View source for Self-adjoint differential equation

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

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where

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<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.

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

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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

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(see [[#References|[1]]]–[[#References|[3]]]).

A linear system of differential equations

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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

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which is equivalent to the self-adjoint differential equation

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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" />.

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>

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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]])

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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).

All the eigenvalues of the self-adjoint problem

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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:

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The linear boundary value problem

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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

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that is,

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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]]]).

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>