Namespaces
Variants
Actions

Difference between revisions of "Self-adjoint differential equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A linear ordinary differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s0838701.png" /> that coincides with the [[Adjoint differential equation|adjoint differential equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s0838702.png" />. Here
+
<!--
 +
s0838701.png
 +
$#A+1 = 47 n = 0
 +
$#C+1 = 47 : ~/encyclopedia/old_files/data/S083/S.0803870 Self\AAhadjoint differential equation
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/s/s083/s083870/s0838703.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/s/s083/s083870/s0838704.png" /></td> </tr></table>
+
A linear ordinary differential equation  $  l ( y) = 0 $
 +
that coincides with the [[Adjoint differential equation|adjoint differential equation]]  $  l  ^ {*} ( y) = 0 $.
 +
Here
 +
 
 +
$$
 +
l ( y)  \equiv  a _ {0} ( t) y  ^ {(} n) + \dots + a _ {n} ( t) y,
 +
$$
 +
 
 +
$$
 +
l  ^ {*} ( y)  \equiv  (- 1)  ^ {n} ( \overline{a}\; _ {0} ( t) y)
 +
^ {(} n) + \dots + (- 1)  ^ {0} \overline{a}\; _ {n} ( t) y,
 +
$$
  
 
where
 
where
  
<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/s/s083/s083870/s0838705.png" /></td> </tr></table>
+
$$
 +
y ^ {( \nu ) }  = \
  
<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/s/s083/s083870/s0838706.png" /></td> </tr></table>
+
\frac{d  ^  \nu  y }{dt  ^  \nu  }
 +
,\ \
 +
y ( \cdot )  \in  C  ^ {n} ( I),\ \
 +
a _ {k} ( \cdot )  \in  C ^ {n - k } ( I),
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s0838707.png" /> is the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s0838708.png" />-times continuously-differentiable complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s0838709.png" />, and the bar denotes complex conjugation.
+
$$
 +
a _ {0} ( t)  \neq  0,\  t  \in  I,
 +
$$
  
The left-hand side of every self-adjoint differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387010.png" /> is a sum of expressions of the form
+
$  C  ^ {m} ( I) $
 +
is the space of  $  m $-
 +
times continuously-differentiable complex-valued functions on  $  I = ( \alpha , \beta ) $,
 +
and the bar denotes complex conjugation.
  
<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/s/s083/s083870/s08387011.png" /></td> </tr></table>
+
The left-hand side of every self-adjoint differential equation  $  l ( y) = 0 $
 +
is a sum of expressions of the form
  
<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/s/s083/s083870/s08387012.png" /></td> </tr></table>
+
$$
 +
l _ {2m} ( y)  = ( p _ {m} y  ^ {(} m) )  ^ {(} m) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387014.png" /> are sufficiently-smooth real-valued functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387015.png" />. A self-adjoint differential equation with real coefficients is necessarily of even order, and has the form
+
$$
 +
l _ {2m - 1 }  ( y)  = {
 +
\frac{1}{2}
 +
} [( iq _ {m} y ^ {( m -
 +
1) } )  ^ {(} m) + ( iq _ {m} y  ^ {(} m) ) ^ {( m - 1) } ],
 +
$$
  
<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/s/s083/s083870/s08387016.png" /></td> </tr></table>
+
where  $  p _ {m} ( t) $
 +
and  $  q _ {m} ( t) $
 +
are sufficiently-smooth real-valued functions and  $  i  ^ {2} = - 1 $.
 +
A self-adjoint differential equation with real coefficients is necessarily of even order, and has the form
 +
 
 +
$$
 +
( p _ {0} y  ^ {(} m) )  ^ {(} m) +
 +
( p _ {1} y ^ {( m - 1) } ) ^ {( m - 1) } + \dots + p _ {m} y  = 0
 +
$$
  
 
(see [[#References|[1]]]–[[#References|[3]]]).
 
(see [[#References|[1]]]–[[#References|[3]]]).
Line 27: Line 72:
 
A linear system of differential equations
 
A linear system of differential equations
  
<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/s/s083/s083870/s08387017.png" /></td> </tr></table>
+
$$
 +
L ( x)  = 0,\ \
 +
L ( x)  \equiv  \dot{x} + A ( t) x,\ \
 +
t \in I,
 +
$$
  
with a continuous complex-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387018.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387019.png" />, is called self-adjoint if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387021.png" /> is the Hermitian conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387022.png" /> (see [[#References|[1]]], [[#References|[4]]], and [[Hermitian operator|Hermitian operator]]). This definition is not consistent with the definition of a self-adjoint differential equation. For example, the system
+
with a continuous complex-valued $  ( n \times n) $-
 +
matrix $  A ( t) $,  
 +
is called self-adjoint if $  A ( t) = - A  ^ {*} ( t) $,  
 +
where $  A  ^ {*} ( t) $
 +
is the Hermitian conjugate of $  A ( t) $(
 +
see [[#References|[1]]], [[#References|[4]]], and [[Hermitian operator|Hermitian operator]]). This definition is not consistent with the definition of a self-adjoint differential equation. For example, the system
  
<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/s/s083/s083870/s08387023.png" /></td> </tr></table>
+
$$
 +
\dot{x} _ {1} - x _ {2}  = 0,\ \
 +
\dot{x} _ {2} + p ( t) x _ {1}  = 0,
 +
$$
  
 
which is equivalent to the self-adjoint differential equation
 
which is equivalent to the self-adjoint differential equation
  
<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/s/s083/s083870/s08387024.png" /></td> </tr></table>
+
$$
 +
\dot{y} dot + p ( t) y  = 0,
 +
$$
  
is self-adjoint as a linear system if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387025.png" />.
+
is self-adjoint as a linear system if and only if $  p ( t) \equiv 1 $.
  
 
The boundary value problem
 
The boundary value problem
  
<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/s/s083/s083870/s08387026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
l ( y)  = 0,\ \
 +
t  \in  \Delta  = \
 +
[ t _ {0} , t _ {1} ],
 +
$$
  
<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/s/s083/s083870/s08387027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
U _ {k} ( y)  = 0 ,\  k = 1 \dots n,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387028.png" /> are linear and linearly independent functionals describing the boundary conditions, is called self-adjoint if it coincides with the adjoint boundary value problem, that is, (1) is a self-adjoint differential equation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387030.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387031.png" /> (see [[#References|[1]]]–[[#References|[3]]], [[#References|[5]]]). If (1), (2) is a self-adjoint boundary value problem, then the equality (see [[Green formulas|Green formulas]])
+
where the $  U _ {k} : C  ^ {(} n) ( \Delta ) \rightarrow \mathbf R  ^ {1} $
 +
are linear and linearly independent functionals describing the boundary conditions, is called self-adjoint if it coincides with the adjoint boundary value problem, that is, (1) is a self-adjoint differential equation and $  U _ {k} ( y) = U _ {k}  ^ {*} ( y) $
 +
for all $  y ( \cdot ) \in C  ^ {n} ( \Delta ) $
 +
and for all $  k = 1 \dots n $(
 +
see [[#References|[1]]]–[[#References|[3]]], [[#References|[5]]]). If (1), (2) is a self-adjoint boundary value problem, then the equality (see [[Green formulas|Green formulas]])
  
<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/s/s083/s083870/s08387032.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
\overline \xi \; l ( y)  dt  = \
 +
\int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
\overline{l}\; ( \xi ) y  dt
 +
$$
  
holds for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387033.png" /> that satisfy the boundary conditions (2).
+
holds for any pair $  y ( \cdot ), \xi ( \cdot ) \in C  ^ {(} n) ( \Delta ) $
 +
that satisfy the boundary conditions (2).
  
 
All the eigenvalues of the self-adjoint problem
 
All the eigenvalues of the self-adjoint problem
  
<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/s/s083/s083870/s08387034.png" /></td> </tr></table>
+
$$
 +
l ( y)  = \lambda y,\ \
 +
U _ {k} ( y)  = 0,\ \
 +
k = 1 \dots n,
 +
$$
  
are real, and the eigenfunctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387035.png" /> corresponding to distinct eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387036.png" /> are orthogonal:
+
are real, and the eigenfunctions $  \phi _ {1} , \phi _ {2} $
 +
corresponding to distinct eigenvalues $  \lambda _ {1} , \lambda _ {2} $
 +
are orthogonal:
  
<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/s/s083/s083870/s08387037.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
\overline \phi \; _ {1} \phi _ {2} ( t)  dt  = 0.
 +
$$
  
 
The linear boundary value problem
 
The linear boundary value problem
  
<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/s/s083/s083870/s08387038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
L ( x)  \equiv  \dot{x} + A ( t) x  = 0,\ \
 +
U ( x) =  0,\ \
 +
t \in \Delta ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387039.png" /> is a continuous complex-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387040.png" />-matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387041.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387042.png" />-vector functional on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387043.png" /> of continuous complex-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387044.png" />, is called self-adjoint if it coincides with its adjoint boundary value problem
+
where $  A ( t) $
 +
is a continuous complex-valued $  ( n \times n) $-
 +
matrix and $  U $
 +
is an $  n $-
 +
vector functional on the space $  C _ {n}  ^ {1} ( \Delta ) $
 +
of continuous complex-valued functions $  x: \Delta \rightarrow \mathbf R  ^ {n} $,  
 +
is called self-adjoint if it coincides with its adjoint boundary value problem
  
<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/s/s083/s083870/s08387045.png" /></td> </tr></table>
+
$$
 +
L  ^ {*} ( x)  = 0,\ \
 +
U  ^ {*} ( x)  = 0,\ \
 +
t \in \Delta ,
 +
$$
  
 
that is,
 
that is,
  
<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/s/s083/s083870/s08387046.png" /></td> </tr></table>
+
$$
 +
L ( x)  = - L  ^ {*} ( x),\ \
 +
U ( x)  = U  ^ {*} ( x)
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083870/s08387047.png" />. A self-adjoint boundary value problem has properties analogous to those of the problem (1), (2) (see [[#References|[4]]]).
+
for all $  x ( \cdot ) \in C _ {n}  ^ {1} ( \Delta ) $.  
 +
A self-adjoint boundary value problem has properties analogous to those of the problem (1), (2) (see [[#References|[4]]]).
  
 
The concepts of a self-adjoint differential equation and of a self-adjoint boundary value problem are closely connected with that of a [[Self-adjoint operator|self-adjoint operator]] [[#References|[6]]] (cf. also [[Spectral theory of differential operators|Spectral theory of differential operators]]). Self-adjointness and a self-adjoint boundary value problem are also defined for a linear partial differential equation (see [[#References|[5]]], [[#References|[7]]]).
 
The concepts of a self-adjoint differential equation and of a self-adjoint boundary value problem are closely connected with that of a [[Self-adjoint operator|self-adjoint operator]] [[#References|[6]]] (cf. also [[Spectral theory of differential operators|Spectral theory of differential operators]]). Self-adjointness and a self-adjoint boundary value problem are also defined for a linear partial differential equation (see [[#References|[5]]], [[#References|[7]]]).
Line 77: Line 179:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,  "Linear differential operators" , '''1–2''' , Harrap  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.P. Mikhailov,  "Partial differential equations" , MIR  (1978)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,  "Linear differential operators" , '''1–2''' , Harrap  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.P. Mikhailov,  "Partial differential equations" , MIR  (1978)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:13, 6 June 2020


A linear ordinary differential equation $ l ( y) = 0 $ that coincides with the adjoint differential equation $ l ^ {*} ( y) = 0 $. Here

$$ l ( y) \equiv a _ {0} ( t) y ^ {(} n) + \dots + a _ {n} ( t) y, $$

$$ l ^ {*} ( y) \equiv (- 1) ^ {n} ( \overline{a}\; _ {0} ( t) y) ^ {(} n) + \dots + (- 1) ^ {0} \overline{a}\; _ {n} ( t) y, $$

where

$$ y ^ {( \nu ) } = \ \frac{d ^ \nu y }{dt ^ \nu } ,\ \ y ( \cdot ) \in C ^ {n} ( I),\ \ a _ {k} ( \cdot ) \in C ^ {n - k } ( I), $$

$$ a _ {0} ( t) \neq 0,\ t \in I, $$

$ C ^ {m} ( I) $ is the space of $ m $- times continuously-differentiable complex-valued functions on $ I = ( \alpha , \beta ) $, and the bar denotes complex conjugation.

The left-hand side of every self-adjoint differential equation $ l ( y) = 0 $ is a sum of expressions of the form

$$ l _ {2m} ( y) = ( p _ {m} y ^ {(} m) ) ^ {(} m) , $$

$$ l _ {2m - 1 } ( y) = { \frac{1}{2} } [( iq _ {m} y ^ {( m - 1) } ) ^ {(} m) + ( iq _ {m} y ^ {(} m) ) ^ {( m - 1) } ], $$

where $ p _ {m} ( t) $ and $ q _ {m} ( t) $ are sufficiently-smooth real-valued functions and $ i ^ {2} = - 1 $. A self-adjoint differential equation with real coefficients is necessarily of even order, and has the form

$$ ( p _ {0} y ^ {(} m) ) ^ {(} m) + ( p _ {1} y ^ {( m - 1) } ) ^ {( m - 1) } + \dots + p _ {m} y = 0 $$

(see [1][3]).

A linear system of differential equations

$$ L ( x) = 0,\ \ L ( x) \equiv \dot{x} + A ( t) x,\ \ t \in I, $$

with a continuous complex-valued $ ( n \times n) $- matrix $ A ( t) $, is called self-adjoint if $ A ( t) = - A ^ {*} ( t) $, where $ A ^ {*} ( t) $ is the Hermitian conjugate of $ A ( t) $( see [1], [4], and Hermitian operator). This definition is not consistent with the definition of a self-adjoint differential equation. For example, the system

$$ \dot{x} _ {1} - x _ {2} = 0,\ \ \dot{x} _ {2} + p ( t) x _ {1} = 0, $$

which is equivalent to the self-adjoint differential equation

$$ \dot{y} dot + p ( t) y = 0, $$

is self-adjoint as a linear system if and only if $ p ( t) \equiv 1 $.

The boundary value problem

$$ \tag{1 } l ( y) = 0,\ \ t \in \Delta = \ [ t _ {0} , t _ {1} ], $$

$$ \tag{2 } U _ {k} ( y) = 0 ,\ k = 1 \dots n, $$

where the $ U _ {k} : C ^ {(} n) ( \Delta ) \rightarrow \mathbf R ^ {1} $ are linear and linearly independent functionals describing the boundary conditions, is called self-adjoint if it coincides with the adjoint boundary value problem, that is, (1) is a self-adjoint differential equation and $ U _ {k} ( y) = U _ {k} ^ {*} ( y) $ for all $ y ( \cdot ) \in C ^ {n} ( \Delta ) $ and for all $ k = 1 \dots n $( see [1][3], [5]). If (1), (2) is a self-adjoint boundary value problem, then the equality (see Green formulas)

$$ \int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline \xi \; l ( y) dt = \ \int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline{l}\; ( \xi ) y dt $$

holds for any pair $ y ( \cdot ), \xi ( \cdot ) \in C ^ {(} n) ( \Delta ) $ that satisfy the boundary conditions (2).

All the eigenvalues of the self-adjoint problem

$$ l ( y) = \lambda y,\ \ U _ {k} ( y) = 0,\ \ k = 1 \dots n, $$

are real, and the eigenfunctions $ \phi _ {1} , \phi _ {2} $ corresponding to distinct eigenvalues $ \lambda _ {1} , \lambda _ {2} $ are orthogonal:

$$ \int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline \phi \; _ {1} \phi _ {2} ( t) dt = 0. $$

The linear boundary value problem

$$ \tag{3 } L ( x) \equiv \dot{x} + A ( t) x = 0,\ \ U ( x) = 0,\ \ t \in \Delta , $$

where $ A ( t) $ is a continuous complex-valued $ ( n \times n) $- matrix and $ U $ is an $ n $- vector functional on the space $ C _ {n} ^ {1} ( \Delta ) $ of continuous complex-valued functions $ x: \Delta \rightarrow \mathbf R ^ {n} $, is called self-adjoint if it coincides with its adjoint boundary value problem

$$ L ^ {*} ( x) = 0,\ \ U ^ {*} ( x) = 0,\ \ t \in \Delta , $$

that is,

$$ L ( x) = - L ^ {*} ( x),\ \ U ( x) = U ^ {*} ( x) $$

for all $ x ( \cdot ) \in C _ {n} ^ {1} ( \Delta ) $. A self-adjoint boundary value problem has properties analogous to those of the problem (1), (2) (see [4]).

The concepts of a self-adjoint differential equation and of a self-adjoint boundary value problem are closely connected with that of a self-adjoint operator [6] (cf. also Spectral theory of differential operators). Self-adjointness and a self-adjoint boundary value problem are also defined for a linear partial differential equation (see [5], [7]).

References

[1] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
[2] M.A. Naimark, "Linear differential operators" , 1–2 , Harrap (1968) (Translated from Russian)
[3] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[4] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[5] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[6] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
[7] V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian)

Comments

In general, the system of eigenfunctions is complete.

References

[a1] S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965)
How to Cite This Entry:
Self-adjoint differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-adjoint_differential_equation&oldid=48648
This article was adapted from an original article by E.L. Tonkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article