View source for Self-adjoint differential equation

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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

$$ 
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 [[#References|[1]]]–[[#References|[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 [[#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

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

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 [[#References|[1]]]–[[#References|[3]]], [[#References|[5]]]). If (1), (2) is a self-adjoint boundary value problem, then the equality (see [[Green formulas|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 [[#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]]]).

====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>

====Comments====
In general, the system of eigenfunctions is complete.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Agmon,   "Lectures on elliptic boundary value problems" , v. Nostrand  (1965)</TD></TR></table>