E $,  
 +
the vector-valued functions $  f _ {k} $
 +
converge uniformly to $  f $
 +
together with all partial derivatives with respect to local coordinates. The space of all linear functionals is called the space of generalized sections of $  E $
 +
and is denoted by $  D  ^  \prime  ( E) $.  
 +
The operator $  {}  ^ {t} A $
 +
takes convergent sequences to convergent sequences and therefore generates an adjoint operator $  D  ^  \prime  ( E) \rightarrow D  ^  \prime  ( F  ) $.  
 +
The latter coincides with $  A $
 +
on the subspace $  \Gamma ( X , E ) $
 +
and is called the extension of the given linear differential operator to the space of generalized sections. One also considers other extensions of linear differential operators, to spaces of generalized sections of infinite order, to the space of hyperfunctions, etc.
  
A linear differential operator of infinite order is understood to be an operator that acts in some space of analytic functions (sections) and is defined by (1), in which the summation is over an infinite set of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917078.png" />.
+
A linear differential operator of infinite order is understood to be an operator that acts in some space of analytic functions (sections) and is defined by (1), in which the summation is over an infinite set of indices $  i _ {1} \dots i _ {n} , \dots $.
  
The following property characterizes linear differential operators. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917079.png" /> is said to converge to a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917080.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917081.png" /> tends uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917082.png" /> together with all partial derivatives in any coordinate neighbourhood that has compact closure. A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917083.png" /> that takes convergent sequences to convergent sequences is a linear differential operator of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917084.png" /> if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917085.png" /> the function
+
The following property characterizes linear differential operators. A sequence $  \{ f _ {k} \} \subset  \Gamma ( X , E ) $
 +
is said to converge to a section $  f $
 +
if $  f _ {k} $
 +
tends uniformly to $  f $
 +
together with all partial derivatives in any coordinate neighbourhood that has compact closure. A linear operator $  A:  \Gamma _ {0} ( X, E) \rightarrow \Gamma ( X, F  ) $
 +
that takes convergent sequences to convergent sequences is a linear differential operator of order at most $  m $
 +
if and only if for any $  f , g \in C  ^  \infty  ( X) $
 +
the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917086.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\mathop{\rm exp} ( - i \lambda g )
 +
A ( f  \mathop{\rm exp} ( i \lambda g ) )
 +
$$
  
is a polynomial in the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917087.png" /> of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917088.png" />. If this condition is replaced by the assumption that (2) is represented by an [[Asymptotic power series|asymptotic power series]], then one obtains a definition of a linear [[Pseudo-differential operator|pseudo-differential operator]].
+
is a polynomial in the parameter $  \lambda $
 +
of degree at most $  m $.  
 +
If this condition is replaced by the assumption that (2) is represented by an [[Asymptotic power series|asymptotic power series]], then one obtains a definition of a linear [[Pseudo-differential operator|pseudo-differential operator]].
  
Suppose that the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917089.png" /> and also the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917091.png" /> are endowed with a [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917092.png" />-structure]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917093.png" /> is a group. Then the action of this group on any linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917094.png" /> is defined by the formula
+
Suppose that the manifold $  X $
 +
and also the bundles $  E $
 +
and $  F $
 +
are endowed with a [[G-structure| $  G $-
 +
structure]], where $  G $
 +
is a group. Then the action of this group on any linear differential operator $  A : E \rightarrow F $
 +
is defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917095.png" /></td> </tr></table>
+
$$
 +
g  ^ {*} ( A) ( u)  = g
 +
( A ( g  ^ {-} 1 ( u) ) ) .
 +
$$
  
A linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917096.png" /> is said to be invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917097.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917098.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l05917099.png" />.
+
A linear differential operator $  A $
 +
is said to be invariant with respect to $  G $
 +
if $  g  ^ {*} ( A) = A $
 +
for all $  g \in G $.
  
A bundle of jets is an object dual to the space of a linear differential operator. Again suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170100.png" /> is a vector bundle on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170101.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170102.png" />. A bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170104.png" />-jets of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170105.png" /> is a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170106.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170107.png" /> whose fibre over a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170108.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170110.png" /> is a fibre of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170111.png" /> of germs of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170113.png" /> is the subspace of this fibre consisting of germs of sections for which all derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170114.png" /> inclusive vanish at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170115.png" />. The linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170116.png" /> that acts according to the rule: the value of the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170117.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170118.png" /> is equal to the image of the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170119.png" /> in the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170120.png" />, is said to be universal. Next, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170121.png" /> is a bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170122.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170123.png" /> is a bundle homomorphism, that is, a linear differential operator of order zero. The composite
+
A bundle of jets is an object dual to the space of a linear differential operator. Again suppose that $  E $
 +
is a vector bundle on a manifold $  X $
 +
of class $  C  ^  \infty  $.  
 +
A bundle of $  m $-
 +
jets of sections of $  E $
 +
is a vector bundle $  J _ {m} ( E) $
 +
on $  X $
 +
whose fibre over a point $  x $
 +
is equal to $  \widetilde{E}  _ {x} / \widetilde{E}  _ {x} ( m) $,  
 +
where $  \widetilde{E}  _ {x} $
 +
is a fibre of the bundle $  \widetilde{E}  $
 +
of germs of sections of $  E $
 +
and $  \widetilde{E}  _ {x} ( m) $
 +
is the subspace of this fibre consisting of germs of sections for which all derivatives up to order $  m $
 +
inclusive vanish at $  x $.  
 +
The linear differential operator $  d _ {m} : E \rightarrow J _ {m} ( E) $
 +
that acts according to the rule: the value of the section $  d _ {m} ( u) $
 +
at $  x $
 +
is equal to the image of the section $  u $
 +
in the quotient space $  \widetilde{E}  _ {x} / \widetilde{E}  _ {x} ( m) $,  
 +
is said to be universal. Next, suppose that $  F $
 +
is a bundle on $  X $
 +
and that $  A : J _ {m} ( E) \rightarrow F $
 +
is a bundle homomorphism, that is, a linear differential operator of order zero. The composite
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170124.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
E  \rightarrow ^ { {d _ m} }  J _ {m} ( E) \rightarrow ^ { a }  F
 +
$$
  
is a linear differential operator of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170125.png" />. Conversely, every linear differential operator of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170126.png" /> can be represented uniquely as a composition (3).
+
is a linear differential operator of order at most $  m $.  
 +
Conversely, every linear differential operator of order at most $  m $
 +
can be represented uniquely as a composition (3).
  
The symbol (principal system) of a linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170127.png" /> is the family of linear mappings
+
The symbol (principal system) of a linear differential operator $  A : E \rightarrow F $
 +
is the family of linear mappings
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170128.png" /></td> </tr></table>
+
$$
 +
\sigma _ {A} ( x , \xi ) : E _ {x}  \rightarrow  F _ {x} ,
 +
$$
  
depending on a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170129.png" /> of the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170130.png" />. They act according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170131.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170132.png" /> is the homomorphism involved in (3), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170133.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170134.png" /> is the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170135.png" /> equal to the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170136.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170137.png" /> is the germ of a function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170138.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170140.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170141.png" /> has the form (1), then
+
depending on a point $  ( x , \xi ) $
 +
of the cotangent bundle $  T  ^ {*} ( X) $.  
 +
They act according to the formula $  e \rightarrow a ( \xi  ^ {m} e ) / m ! $,  
 +
where $  a $
 +
is the homomorphism involved in (3), $  e \in \widetilde{E}  _ {x} $,  
 +
and $  \xi  ^ {m} e $
 +
is the element of $  J _ {m} ( E) _ {x} $
 +
equal to the image of $  f ^ { m } e $,  
 +
where $  f $
 +
is the germ of a function of class $  C  ^  \infty  $
 +
such that $  f ( x) = 0 $,  
 +
$  d f ( x) = \xi $.  
 +
If $  A $
 +
has the form (1), then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170142.png" /></td> </tr></table>
+
$$
 +
\sigma _ {A} ( x , \xi )  = \
 +
\sum _ {i _ {n} + \dots + i _ {n} = m }
 +
a _ {i _ {1}  \dots i _ {n} } ( x)
 +
\xi _ {1} ^ {i _ {1} } \dots \xi _ {n} ^ {i _ {n} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170143.png" /> are the coordinates in a fibre of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170144.png" />; thus, the symbol is a form of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170145.png" />, homogeneous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170146.png" />. In accordance with this construction of the symbol one introduces the concept of a characteristic. A characteristic of a linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170147.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170148.png" /> at which the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170149.png" /> has non-zero kernel.
+
where $  \xi _ {1} \dots \xi _ {n} $
 +
are the coordinates in a fibre of the bundle $  T  ^ {*} ( U) \cong U \times k  ^ {n} $;  
 +
thus, the symbol is a form of degree $  m $,  
 +
homogeneous in $  \xi $.  
 +
In accordance with this construction of the symbol one introduces the concept of a characteristic. A characteristic of a linear differential operator $  A $
 +
is a point $  ( x , \xi ) \in T  ^ {*} ( X) $
 +
at which the symbol $  \sigma _ {A} $
 +
has non-zero kernel.
  
The classification adopted in the theory of linear differential operators refers mainly to linear differential operators that act in bundles of the same dimension, in fact to operators of the form (1) where the coefficients are square matrices. A linear differential operator is said to be elliptic if it does not have real characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170150.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170151.png" /> (cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]). This class is characterized by the best local properties of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170152.png" />, and also by the fact that boundary value problems in bounded domains are well-posed. The class of hyperbolic linear differential operators is also distinguished by a condition imposed only on the characteristics (cf. [[Hyperbolic partial differential equation|Hyperbolic partial differential equation]]). The property of being hyperbolic is closely connected with the well-posedness of the Cauchy problem with non-analytic data. The class of linear differential operators of principal type is specified by a condition imposed only on the symbol (cf. [[Principal type, partial differential operator of|Principal type, partial differential operator of]]). A theory of local solvability and smoothness of solutions has been developed for such operators. The class of parabolic linear differential operators is distinguished by a condition related not only to the symbol but also to some lower-order terms (cf. [[Parabolic partial differential equation|Parabolic partial differential equation]]). Typical for parabolic linear differential operators are the [[Mixed problem|mixed problem]] and the [[Cauchy problem|Cauchy problem]] with conditions at infinity. The class of hypo-elliptic linear differential operators is specified by the following informal condition: Every a priori generalized solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170153.png" /> with right-hand side from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170154.png" /> itself belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170155.png" />. A number of formal conditions on the expression (1) that guarantee that the operator is hypo-elliptic are known.
+
The classification adopted in the theory of linear differential operators refers mainly to linear differential operators that act in bundles of the same dimension, in fact to operators of the form (1) where the coefficients are square matrices. A linear differential operator is said to be elliptic if it does not have real characteristics $  ( x , \xi ) $
 +
with $  \xi \neq 0 $(
 +
cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]). This class is characterized by the best local properties of solutions of the equation $  Au = w $,  
 +
and also by the fact that boundary value problems in bounded domains are well-posed. The class of hyperbolic linear differential operators is also distinguished by a condition imposed only on the characteristics (cf. [[Hyperbolic partial differential equation|Hyperbolic partial differential equation]]). The property of being hyperbolic is closely connected with the well-posedness of the Cauchy problem with non-analytic data. The class of linear differential operators of principal type is specified by a condition imposed only on the symbol (cf. [[Principal type, partial differential operator of|Principal type, partial differential operator of]]). A theory of local solvability and smoothness of solutions has been developed for such operators. The class of parabolic linear differential operators is distinguished by a condition related not only to the symbol but also to some lower-order terms (cf. [[Parabolic partial differential equation|Parabolic partial differential equation]]). Typical for parabolic linear differential operators are the [[Mixed problem|mixed problem]] and the [[Cauchy problem|Cauchy problem]] with conditions at infinity. The class of hypo-elliptic linear differential operators is specified by the following informal condition: Every a priori generalized solution of the equation $  Au = w $
 +
with right-hand side from $  C  ^  \infty  $
 +
itself belongs to $  C  ^  \infty  $.  
 +
A number of formal conditions on the expression (1) that guarantee that the operator is hypo-elliptic are known.
  
 
Apart from these fundamental types of linear differential operators, one sometimes talks about linear differential operators of mixed or variable type (cf. also [[Mixed-type differential equation|Mixed-type differential equation]]), of linear differential operators of composite type, etc. One also considers problems in unbounded domains with conditions at infinity, boundary value problems with a free boundary, problems of spectral theory, problems of optimal control, etc.
 
Apart from these fundamental types of linear differential operators, one sometimes talks about linear differential operators of mixed or variable type (cf. also [[Mixed-type differential equation|Mixed-type differential equation]]), of linear differential operators of composite type, etc. One also considers problems in unbounded domains with conditions at infinity, boundary value problems with a free boundary, problems of spectral theory, problems of optimal control, etc.
Line 75: Line 267:
 
A complex of linear differential operators is a sequence of linear differential operators
 
A complex of linear differential operators is a sequence of linear differential operators
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170156.png" /></td> </tr></table>
+
$$
 +
E  ^ {*} :\  \dots \rightarrow  E _ {k}  \rightarrow ^ { {A _ k} }  E _ {k+} 1
 +
\rightarrow ^ { {A _ k+} 1 }  E _ {k+} 2  \rightarrow \dots
 +
$$
  
in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170157.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170158.png" />. The cohomology of a complex of linear differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170159.png" /> is the cohomology of the complex of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170160.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170161.png" /> be the cohomology of this complex at the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170162.png" />-th term. The sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170163.png" /> is called the index of the complex of linear differential operators. Thus, the index of an elliptic complex of linear differential operators (that is, such that only finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170164.png" /> are non-zero, and the complex formed by the symbols of the linear differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170165.png" /> is exact at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170166.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170167.png" />) is finite in the case of compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170168.png" />, and the search for formulas that express the index of such a complex in terms of its symbol is the content of a number of investigations that combine the theory of linear differential operators with algebraic geometry and algebraic topology (see [[Index formulas|Index formulas]]).
+
in which $  A _ {k+} 1 A _ {k} = 0 $
 +
for all $  k $.  
 +
The cohomology of a complex of linear differential operators $  E  ^ {*} $
 +
is the cohomology of the complex of vector spaces $  \Gamma ( X , E  ^ {*} ) $.  
 +
Let $  H  ^ {k} $
 +
be the cohomology of this complex at the $  k $-
 +
th term. The sum $  \sum ( - 1 )  ^ {k}  \mathop{\rm dim}  H  ^ {k} $
 +
is called the index of the complex of linear differential operators. Thus, the index of an elliptic complex of linear differential operators (that is, such that only finitely many $  E _ {k} $
 +
are non-zero, and the complex formed by the symbols of the linear differential operators $  A _ {k} $
 +
is exact at all points $  ( x , \xi ) \in T  ^ {*} ( X), $
 +
$  \xi \neq 0 $)  
 +
is finite in the case of compact $  X $,  
 +
and the search for formulas that express the index of such a complex in terms of its symbol is the content of a number of investigations that combine the theory of linear differential operators with algebraic geometry and algebraic topology (see [[Index formulas|Index formulas]]).
  
The definition of a symbol (and of characteristics) described above is not entirely satisfactory for linear differential operators that act in bundles of dimension greater than 1. One of the reasons for this is the fact that the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170169.png" /> may be violated. The following complicated construction, which replaces the concept of the symbol, is more adequate. For every bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170170.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170171.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170172.png" /> one considers the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170173.png" /> of germs of linear differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170174.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170175.png" /> is the one-dimensional trivial bundle. By definition, the value of this sheaf on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170176.png" /> is the totality of all linear differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170177.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170178.png" /> be the subsheaf of it formed by the operators of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170179.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170180.png" /> there is a structure of a sheaf of (non-commutative) algebras, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170181.png" /> has a structure of a left module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170182.png" />, where the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170183.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170184.png" /> is equal to the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170185.png" />. A given linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170186.png" /> determines a morphism of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170187.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170188.png" /> according to the law of composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170189.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170190.png" /> be the cokernel of this morphism. There is an exact sequence of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170191.png" />-modules
+
The definition of a symbol (and of characteristics) described above is not entirely satisfactory for linear differential operators that act in bundles of dimension greater than 1. One of the reasons for this is the fact that the equality $  \sigma _ {AB} = \sigma _ {A} \circ \sigma _ {B} $
 +
may be violated. The following complicated construction, which replaces the concept of the symbol, is more adequate. For every bundle $  E $
 +
on a manifold $  X $
 +
of class $  C  ^  \infty  $
 +
one considers the sheaf $  D ( E) $
 +
of germs of linear differential operators $  E \rightarrow I $,  
 +
where $  I $
 +
is the one-dimensional trivial bundle. By definition, the value of this sheaf on an open set $  U \subset  X $
 +
is the totality of all linear differential operators $  E | _ {U} \rightarrow I | _ {U} $.  
 +
Let $  D _ {k} ( E) $
 +
be the subsheaf of it formed by the operators of order at most $  k $.  
 +
On $  D \equiv D ( I) $
 +
there is a structure of a sheaf of (non-commutative) algebras, and $  D ( E) $
 +
has a structure of a left module over $  D $,  
 +
where the action of $  a \in D $
 +
on $  b \in D ( E) $
 +
is equal to the composite $  ab $.  
 +
A given linear differential operator $  A : E \rightarrow F $
 +
determines a morphism of left $  D $-
 +
modules $  A  ^  \prime  : D ( F  ) \rightarrow D ( E) $
 +
according to the law of composition $  a \rightarrow aA $.  
 +
Let $  M ( A) $
 +
be the cokernel of this morphism. There is an exact sequence of left $  D $-
 +
modules
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170192.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
D ( F)  \rightarrow ^ { {A  ^  \prime } }  D ( E)  \rightarrow ^ { P }  M ( A) \rightarrow  0 ,
 +
$$
  
and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170193.png" />-submodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170194.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170195.png" /> form an increasing filtration in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170196.png" />. The graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170197.png" />-module
+
and the $  O ( X) $-
 +
submodules $  M _ {k} \equiv p ( D _ {k} ( E)) $,
 +
$  k = 0 , 1 \dots $
 +
form an increasing filtration in $  M ( A) $.  
 +
The graded $  O ( X) $-
 +
module
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170198.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm gr}  M ( A)  = \oplus _ { 0 } ^  \infty  M _ {k} / M _ {k-} 1
 +
,\  M _ {-} 1  = 0 ,
 +
$$
  
is called the symbol module of the linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170199.png" />. Since for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170200.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170201.png" /> the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170202.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170203.png" /> takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170204.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170205.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170206.png" /> there is a structure of a [[Graded module|graded module]] over the graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170207.png" />. The annihilator of this module is a homogeneous ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170208.png" />. The characteristic manifold of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170209.png" /> is the set of zeros of this ideal. Since the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170210.png" /> is isomorphic to the symmetric algebra of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170211.png" />, the characteristic manifold is canonically imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170212.png" />, and its intersection with every fibre is an algebraic cone.
+
is called the symbol module of the linear differential operator $  A $.  
 +
Since for any $  k $
 +
and l $
 +
the action of $  D _ {k} $
 +
on $  M ( A) $
 +
takes $  M _ {l} $
 +
into $  M _ {l+} k $,  
 +
in $  \mathop{\rm gr}  M ( A) $
 +
there is a structure of a [[Graded module|graded module]] over the graded algebra $  \mathop{\rm gr}  D \equiv \oplus _ {0}  ^  \infty  D _ {k} / D _ {k-} 1 $.  
 +
The annihilator of this module is a homogeneous ideal in $  \mathop{\rm gr}  D $.  
 +
The characteristic manifold of the operator $  A $
 +
is the set of zeros of this ideal. Since the algebra $  \mathop{\rm gr}  D $
 +
is isomorphic to the symmetric algebra of the tangent bundle $  T ( X) $,  
 +
the characteristic manifold is canonically imbedded in $  T  ^ {*} ( X) $,  
 +
and its intersection with every fibre is an algebraic cone.
  
If the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170213.png" /> and the given bundles have real or complex analytic structure, then the characteristic manifold coincides with the set of roots of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170214.png" />. In this case it is a closed analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170215.png" />, and if it is not empty its dimension is at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170216.png" />. In the case when this dimension is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170217.png" />, the linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170218.png" /> is said to be maximally overdetermined, or holonomic.
+
If the manifold $  X $
 +
and the given bundles have real or complex analytic structure, then the characteristic manifold coincides with the set of roots of the ideal $  \mathop{\rm gr} (  \mathop{\rm ann}  M ( A)) $.  
 +
In this case it is a closed analytic subset of $  T  ^ {*} ( X) $,  
 +
and if it is not empty its dimension is at least $  \mathop{\rm dim}  X $.  
 +
In the case when this dimension is equal to $  \mathop{\rm dim}  X $,  
 +
the linear differential operator $  A $
 +
is said to be maximally overdetermined, or holonomic.
  
The formal theory of general linear differential operators is concerned with the concepts of formal integrability and the resolvent. The property of formal integrability, formalized in the dual terminology of jets, is equivalent to the condition that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170219.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170220.png" /> is locally free. The resolvent of a linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170221.png" /> is understood to be the sequence, extending (4),
+
The formal theory of general linear differential operators is concerned with the concepts of formal integrability and the resolvent. The property of formal integrability, formalized in the dual terminology of jets, is equivalent to the condition that the $  O ( X) $-
 +
module $  \mathop{\rm gr}  M ( A) $
 +
is locally free. The resolvent of a linear differential operator $  A $
 +
is understood to be the sequence, extending (4),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170222.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  D ( F _ {1} )  \rightarrow ^ { {A _ 1}  ^  \prime  }  D ( F  )  \mathop \rightarrow \limits ^ { {A
 +
^  \prime  }}  D ( E)  \rightarrow  M ( A) ,
 +
$$
  
in which all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170223.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170224.png" /> are linear differential operators. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170225.png" /> is called the compatibility operator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170226.png" />. Formal integrability ensures the local existence of the resolvent.
+
in which all the $  A _ {k} $,
 +
$  k = 1 , 2 \dots $
 +
are linear differential operators. In particular, $  A _ {1} $
 +
is called the compatibility operator for $  A $.  
 +
Formal integrability ensures the local existence of the resolvent.
  
In the literature use is made of the terms "overdetermined" and "underdetermined" for systems of differential equations; however, there is no satisfactory general definition. The following could serve as an approximation to such a definition: There is a non-zero linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170227.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170228.png" /> (overdetermination), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170229.png" /> (underdetermination). For example, the linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170230.png" /> equal to the restriction of the operator of exterior differentiation to forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170231.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170232.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170233.png" /> is underdetermined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170234.png" />, overdetermined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170235.png" /> and holonomic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170236.png" />.
+
In the literature use is made of the terms "overdetermined" and "underdetermined" for systems of differential equations; however, there is no satisfactory general definition. The following could serve as an approximation to such a definition: There is a non-zero linear differential operator $  B $
 +
such that $  BA = 0 $(
 +
overdetermination), $  AB = 0 $(
 +
underdetermination). For example, the linear differential operator $  d $
 +
equal to the restriction of the operator of exterior differentiation to forms of degree $  k $
 +
on a manifold $  X $
 +
of dimension $  n $
 +
is underdetermined for $  k > 0 $,  
 +
overdetermined for $  k < n $
 +
and holonomic for $  k = 0 $.
  
The main problems studied for general linear differential operators are the following: The solvability of an equation with right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170237.png" /> if a compatibility condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170238.png" /> is satisfied; the possibility of extending solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170239.png" /> to a larger domain (an effect connected with overdetermination); and the representation of the general solution in terms of a solution of special form. The last problem can be stated more specifically for invariant operators, for example for linear differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170240.png" /> with constant or periodic coefficients: To describe a representation of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170241.png" /> in the space of solutions as an integral (in some sense) over all indecomposable subrepresentations. In determining operators with constant coefficients such a representation is specified by an integral with respect to exponents (exponential representation), and for operators with periodic coefficients by an integral with respect to Floquet-generalized solutions.
+
The main problems studied for general linear differential operators are the following: The solvability of an equation with right-hand side $  Au = w $
 +
if a compatibility condition $  A _ {1} u = 0 $
 +
is satisfied; the possibility of extending solutions of the equation $  Au = 0 $
 +
to a larger domain (an effect connected with overdetermination); and the representation of the general solution in terms of a solution of special form. The last problem can be stated more specifically for invariant operators, for example for linear differential operators in $  \mathbf R  ^ {n} $
 +
with constant or periodic coefficients: To describe a representation of a group $  G $
 +
in the space of solutions as an integral (in some sense) over all indecomposable subrepresentations. In determining operators with constant coefficients such a representation is specified by an integral with respect to exponents (exponential representation), and for operators with periodic coefficients by an integral with respect to Floquet-generalized solutions.
  
Linear differential operators are also defined on arbitrary algebraic structures. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170242.png" /> be a commutative ring and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170243.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170244.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170245.png" />-modules. A mapping of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170246.png" /> is called a linear differential operator of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170247.png" /> if it is additive and for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170248.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170249.png" /> is a linear differential operator of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170250.png" />. A linear differential operator of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170251.png" /> means the zero mapping. In particular, a linear differential operator of order zero is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170252.png" />-modules, and conversely. Every derivation (cf. [[Derivation in a ring|Derivation in a ring]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170253.png" /> is a linear differential operator of the first order (or equal to zero). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170254.png" /> is an algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170255.png" />, then a linear differential operator over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170256.png" /> is a linear differential operator over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170257.png" /> that is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170258.png" />-linear mapping. Such a linear differential operator has a number of the formal properties of ordinary linear differential operators. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170259.png" /> is the algebra of all formal power series over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170260.png" /> or the algebra of convergent power series over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170261.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170262.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170263.png" /> are free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170264.png" />-modules of finite type, then every linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170265.png" /> of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170266.png" /> can be written uniquely in the form (1).
+
Linear differential operators are also defined on arbitrary algebraic structures. Let $  R $
 +
be a commutative ring and let $  E $
 +
and $  F $
 +
be $  R $-
 +
modules. A mapping of sets $  A : E \rightarrow F $
 +
is called a linear differential operator of order at most $  m $
 +
if it is additive and for any element $  a \in R $
 +
the mapping $  aA- Aa $
 +
is a linear differential operator of order at most $  m- 1 $.  
 +
A linear differential operator of order at most $  - 1 $
 +
means the zero mapping. In particular, a linear differential operator of order zero is a homomorphism of $  R $-
 +
modules, and conversely. Every derivation (cf. [[Derivation in a ring|Derivation in a ring]]) $  v : R \rightarrow F $
 +
is a linear differential operator of the first order (or equal to zero). If $  R $
 +
is an algebra over a field $  k $,  
 +
then a linear differential operator over $  R $
 +
is a linear differential operator over the ring $  R $
 +
that is a $  k $-
 +
linear mapping. Such a linear differential operator has a number of the formal properties of ordinary linear differential operators. If $  R $
 +
is the algebra of all formal power series over $  k $
 +
or the algebra of convergent power series over $  k $,  
 +
and if $  E $
 +
and $  F $
 +
are free $  R $-
 +
modules of finite type, then every linear differential operator $  A : E \rightarrow F $
 +
of order at most $  m $
 +
can be written uniquely in the form (1).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170267.png" /> be a [[Ringed space|ringed space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170268.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170269.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170270.png" />-modules. A linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170271.png" /> is any sheaf morphism that acts in the fibres over every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170272.png" /> like a linear differential operator over the ring (algebra) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170273.png" />. Linear differential operators that act in modules or sheaves of modules have been used in a number of questions in algebraic geometry.
+
Let $  ( X , {\mathcal O} ) $
 +
be a [[Ringed space|ringed space]] and let $  E $
 +
and $  F $
 +
be $  {\mathcal O} $-
 +
modules. A linear differential operator $  A : E \rightarrow F $
 +
is any sheaf morphism that acts in the fibres over every point $  x \in X $
 +
like a linear differential operator over the ring (algebra) $  {\mathcal O} _ {x} $.  
 +
Linear differential operators that act in modules or sheaves of modules have been used in a number of questions in algebraic geometry.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Peetre,   "Uniqueness in the Cauchy problem for elliptic equations with double characteristics" ''Math. Scand.'' , '''8''' (1960) pp. 116–120</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Hörmander, , ''Pseudo-differential operators'' , Moscow (1967) pp. 63–87; 166–296; 297–367 (In Russian; translated from English)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Bernshtein,   "The analytic continuation of generalized functions with respect to a parameter" ''Funct. Anal. Appl.'' , '''6''' : 4 (1972) pp. 273–285 ''Funktsional. Anal. i Prilozhen.'' , '''6''' : 4 (1972) pp. 26–40</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1–4''' , Springer (1983–1985)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.P. Palamodov,   "Linear differential operators with constant coefficients" , Springer (1970) (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> P. Hartman,   "Ordinary differential equations" , Birkhäuser (1982)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> R.S. Palais,   "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> V.P. Palamodov,   "Systems of linear differential equations" ''Itogi Nauk. Mat. Anal. 1968'' (1969) pp. 5–37 (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> J.-E. Björk,   "Rings of differential operators" , North-Holland (1979)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M. Kashiwara,   "Microfunctions and pseudodifferential equations" H. Komatsu (ed.) , ''Hyperfunctions and pseudodifferential equations. Proc. Conf. Katata, 1971'' , ''Lect. notes in math.'' , '''287''' , Springer (1973) pp. 265–529</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Peetre, "Uniqueness in the Cauchy problem for elliptic equations with double characteristics" ''Math. Scand.'' , '''8''' (1960) pp. 116–120 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Hörmander, , ''Pseudo-differential operators'' , Moscow (1967) pp. 63–87; 166–296; 297–367 (In Russian; translated from English) {{MR|0383152}} {{ZBL|0167.09603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Bernshtein, "The analytic continuation of generalized functions with respect to a parameter" ''Funct. Anal. Appl.'' , '''6''' : 4 (1972) pp. 273–285 ''Funktsional. Anal. i Prilozhen.'' , '''6''' : 4 (1972) pp. 26–40 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German) {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) {{MR|104888}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''1–4''' , Springer (1983–1985) {{MR|2512677}} {{MR|2304165}} {{MR|2108588}} {{MR|1996773}} {{MR|1481433}} {{MR|1313500}} {{MR|1065993}} {{MR|1065136}} {{MR|0961959}} {{MR|0925821}} {{MR|0881605}} {{MR|0862624}} {{MR|1540773}} {{MR|0781537}} {{MR|0781536}} {{MR|0717035}} {{MR|0705278}} {{ZBL|1178.35003}} {{ZBL|1115.35005}} {{ZBL|1062.35004}} {{ZBL|1028.35001}} {{ZBL|0712.35001}} {{ZBL|0687.35002}} {{ZBL|0619.35002}} {{ZBL|0619.35001}} {{ZBL|0612.35001}} {{ZBL|0601.35001}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.P. Palamodov, "Linear differential operators with constant coefficients" , Springer (1970) (Translated from Russian) {{MR|0264197}} {{ZBL|0191.43401}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) {{MR|0658490}} {{ZBL|0476.34002}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> R.S. Palais, "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965) {{MR|0198494}} {{ZBL|0137.17002}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> V.P. Palamodov, "Systems of linear differential equations" ''Itogi Nauk. Mat. Anal. 1968'' (1969) pp. 5–37 (In Russian) {{MR|0369889}} {{ZBL|0245.35002}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> J.-E. Björk, "Rings of differential operators" , North-Holland (1979) {{MR|0549189}} {{ZBL|0499.13009}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M. Kashiwara, "Microfunctions and pseudodifferential equations" H. Komatsu (ed.) , ''Hyperfunctions and pseudodifferential equations. Proc. Conf. Katata, 1971'' , ''Lect. notes in math.'' , '''287''' , Springer (1973) pp. 265–529 {{MR|420735}} {{ZBL|}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
Cf. also [[Differential operator|Differential operator]]; [[Differential operator on a module|Differential operator on a module]]; [[D-module|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059170/l059170274.png" />-module]]; [[Hyperfunction|Hyperfunction]]; [[Microlocal analysis|Microlocal analysis]].
+
Cf. also [[Differential operator|Differential operator]]; [[Differential operator on a module|Differential operator on a module]]; [[D-module| $  D $-
 +
module]]; [[Hyperfunction|Hyperfunction]]; [[Microlocal analysis|Microlocal analysis]].

Latest revision as of 22:17, 5 June 2020


in the narrow sense

An operator $ A $ that acts on $ k $- valued functions ( $ k = \mathbf R $ or $ k = \mathbf C $) defined on an open set $ U \subset \mathbf R ^ {n} $, according to the formula

$$ \tag{1 } A u = v \equiv \ \sum _ {i _ {1} + \dots + i _ {n} \leq m } a _ {i _ {1} \dots i _ {n} } \frac{\partial ^ {i _ {1} + \dots + i _ {n } } u }{\partial x _ {1} ^ {i _ {1} } \dots \partial x _ {n} ^ {i _ {n} } } . $$

Here $ a _ {i _ {1} \dots i _ {n} } $ are functions with values in the same field, called the coefficients of $ A $. If the coefficients take values in the set of $ ( t \times s ) $- dimensional matrices over $ k $, then the linear differential operator $ A $ is defined on vector-valued functions $ u = ( u _ {1} \dots u _ {s} ) $ and transforms them into vector-valued functions $ v = ( v _ {1} \dots v _ {t} ) $. In the case $ n = 1 $ it is called a linear ordinary differential operator, and in the case $ n > 1 $ it is called a linear partial differential operator.

Let $ X $ be a differentiable manifold and let $ E $ and $ F $ be finite-dimensional vector bundles on $ X $( all of class $ C ^ \infty $, cf. Vector bundle). Let $ \widetilde{E} \rightarrow \widetilde{F} $ be the sheaves (cf. Sheaf) of germs of sections of these bundles of the corresponding smoothness class. A linear differential operator in the wide sense $ A: E\rightarrow F $ is a sheaf mapping $ \widetilde{E} \rightarrow \widetilde{F} $ satisfying the following condition: Every point $ x \in X $ has a coordinate neighbourhood $ U $ within which the bundles are trivial, while the mapping

$$ A : \Gamma ( U , E ) \rightarrow \Gamma ( U , F ) , $$

where $ \Gamma ( U , E ) $ is the space of sections of $ E $ over $ U $, acts according to (1), in which local coordinates $ x _ {1} \dots x _ {n} $ and the trivializations

$$ E \mid _ {U} \cong U \times k ^ {s} ,\ \ F \mid _ {U} \cong U \times k ^ {t} $$

are used. The smallest number $ m $ such that (1) is suitable at all points $ x \in X $ is called the order of the linear differential operator $ A $. For example, every non-zero connection on $ E $ is a linear differential operator $ d : E \rightarrow E \otimes \Omega ^ {1} ( X) $ of the first order. Another equivalent definition of a linear differential operator $ A : E \rightarrow F $ is the following: It is a linear operator $ A : \Gamma ( X , E ) \rightarrow \Gamma ( X , F ) $ satisfying the condition $ \supp Au \subset \supp u $, where $ \supp u $ is the support of $ u $.

A linear differential operator can be defined on wider function spaces. For example, if a positive metric is defined on $ X $ and a scalar product is defined on the bundles $ E $ and $ F $, then the spaces of square-integrable sections of these bundles are defined. A linear differential operator defined by the local expressions (1) determines a linear unbounded operator $ A : L _ {2} ( E) \rightarrow L _ {2} ( F ) $. Under certain weak assumptions the latter may be closed as an operator on Hilbert spaces. This closure is also called a linear differential operator. In a similar way one can construct an operator that acts on Sobolev spaces or on spaces of more general scales.

A linear differential operator of class $ C ^ \infty $ can be extended to an operator on spaces of generalized sections. Such an extension can be constructed by means of a formally adjoint operator. Let $ E ^ \prime $ be the bundle dual to $ E $( that is, $ E ^ \prime = \mathop{\rm Hom} ( E , I ) $, where $ I $ is the trivial one-dimensional bundle) and let $ \Omega $ be the bundle of differential forms on $ X $ of maximal degree. There is defined a bilinear mapping

$$ ( \cdot , \cdot ) _ {E} : \Gamma ( X, E) \times \Gamma _ {0} ( X , E ^ \prime \otimes \Omega ) \rightarrow k , $$

which involves integration over $ X $. Here $ \Gamma _ {0} ( \cdot ) $ is the space of sections with compact support. The formula

$$ ( {} ^ {t} A v , u ) _ {E} = \ ( v , A u ) _ {F} $$

uniquely defines a linear operator

$$ {} ^ {t} A : \Gamma _ {0} ( X , F ^ \prime \otimes \Omega ) \rightarrow \Gamma _ {0} ( X , E ^ \prime \otimes \Omega ) . $$

It is induced by the linear differential operator $ {} ^ {t} A : F ^ { \prime } \otimes \Omega \rightarrow E ^ \prime \otimes \Omega $ which inside the coordinate neighbourhood $ U $ has the expression

$$ {} ^ {t} A u = \ \sum (- 1) ^ {i _ {1} + \dots + i _ {n} } \frac{\partial ^ {i _ {1} + \dots + i _ {n} } ( {} ^ {t} a _ {i _ {1} \dots i _ {n} } u ) }{\partial x _ {1} ^ {i _ {1} } \dots \partial x _ {n} ^ {i _ {n} } } , $$

if the bundle $ \Omega $ is trivialized by the choice of the section $ d x _ {1} \wedge \dots \wedge d x _ {n} $. The linear differential operator $ {} ^ {t} A $ is said to be formally adjoint with respect to $ A $.

In the space $ \Gamma _ {0} ( X , E ^ \prime \otimes \Omega ) $ convergence is defined according to the following rule: $ f _ {k} \rightarrow f $ if the union of the supports of the sections $ f _ {k} $ belongs to a compact set and if in any coordinate neighbourhood $ U \subset X $ over which there is a trivialization of $ E $, the vector-valued functions $ f _ {k} $ converge uniformly to $ f $ together with all partial derivatives with respect to local coordinates. The space of all linear functionals is called the space of generalized sections of $ E $ and is denoted by $ D ^ \prime ( E) $. The operator $ {} ^ {t} A $ takes convergent sequences to convergent sequences and therefore generates an adjoint operator $ D ^ \prime ( E) \rightarrow D ^ \prime ( F ) $. The latter coincides with $ A $ on the subspace $ \Gamma ( X , E ) $ and is called the extension of the given linear differential operator to the space of generalized sections. One also considers other extensions of linear differential operators, to spaces of generalized sections of infinite order, to the space of hyperfunctions, etc.

A linear differential operator of infinite order is understood to be an operator that acts in some space of analytic functions (sections) and is defined by (1), in which the summation is over an infinite set of indices $ i _ {1} \dots i _ {n} , \dots $.

The following property characterizes linear differential operators. A sequence $ \{ f _ {k} \} \subset \Gamma ( X , E ) $ is said to converge to a section $ f $ if $ f _ {k} $ tends uniformly to $ f $ together with all partial derivatives in any coordinate neighbourhood that has compact closure. A linear operator $ A: \Gamma _ {0} ( X, E) \rightarrow \Gamma ( X, F ) $ that takes convergent sequences to convergent sequences is a linear differential operator of order at most $ m $ if and only if for any $ f , g \in C ^ \infty ( X) $ the function

$$ \tag{2 } \mathop{\rm exp} ( - i \lambda g ) A ( f \mathop{\rm exp} ( i \lambda g ) ) $$

is a polynomial in the parameter $ \lambda $ of degree at most $ m $. If this condition is replaced by the assumption that (2) is represented by an asymptotic power series, then one obtains a definition of a linear pseudo-differential operator.

Suppose that the manifold $ X $ and also the bundles $ E $ and $ F $ are endowed with a $ G $- structure, where $ G $ is a group. Then the action of this group on any linear differential operator $ A : E \rightarrow F $ is defined by the formula

$$ g ^ {*} ( A) ( u) = g ( A ( g ^ {-} 1 ( u) ) ) . $$

A linear differential operator $ A $ is said to be invariant with respect to $ G $ if $ g ^ {*} ( A) = A $ for all $ g \in G $.

A bundle of jets is an object dual to the space of a linear differential operator. Again suppose that $ E $ is a vector bundle on a manifold $ X $ of class $ C ^ \infty $. A bundle of $ m $- jets of sections of $ E $ is a vector bundle $ J _ {m} ( E) $ on $ X $ whose fibre over a point $ x $ is equal to $ \widetilde{E} _ {x} / \widetilde{E} _ {x} ( m) $, where $ \widetilde{E} _ {x} $ is a fibre of the bundle $ \widetilde{E} $ of germs of sections of $ E $ and $ \widetilde{E} _ {x} ( m) $ is the subspace of this fibre consisting of germs of sections for which all derivatives up to order $ m $ inclusive vanish at $ x $. The linear differential operator $ d _ {m} : E \rightarrow J _ {m} ( E) $ that acts according to the rule: the value of the section $ d _ {m} ( u) $ at $ x $ is equal to the image of the section $ u $ in the quotient space $ \widetilde{E} _ {x} / \widetilde{E} _ {x} ( m) $, is said to be universal. Next, suppose that $ F $ is a bundle on $ X $ and that $ A : J _ {m} ( E) \rightarrow F $ is a bundle homomorphism, that is, a linear differential operator of order zero. The composite

$$ \tag{3 } E \rightarrow ^ { {d _ m} } J _ {m} ( E) \rightarrow ^ { a } F $$

is a linear differential operator of order at most $ m $. Conversely, every linear differential operator of order at most $ m $ can be represented uniquely as a composition (3).

The symbol (principal system) of a linear differential operator $ A : E \rightarrow F $ is the family of linear mappings

$$ \sigma _ {A} ( x , \xi ) : E _ {x} \rightarrow F _ {x} , $$

depending on a point $ ( x , \xi ) $ of the cotangent bundle $ T ^ {*} ( X) $. They act according to the formula $ e \rightarrow a ( \xi ^ {m} e ) / m ! $, where $ a $ is the homomorphism involved in (3), $ e \in \widetilde{E} _ {x} $, and $ \xi ^ {m} e $ is the element of $ J _ {m} ( E) _ {x} $ equal to the image of $ f ^ { m } e $, where $ f $ is the germ of a function of class $ C ^ \infty $ such that $ f ( x) = 0 $, $ d f ( x) = \xi $. If $ A $ has the form (1), then

$$ \sigma _ {A} ( x , \xi ) = \ \sum _ {i _ {n} + \dots + i _ {n} = m } a _ {i _ {1} \dots i _ {n} } ( x) \xi _ {1} ^ {i _ {1} } \dots \xi _ {n} ^ {i _ {n} } , $$

where $ \xi _ {1} \dots \xi _ {n} $ are the coordinates in a fibre of the bundle $ T ^ {*} ( U) \cong U \times k ^ {n} $; thus, the symbol is a form of degree $ m $, homogeneous in $ \xi $. In accordance with this construction of the symbol one introduces the concept of a characteristic. A characteristic of a linear differential operator $ A $ is a point $ ( x , \xi ) \in T ^ {*} ( X) $ at which the symbol $ \sigma _ {A} $ has non-zero kernel.

The classification adopted in the theory of linear differential operators refers mainly to linear differential operators that act in bundles of the same dimension, in fact to operators of the form (1) where the coefficients are square matrices. A linear differential operator is said to be elliptic if it does not have real characteristics $ ( x , \xi ) $ with $ \xi \neq 0 $( cf. also Elliptic partial differential equation). This class is characterized by the best local properties of solutions of the equation $ Au = w $, and also by the fact that boundary value problems in bounded domains are well-posed. The class of hyperbolic linear differential operators is also distinguished by a condition imposed only on the characteristics (cf. Hyperbolic partial differential equation). The property of being hyperbolic is closely connected with the well-posedness of the Cauchy problem with non-analytic data. The class of linear differential operators of principal type is specified by a condition imposed only on the symbol (cf. Principal type, partial differential operator of). A theory of local solvability and smoothness of solutions has been developed for such operators. The class of parabolic linear differential operators is distinguished by a condition related not only to the symbol but also to some lower-order terms (cf. Parabolic partial differential equation). Typical for parabolic linear differential operators are the mixed problem and the Cauchy problem with conditions at infinity. The class of hypo-elliptic linear differential operators is specified by the following informal condition: Every a priori generalized solution of the equation $ Au = w $ with right-hand side from $ C ^ \infty $ itself belongs to $ C ^ \infty $. A number of formal conditions on the expression (1) that guarantee that the operator is hypo-elliptic are known.

Apart from these fundamental types of linear differential operators, one sometimes talks about linear differential operators of mixed or variable type (cf. also Mixed-type differential equation), of linear differential operators of composite type, etc. One also considers problems in unbounded domains with conditions at infinity, boundary value problems with a free boundary, problems of spectral theory, problems of optimal control, etc.

A complex of linear differential operators is a sequence of linear differential operators

$$ E ^ {*} :\ \dots \rightarrow E _ {k} \rightarrow ^ { {A _ k} } E _ {k+} 1 \rightarrow ^ { {A _ k+} 1 } E _ {k+} 2 \rightarrow \dots $$

in which $ A _ {k+} 1 A _ {k} = 0 $ for all $ k $. The cohomology of a complex of linear differential operators $ E ^ {*} $ is the cohomology of the complex of vector spaces $ \Gamma ( X , E ^ {*} ) $. Let $ H ^ {k} $ be the cohomology of this complex at the $ k $- th term. The sum $ \sum ( - 1 ) ^ {k} \mathop{\rm dim} H ^ {k} $ is called the index of the complex of linear differential operators. Thus, the index of an elliptic complex of linear differential operators (that is, such that only finitely many $ E _ {k} $ are non-zero, and the complex formed by the symbols of the linear differential operators $ A _ {k} $ is exact at all points $ ( x , \xi ) \in T ^ {*} ( X), $ $ \xi \neq 0 $) is finite in the case of compact $ X $, and the search for formulas that express the index of such a complex in terms of its symbol is the content of a number of investigations that combine the theory of linear differential operators with algebraic geometry and algebraic topology (see Index formulas).

The definition of a symbol (and of characteristics) described above is not entirely satisfactory for linear differential operators that act in bundles of dimension greater than 1. One of the reasons for this is the fact that the equality $ \sigma _ {AB} = \sigma _ {A} \circ \sigma _ {B} $ may be violated. The following complicated construction, which replaces the concept of the symbol, is more adequate. For every bundle $ E $ on a manifold $ X $ of class $ C ^ \infty $ one considers the sheaf $ D ( E) $ of germs of linear differential operators $ E \rightarrow I $, where $ I $ is the one-dimensional trivial bundle. By definition, the value of this sheaf on an open set $ U \subset X $ is the totality of all linear differential operators $ E | _ {U} \rightarrow I | _ {U} $. Let $ D _ {k} ( E) $ be the subsheaf of it formed by the operators of order at most $ k $. On $ D \equiv D ( I) $ there is a structure of a sheaf of (non-commutative) algebras, and $ D ( E) $ has a structure of a left module over $ D $, where the action of $ a \in D $ on $ b \in D ( E) $ is equal to the composite $ ab $. A given linear differential operator $ A : E \rightarrow F $ determines a morphism of left $ D $- modules $ A ^ \prime : D ( F ) \rightarrow D ( E) $ according to the law of composition $ a \rightarrow aA $. Let $ M ( A) $ be the cokernel of this morphism. There is an exact sequence of left $ D $- modules

$$ \tag{4 } D ( F) \rightarrow ^ { {A ^ \prime } } D ( E) \rightarrow ^ { P } M ( A) \rightarrow 0 , $$

and the $ O ( X) $- submodules $ M _ {k} \equiv p ( D _ {k} ( E)) $, $ k = 0 , 1 \dots $ form an increasing filtration in $ M ( A) $. The graded $ O ( X) $- module

$$ \mathop{\rm gr} M ( A) = \oplus _ { 0 } ^ \infty M _ {k} / M _ {k-} 1 ,\ M _ {-} 1 = 0 , $$

is called the symbol module of the linear differential operator $ A $. Since for any $ k $ and $ l $ the action of $ D _ {k} $ on $ M ( A) $ takes $ M _ {l} $ into $ M _ {l+} k $, in $ \mathop{\rm gr} M ( A) $ there is a structure of a graded module over the graded algebra $ \mathop{\rm gr} D \equiv \oplus _ {0} ^ \infty D _ {k} / D _ {k-} 1 $. The annihilator of this module is a homogeneous ideal in $ \mathop{\rm gr} D $. The characteristic manifold of the operator $ A $ is the set of zeros of this ideal. Since the algebra $ \mathop{\rm gr} D $ is isomorphic to the symmetric algebra of the tangent bundle $ T ( X) $, the characteristic manifold is canonically imbedded in $ T ^ {*} ( X) $, and its intersection with every fibre is an algebraic cone.

If the manifold $ X $ and the given bundles have real or complex analytic structure, then the characteristic manifold coincides with the set of roots of the ideal $ \mathop{\rm gr} ( \mathop{\rm ann} M ( A)) $. In this case it is a closed analytic subset of $ T ^ {*} ( X) $, and if it is not empty its dimension is at least $ \mathop{\rm dim} X $. In the case when this dimension is equal to $ \mathop{\rm dim} X $, the linear differential operator $ A $ is said to be maximally overdetermined, or holonomic.

The formal theory of general linear differential operators is concerned with the concepts of formal integrability and the resolvent. The property of formal integrability, formalized in the dual terminology of jets, is equivalent to the condition that the $ O ( X) $- module $ \mathop{\rm gr} M ( A) $ is locally free. The resolvent of a linear differential operator $ A $ is understood to be the sequence, extending (4),

$$ \dots \rightarrow D ( F _ {1} ) \rightarrow ^ { {A _ 1} ^ \prime } D ( F ) \mathop \rightarrow \limits ^ { {A ^ \prime }} D ( E) \rightarrow M ( A) , $$

in which all the $ A _ {k} $, $ k = 1 , 2 \dots $ are linear differential operators. In particular, $ A _ {1} $ is called the compatibility operator for $ A $. Formal integrability ensures the local existence of the resolvent.

In the literature use is made of the terms "overdetermined" and "underdetermined" for systems of differential equations; however, there is no satisfactory general definition. The following could serve as an approximation to such a definition: There is a non-zero linear differential operator $ B $ such that $ BA = 0 $( overdetermination), $ AB = 0 $( underdetermination). For example, the linear differential operator $ d $ equal to the restriction of the operator of exterior differentiation to forms of degree $ k $ on a manifold $ X $ of dimension $ n $ is underdetermined for $ k > 0 $, overdetermined for $ k < n $ and holonomic for $ k = 0 $.

The main problems studied for general linear differential operators are the following: The solvability of an equation with right-hand side $ Au = w $ if a compatibility condition $ A _ {1} u = 0 $ is satisfied; the possibility of extending solutions of the equation $ Au = 0 $ to a larger domain (an effect connected with overdetermination); and the representation of the general solution in terms of a solution of special form. The last problem can be stated more specifically for invariant operators, for example for linear differential operators in $ \mathbf R ^ {n} $ with constant or periodic coefficients: To describe a representation of a group $ G $ in the space of solutions as an integral (in some sense) over all indecomposable subrepresentations. In determining operators with constant coefficients such a representation is specified by an integral with respect to exponents (exponential representation), and for operators with periodic coefficients by an integral with respect to Floquet-generalized solutions.

Linear differential operators are also defined on arbitrary algebraic structures. Let $ R $ be a commutative ring and let $ E $ and $ F $ be $ R $- modules. A mapping of sets $ A : E \rightarrow F $ is called a linear differential operator of order at most $ m $ if it is additive and for any element $ a \in R $ the mapping $ aA- Aa $ is a linear differential operator of order at most $ m- 1 $. A linear differential operator of order at most $ - 1 $ means the zero mapping. In particular, a linear differential operator of order zero is a homomorphism of $ R $- modules, and conversely. Every derivation (cf. Derivation in a ring) $ v : R \rightarrow F $ is a linear differential operator of the first order (or equal to zero). If $ R $ is an algebra over a field $ k $, then a linear differential operator over $ R $ is a linear differential operator over the ring $ R $ that is a $ k $- linear mapping. Such a linear differential operator has a number of the formal properties of ordinary linear differential operators. If $ R $ is the algebra of all formal power series over $ k $ or the algebra of convergent power series over $ k $, and if $ E $ and $ F $ are free $ R $- modules of finite type, then every linear differential operator $ A : E \rightarrow F $ of order at most $ m $ can be written uniquely in the form (1).

Let $ ( X , {\mathcal O} ) $ be a ringed space and let $ E $ and $ F $ be $ {\mathcal O} $- modules. A linear differential operator $ A : E \rightarrow F $ is any sheaf morphism that acts in the fibres over every point $ x \in X $ like a linear differential operator over the ring (algebra) $ {\mathcal O} _ {x} $. Linear differential operators that act in modules or sheaves of modules have been used in a number of questions in algebraic geometry.

References

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[3] I.N. Bernshtein, "The analytic continuation of generalized functions with respect to a parameter" Funct. Anal. Appl. , 6 : 4 (1972) pp. 273–285 Funktsional. Anal. i Prilozhen. , 6 : 4 (1972) pp. 26–40
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Comments

Cf. also Differential operator; Differential operator on a module; $ D $- module; Hyperfunction; Microlocal analysis.

How to Cite This Entry:
Linear differential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_differential_operator&oldid=18859
This article was adapted from an original article by V.P. Palamodov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article