Spectral theory of differential operators

The branch of the general 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 be a domain in , let be its boundary, let

 (1)

be a linear differential operator, and let

 (2)

be the boundary conditions, defined by linear differential operators .

Here

the are non-negative integers, , , and and are functions defined in and on , respectively. Unless otherwise stated, in the sequel it is assumed that and are sufficiently smooth functions when , and that for all , where if .

Let be the differential operator given by (1) on functions in , that is, functions having derivatives of arbitrary order and vanishing outside a compact set lying inside . If

 (3)

for any pair of functions and in , then is called a symmetric differential operator, and a formally self-adjoint differential operator (cf. also Self-adjoint differential equation; Self-adjoint operator). Let be the closure of in (cf. Closed operator). Then and its adjoint (cf. Adjoint operator) are called the minimal and maximal operators, respectively, generated by ; is an extension of . An important problem in the theory of differential operators is to describe and , and also to describe all self-adjoint extensions of (cf. Self-adjoint operator).

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

Let

 (4)

be the deficiency subspaces (cf. Deficiency subspace) of the operator . If , then , and is said to be essentially self-adjoint. Any of the following conditions are sufficient for to be essentially self-adjoint on : The formally self-adjoint differential operator has the form

 (5)

with real coefficients, and is bounded from below; it has the form (5), is elliptic, the are constants, and , where does not decrease monotonically, while the integral

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).

Let have finite deficiency indices , which is typical for ordinary differential operators. In this case the numbers coincide with the dimensions of the subspaces of solutions of the equations in . Therefore , 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 and . If , then does not even have one self-adjoint extension. If , then for the self-adjointness of extensions of it is necessary to give boundary conditions, and these have been completely described. Boundary value problems take a simple form when the expression has two regular end-points, or has one regular end-point but and . An end-point is called regular if and , , , are summable on for any .

There are examples of partial differential operators on , , 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 be a formally self-adjoint elliptic differential operator of even order with real coefficients, and let be the set of all functions having derivatives of arbitrary order in the bounded closed domain and satisfying Dirichlet-type boundary conditions , , . Then the differential operator defined by with domain of definition is symmetric, and its closure 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.

 (6)

where is a 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 in terms of eigenfunctions. If a self-adjoint differential operator has discrete spectrum with corresponding orthonormalized eigenfunctions , then is an integral operator with (spectral) kernel

 (7)

In the case of a continuous spectrum of a differential operator, the question becomes complicated: For continuous spectra there are no eigenfunctions in . However, the following results are true.

Let be an ordinary self-adjoint differential operator of the form (1) on , and let be a fundamental system of solutions of the equations . Then there is a monotone matrix function (a spectral measure) such that the resolution of the identity of is given by the kernel

 (8)

Moreover, for any function in the integral

 (9)

converges in the space of vector functions generated by the measure , and, conversely, the integral

converges to in . If (1) has a regular end-point and , and the deficiency indices , then the functions can be chosen to form a fundamental system in the class of solutions of the equation satisfying boundary conditions at , and in this case the order of the spectral measure is equal to .

Let be a self-adjoint elliptic differential operator on . Then its resolution of the identity is an integral operator with kernel , and there is a non-decreasing function such that for all numbers and ,

 (10)

where, for every , there is a finite or infinite system of solutions of the equation and

 (11)

For the Schrödinger operator , , under the condition , the kernel can be explicitly expressed in terms of the solutions of the dispersion equation.

The formulas (10), (11) also hold for arbitrary self-adjoint partial differential operators, and in this case the 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 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 , then the spectrum of the differential operator generated by the expression on is discrete if and only if for any ,

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 as . The self-adjoint differential operator 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 having a common period, then its spectrum is continuous and consists of a sequence of disjoint intervals whose end-points tend to or ; 2) if a differential operator is defined by the expression on and as , then its continuous spectrum fills , while its negative spectrum is discrete and can have a limit point at zero only. If , and

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 be an elliptic differential operator of order . The resolvent when , and the functions and when , have been thoroughly studied. The latter are solution operators for the generalized heat equation , , and the generalized wave equation , , respectively. All three operator-functions are integral functions, and have kernels , , (Green functions), respectively. The formula

 (12)

establishes a connection between and . Some properties of are: If is an elliptic self-adjoint differential operator of order on , then to there corresponds a kernel of Carleman type when ; when , is nuclear, and therefore

 (13)

where are the eigenvalues of . There are also other tests of nuclearity for on .

The analytic and asymptotic properties of Green functions give useful information about the spectral nature of a differential operator . For example, if in (13) the behaviour of as is known, then the application of Tauberian theorems enables one to find the asymptotics of . The same can be done if one knows the asymptotics of as . The asymptotics of and can be established, for example, by the method of parametrics, by the method of potentials, etc. The asymptotics of have thus been found for an extensive class of elliptic differential operators. To determine the asymptotics of the spectral kernel of an elliptic differential operator, the study of the asymptotics of the kernel as has proved effective in conjunction with subsequent application of various Tauberian theorems. In particular, when , ,

Most complete results have been obtained for ordinary differential operators on a finite interval. Let be the differential operator defined by (1) when and on functions having absolutely-continuous derivatives and satisfying the boundary conditions:

Here , and do not simultaneously vanish. Let the boundary conditions (2) be regular. This holds for boundary conditions of Sturm–Liouville type (, , ), and also for periodic-type boundary conditions . Then has an infinite number of eigenvalues, which have precise asymptotics; the system consisting of the eigenfunctions of and of their associates is complete in ; the expansion of functions in into eigenfunctions of and their associates converges uniformly on . The system of eigenfunctions and their associates may also be complete under certain non-regular boundary conditions, in particular of splitting type (, , , , , ). However, the convergence of the expansion into a series of eigenfunctions and their associates holds only for a narrow class of (-analytic) functions.

Let be a self-adjoint operator on a separable Hilbert space with eigenvalues , and suppose that the operator is nuclear for a certain . Let be another operator such that is compact. Then the system consisting of the eigenvectors of and of their associates is complete in (Keldysh' theorem). The application of this theorem gives classes of differential operators which have a complete system of eigenfunctions and their associates.

Let be a differential operator on and let

Then the system consisting of the eigenfunctions of and of their associates is complete in . 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 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 where or and is a complex-valued function. Let be the solution of the equation for satisfying the initial conditions , . Let and be functions in of compact support and let

Then there is a linear functional on a linear topological space such that and

The space 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 , then 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

on , where 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) 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

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