View source for Spectral theory of differential operators

The branch of the general [[Spectral theory|spectral theory]] of operators in which one investigates the spectral properties of differential operators on various function spaces, especially on Hilbert spaces of measurable functions.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s0865301.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s0865302.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s0865303.png" /> be its boundary, let

<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/s086/s086530/s0865304.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

be a linear differential operator, and let

<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/s086/s086530/s0865305.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

be the boundary conditions, defined by linear differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s0865306.png" />.

Here

<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/s086/s086530/s0865307.png" /></td> </tr></table>

<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/s086/s086530/s0865308.png" /></td> </tr></table>

the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s0865309.png" /> are non-negative integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653013.png" /> are functions defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653014.png" /> and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653015.png" />, respectively. Unless otherwise stated, in the sequel it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653017.png" /> are sufficiently smooth functions when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653018.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653022.png" />.

==Self-adjoint extensions of differential operators.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653023.png" /> be the differential operator given by (1) on functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653024.png" />, that is, functions having derivatives of arbitrary order and vanishing outside a compact set lying inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653025.png" />. If

<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/s086/s086530/s08653026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

for any pair of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653030.png" /> is called a symmetric differential operator, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653031.png" /> a formally self-adjoint differential operator (cf. also [[Self-adjoint differential equation|Self-adjoint differential equation]]; [[Self-adjoint operator|Self-adjoint operator]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653032.png" /> be the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653034.png" /> (cf. [[Closed operator|Closed operator]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653035.png" /> and its adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653036.png" /> (cf. [[Adjoint operator|Adjoint operator]]) are called the minimal and maximal operators, respectively, generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653037.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653038.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653039.png" />. An important problem in the theory of differential operators is to describe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653041.png" />, and also to describe all self-adjoint extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653042.png" /> (cf. [[Self-adjoint operator|Self-adjoint operator]]).

Here one can apply the abstract theory of extensions of symmetric operators (cf. [[Extension of an operator|Extension of an operator]]). However, for differential operators, self-adjoint extensions can often be successfully described in terms of boundary conditions.

Let

<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/s086/s086530/s08653043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

be the deficiency subspaces (cf. [[Deficiency subspace|Deficiency subspace]]) of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653044.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653046.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653047.png" /> is said to be essentially self-adjoint. Any of the following conditions are sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653048.png" /> to be essentially self-adjoint on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653049.png" />: The formally self-adjoint differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653050.png" /> has 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/s086/s086530/s08653051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

with real coefficients, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653052.png" /> is bounded from below; it has the form (5), is elliptic, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653053.png" /> are constants, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653055.png" /> does not decrease monotonically, while the integral

<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/s086/s086530/s08653056.png" /></td> </tr></table>

it has constant real coefficients; it has bounded coefficients and the principal part is of elliptic type with real constant coefficients (cf. [[Principal part of a differential operator|Principal part of a differential operator]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653057.png" /> have finite deficiency indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653058.png" />, which is typical for ordinary differential operators. In this case the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653059.png" /> coincide with the dimensions of the subspaces of solutions of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653061.png" />. Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653062.png" />, and the calculation of the deficiency indices of a differential operator is connected with the qualitative theory and asymptotic methods of ordinary differential equations.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653064.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653066.png" /> does not even have one self-adjoint extension. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653067.png" />, then for the self-adjointness of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653068.png" /> it is necessary to give <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653069.png" /> boundary conditions, and these have been completely described. Boundary value problems take a simple form when the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653070.png" /> has two regular end-points, or has one regular end-point but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653072.png" />. An end-point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653073.png" /> is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653077.png" />, are summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653078.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653079.png" />.

There are examples of partial differential operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653081.png" />, with discontinuous coefficients and with finite deficiency indices, but their theory is still underdeveloped. Not all self-adjoint extensions of symmetric partial differential operators in a bounded domain have been described in terms of boundary conditions, but various extensions with given properties have been described.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653082.png" /> be a formally self-adjoint elliptic differential operator of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653083.png" /> with real coefficients, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653084.png" /> be the set of all functions having derivatives of arbitrary order in the bounded closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653085.png" /> and satisfying Dirichlet-type boundary conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653088.png" />. Then the differential operator defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653089.png" /> with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653090.png" /> is symmetric, and its closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653091.png" /> is self-adjoint. There are other examples of concrete self-adjoint boundary conditions for differential operators. Most complete studies have been made in the case of second-order differential operators with boundary conditions of Dirichlet-type, von Neumann-type or of the third kind.

==Spectral analysis of self-adjoint differential operators.==
Every self-adjoint differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653092.png" /> admits a spectral decomposition 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/s086/s086530/s08653093.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653094.png" /> is a [[Resolution of the identity|resolution of the identity]] (into an orthogonal family of projectors). However, the general formula does not give a direct expansion with respect to the eigenfunctions of a concrete self-adjoint differential operator, and so it is important to be able to express the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653095.png" /> in terms of eigenfunctions. If a self-adjoint differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653096.png" /> has discrete spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653097.png" /> with corresponding orthonormalized eigenfunctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653098.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s08653099.png" /> is an integral operator with (spectral) kernel

<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/s086/s086530/s086530100.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

In the case of a continuous spectrum of a differential operator, the question becomes complicated: For continuous spectra there are no eigenfunctions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530101.png" />. However, the following results are true.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530102.png" /> be an ordinary self-adjoint differential operator of the form (1) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530103.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530104.png" /> be a fundamental system of solutions of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530105.png" />. Then there is a monotone matrix function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530106.png" /> (a spectral measure) such that the resolution of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530107.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530108.png" /> is given by the kernel

<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/s086/s086530/s086530109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

Moreover, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530110.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530111.png" /> the integral

<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/s086/s086530/s086530112.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

converges in the space of vector functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530113.png" /> generated by the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530114.png" />, and, conversely, the integral

<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/s086/s086530/s086530115.png" /></td> </tr></table>

converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530116.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530117.png" />. If (1) has a regular end-point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530119.png" />, and the deficiency indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530120.png" />, then the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530121.png" /> can be chosen to form a fundamental system in the class of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530122.png" /> satisfying boundary conditions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530123.png" />, and in this case the order of the spectral measure is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530124.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530125.png" /> be a self-adjoint elliptic differential operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530126.png" />. Then its resolution of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530127.png" /> is an integral operator with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530128.png" />, and there is a non-decreasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530129.png" /> such that for all numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530130.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530131.png" />,

<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/s086/s086530/s086530132.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

where, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530133.png" />, there is a finite or infinite system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530134.png" /> of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530135.png" /> and

<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/s086/s086530/s086530136.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

For the Schrödinger operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530138.png" />, under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530139.png" />, the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530140.png" /> can be explicitly expressed in terms of the solutions of the [[Dispersion equation|dispersion equation]].

The formulas (10), (11) also hold for arbitrary self-adjoint partial differential operators, and in this case the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530141.png" /> may be generalized functions, but they are of finite order.

The nature of the convergence of the expansion into eigenfunctions and the asymptotic properties of the spectral kernel help to justify the [[Fourier method|Fourier method]] for solving the equations of mathematical physics. For ordinary differential operators, there is the following final result, the so-called equiconvergence theorem: The expansion of a given summable function into the eigenfunctions of a differential operator which is bounded from below and the Fourier integral are both convergent or both divergent (i.e. equi-convergent) at any point. For partial differential operators, the question becomes complicated.

==Qualitative theory of the spectrum of a differential operator.==
This theory is concerned with the study of the nature of the spectrum in relation to the behaviour of the coefficients, the geometry of the domain and the boundary conditions.

There is a series of tests for the discreteness of the spectrum of a differential operator. The most general are the following criterion and its generalizations: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530142.png" />, then the spectrum of the differential operator generated by the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530143.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530144.png" /> is discrete if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530145.png" />,

<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/s086/s086530/s086530146.png" /></td> </tr></table>

The generalization of this criterion to partial differential operators takes a more complicated form. There are other, simpler, tests for discreteness of the spectrum of a differential operator. For example, the self-adjoint differential operator generated by (5) has discrete spectrum if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530147.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530148.png" />. The self-adjoint differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530149.png" /> has discrete spectrum.

The study of the nature of a spectrum when there is a continuous part is a difficult problem. Here are some results: 1) if an ordinary differential operator is defined by a formally self-adjoint expression (1) with periodic coefficients on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530150.png" /> having a common period, then its spectrum is continuous and consists of a sequence of disjoint intervals whose end-points tend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530151.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530152.png" />; 2) if a differential operator is defined by the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530153.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530155.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530156.png" />, then its continuous spectrum fills <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530157.png" />, while its negative spectrum is discrete and can have a limit point at zero only. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530159.png" /> and

<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/s086/s086530/s086530160.png" /></td> </tr></table>

then the negative spectrum is finite (there are no eigenvalues in the continuous spectrum).

The nature of the spectrum also depends on the boundary conditions. In a bounded domain, concrete boundary conditions have been described whose fulfillment guarantees that the spectrum of a self-adjoint Laplace operator has a continuous part. This is a consequence of the deficiency indices of the minimal Laplace operator being infinite in a domain with boundary.

==Functions of a self-adjoint differential operator.==
These are studied with the aim of solving mixed problems for differential equations, and also for problems in the theory of differential operators. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530161.png" /> be an elliptic differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530162.png" />. The [[Resolvent|resolvent]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530163.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530164.png" />, and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530165.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530166.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530167.png" />, have been thoroughly studied. The latter are solution operators for the generalized heat equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530169.png" />, and the generalized wave equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530170.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530171.png" />, respectively. All three operator-functions are integral functions, and have kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530172.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530173.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530174.png" /> (Green functions), respectively. 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/s/s086/s086530/s086530175.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

establishes a connection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530176.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530177.png" />. Some properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530178.png" /> are: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530179.png" /> is an elliptic self-adjoint differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530180.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530181.png" />, then to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530182.png" /> there corresponds a kernel of Carleman type when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530183.png" />; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530185.png" /> is nuclear, and therefore

<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/s086/s086530/s086530186.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530187.png" /> are the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530188.png" />. There are also other tests of nuclearity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530189.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530190.png" />.

The analytic and asymptotic properties of Green functions give useful information about the spectral nature of a differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530191.png" />. For example, if in (13) the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530192.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530193.png" /> is known, then the application of Tauberian theorems enables one to find the asymptotics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530194.png" />. The same can be done if one knows the asymptotics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530195.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530196.png" />. The asymptotics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530197.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530198.png" /> can be established, for example, by the method of parametrics, by the method of potentials, etc. The asymptotics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530199.png" /> have thus been found for an extensive class of elliptic differential operators. To determine the asymptotics of the spectral kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530200.png" /> of an elliptic differential operator, the study of the asymptotics of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530201.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530202.png" /> has proved effective in conjunction with subsequent application of various Tauberian theorems. In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530204.png" />,

<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/s086/s086530/s086530205.png" /></td> </tr></table>

==Non-self-adjoint differential operators.==
Most complete results have been obtained for ordinary differential operators on a finite interval. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530206.png" /> be the differential operator defined by (1) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530208.png" /> on functions having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530209.png" /> absolutely-continuous derivatives and satisfying the boundary conditions:

<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/s086/s086530/s086530210.png" /></td> </tr></table>

<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/s086/s086530/s086530211.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530212.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530213.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530214.png" /> do not simultaneously vanish. Let the boundary conditions (2) be regular. This holds for boundary conditions of Sturm–Liouville type (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530215.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530216.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530217.png" />), and also for periodic-type boundary conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530218.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530219.png" /> has an infinite number of eigenvalues, which have precise asymptotics; the system consisting of the eigenfunctions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530220.png" /> and of their associates is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530221.png" />; the expansion of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530222.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530223.png" /> into eigenfunctions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530224.png" /> and their associates converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530225.png" />. The system of eigenfunctions and their associates may also be complete under certain non-regular boundary conditions, in particular of splitting type (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530226.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530227.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530228.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530229.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530230.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530231.png" />). However, the convergence of the expansion into a series of eigenfunctions and their associates holds only for a narrow class of (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530232.png" />-analytic) functions.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530233.png" /> be a self-adjoint operator on a separable Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530234.png" /> with eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530235.png" />, and suppose that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530236.png" /> is nuclear for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530237.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530238.png" /> be another operator such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530239.png" /> is compact. Then the system consisting of the eigenvectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530240.png" /> and of their associates is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530241.png" /> (Keldysh' theorem). The application of this theorem gives classes of differential operators which have a complete system of eigenfunctions and their associates.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530242.png" /> be a differential operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530243.png" /> and let

<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/s086/s086530/s086530244.png" /></td> </tr></table>

Then the system consisting of the eigenfunctions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530245.png" /> and of their associates is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530246.png" />. However, the expansion of a function into a series with respect to this system is divergent, in general, and is conditionally summable by the generalized Abel method.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530247.png" /> is an unbounded domain, then to satisfy the conditions of Keldysh' theorem one must impose further conditions on the growth of the coefficient functions of the differential operator.

Non-self-adjoint differential operators with a continuous part in their spectrum have not been studied much. This is connected with the fact that no analogue of the spectral decomposition theorem exists in this case. An exception is the differential operator generated by the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530248.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530249.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530250.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530251.png" /> is a complex-valued function. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530252.png" /> be the solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530253.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530254.png" /> satisfying the initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530255.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530256.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530257.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530258.png" /> be functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530259.png" /> of compact support and let

<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/s086/s086530/s086530260.png" /></td> </tr></table>

Then there is a linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530261.png" /> on a linear topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530262.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530263.png" /> and

<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/s086/s086530/s086530264.png" /></td> </tr></table>

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530265.png" /> is the set of all even entire functions of order of growth one and of finite type that are summable on the real axis. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530266.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530267.png" /> can be explicitly calculated. In this case, spectral singularities, that is, poles of the kernel of the resolvent, appear in the continuous spectrum, and these are not eigenvalues of the differential operator. Spectral singularities are inherent in non-self-adjoint operators, and because of them, questions of the expansion into eigenfunctions (and convergence problems) become more complicated. For the differential 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/s/s086/s086530/s086530268.png" /></td> </tr></table>

on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530269.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530270.png" /> is a complex-valued function that decreases exponentially, a form of spectral decomposition has also been found by solving a problem in dispersion theory, taking the influence of spectral singularities into account.

==Inverse problems of spectral analysis.==
These arise when one asks for the determination of differential operators by certain spectral characteristics. The problems of determining one-dimensional Schrödinger equations and systems of Dirac type have been completely solved, given the spectra of various extensions, the spectral measure, scattering data (that is, the asymptotic behaviour of normalized eigenfunctions), or other properties. Inverse problems have found applications in the integration of non-linear equations.

The spectral theory of differential operators arose in connection with investigations on vibrating strings and gave birth to the theory of orthogonal expansions (18th century and 19th century). The systematic study of self-adjoint differential operators of the second order on a finite interval dates from 1830 (the [[Sturm–Liouville problem|Sturm–Liouville problem]]) and was the subject of intensive study in the 19th century, in particular in connection with the theory of special functions. However, the completeness of the system of eigenfunctions of the Sturm–Liouville operator was not proved until 1896, when the nature of the convergence of the expansion into eigenfunctions was also investigated. The theory of singular differential operators began in 1909–1910, when the spectral decomposition of a self-adjoint unbounded differential operator of the second order with an arbitrary spectral structure was discovered, and when, in principle, the concept of a deficiency index was introduced, and the first results in the theory of extensions were obtained. Interest in singular differential operators grew from 1920 onwards, along with the rise of quantum mechanics. The systematic investigation of non-self-adjoint singular differential operators began in 1950, when the foundations of the theory of operator pencils were expounded and a method was found for proving the completeness of the system consisting of the eigenfunctions of a differential operator and of their associates.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,   "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Glazman,   "Direct methods of qualitative spectral analysis of singular differential operators" , Israel Program Sci. Transl.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T. Kato,   "Perturbation theory for linear operators" , Springer  (1976)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B.M. Levitan,   I.S. Sargsyan,   "Introduction to spectral theory: selfadjoint ordinary differential operators" , Amer. Math. Soc.  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.A. Marchenko,   "Sturm–Liouville operators and applications" , Birkhäuser  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M.A. Naimark,   "Linear differential operators" , '''1–2''' , F. Ungar  (1967–1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E.C. Titchmarsh,   "Eigenfunction expansions associated with second-order differential equations" , '''1''' , Clarendon Press  (1962)  pp. Chapts. 1; 2</TD></TR><TR><TD valign="top">[9a]</TD> <TD valign="top">  L.D. Faddeev,   "Mathematical aspects of the three-body problem in the quantum scattering theory" , Israel Program Sci. Transl.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9b]</TD> <TD valign="top">  L.D. Faddeev,   "Properties of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530271.png" />-matrix of the one-dimensional Schrödinger equation"  ''Trudy. Mat. Inst. Steklov.'' , '''73'''  (1964)  pp. 314–336  (In Russian)</TD></TR><TR><TD valign="top">[10a]</TD> <TD valign="top">  Sh.A. Alimov,   V.A. Il'in,   E.M. Nikishin,   "Convergence problems of multiple trigonometric series and spectral decompositions I"  ''Russian Math. Surveys'' , '''31''' :  6  (1976)  pp. 29–86  ''Uspekhi Mat. Nauk'' , '''31''' :  6  (1976)  pp. 28–83</TD></TR><TR><TD valign="top">[10b]</TD> <TD valign="top">  Sh.A. Alimov,   V.A. Il'in,   E.M. Nikishin,   "Problems of convergence of multiple trigonometric series and spectral decompositions II"  ''Russian Math. Surveys'' , '''32''' :  1  (1977)  pp. 115–139  ''Uspekhi Mat. Nauk'' , '''32''' :  1  (1977)  pp. 107–130</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  Yu.M. Berezanskii,   "Self-conjugate elliptic operators with singular potential"  ''Ukr. Math. J.'' , '''26''' :  5  (1974)  pp. 475–483  ''Ukrain. Mat. Zh.'' , '''26''' :  5  (1974)  pp. 579–590</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  M.G. Gasymov,   "The inverse scattering problem for a system of Dirac equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086530/s086530272.png" />"  ''Trans. Moscow Math. Soc.'' , '''19'''  (1968)  pp. 41–120  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 41–112</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  B.M. Levitan,   M.G. Gasymov,   "Determination of a differential equation by two of its spectra"  ''Russian Math. Surveys'' , '''19''' :  2  (1964)  pp. 1–63  ''Uspekhi Mat. Nauk'' , '''19''' :  2  (1964)  pp. 3–63</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  B.A. Dubrovin,   V.B. Matveev,   S.P. Novikov,   "Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and abelian varieties"  ''Russian Math. Surveys'' , '''31''' :  1  (1976)  pp. 59–146  ''Uspekhi Mat. Nauk'' , '''31''' :  1  (1976)  pp. 55–136</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  A.G. Kostyuchenko,   "Asymptotic behavior of the spectral function of self-adjoint elliptic operators" , ''4-th Math. Summer School'' , Kiev  (1968)  pp. 42–117  (In Russian)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  L. Hörmander,   "The spectral function of an elliptic operator"  ''Acta Math.'' , '''121'''  (1968)  pp. 193–218</TD></TR></table>



====Comments====
An important recent development is the study of the spectral asymptotics of elliptic operators, see [[#References|[a1]]], Chapt. XXIX, and [[#References|[a4]]] as an introduction; cf. also [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''II, IV''' , Springer  (1983–1985)  pp. Chapts. XIV, XXIX, XXX</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Schechter,   "Spectra of partial differential operators" , North-Holland  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Calogero,   A. Degasperis,   "Nonlinear evolution equations solvable by the inverse spectral transform II"  ''Nuovo Cimento'' , '''39B'''  (1977)  pp. 1–54</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.A. Shubin,   "Pseudodifferential operators and spectral theory" , Springer  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Levendorskii,   "Asymptotic distribution of eigenvalues of differential operators" , Kluwer  (1991)  (Translated from Russian)</TD></TR></table>