Reports of the Academy of Sciences of the USSR
1957. Vol. 115, No. 1

MATHEMATICS

A. V. STRAUS

ON THE SPECTRAL FUNCTIONS OF A DIFFERENTIAL OPERATOR OF EVEN ORDER

(Presented by Academician A. N. Kolmogorov on 23 I 1957)

1. Let \(l[y]\) be a self-adjoint ordinary differential expression of even order with real coefficients, given on the interval \((a,b)\), whose endpoints are regular or singular—indifferently. As is known, the expression \(l[y]\) generates in the Hilbert space \(L^{2}(a,b)\) a symmetric differential operator \(L\) with minimal domain of definition\(^*\). On the basis of the formula, established in the author’s note \((^{4})\), for the generalized resolvents of the operator \(L\), the present paper gives an effective construction of all spectral functions of this operator. Under the assumption that both endpoints of the interval \((a,b)\) are regular, a class of boundary-value problems with boundary conditions depending on a parameter is investigated, and an expansion formula in eigenfunctions of such problems is established.

2. In \((^{4})\) it is shown that every generalized resolvent \(R_\lambda\) \((\operatorname{Im}\lambda \ne 0)\) of the operator \(L\) is an integral operator, and the corresponding formula for its kernel \(K(x,s;\lambda)\) is obtained. Denote by \(K^{[j,k]}(x,s;\lambda)\) the so-called quasi-derivatives of this kernel of order \(j\) with respect to \(x\) and of order \(k\) with respect to \(s\) \((j,k=0,1,\ldots,2n-1)\).

Put

\[ m_{jk}(\lambda)=\frac{1}{2}\left[K^{[j-1,k-1]}(x_0,x_0-0;\lambda)+K^{[j-1,k-1]}(x_0,x_0+0;\lambda)\right] \]

\[ (j,k=1,2,\ldots,2n), \]

where \(x_0\) is any fixed point of the interval \((a,b)\). The matrix function

\[ M(\lambda)=\left\|m_{jk}(\lambda)\right\|_{1}^{2n} \]

will be called the characteristic matrix of the generalized resolvent \(R_\lambda\).\(^{{**}}\)

Theorem 1. The characteristic matrix \(M(\lambda)\) of any generalized resolvent \(R_\lambda\) of the operator \(L\) is a regular function of the parameter \(\lambda\) in the upper and lower half-planes, and

\[ M(\overline{\lambda})=M^{*}(\lambda). \]

For any \(\lambda\) in the upper half-plane the matrix

\[ \operatorname{Im} M(\lambda)=\frac{1}{2i}\left[M(\lambda)-M^{*}(\lambda)\right] \]

is Hermitian nonnegative, \(\operatorname{Im} M(\lambda)\geq 0\).

\(^*\) The basic definitions and facts to which we refer here are set out, for example, in \((^{1,2})\).

\(^{{**}}\) In \((^{4})\) the matrix \(M(\lambda)\) is defined directly by formula (9). We note that in that formula the factor \(1/2\) is omitted at the matrix \(Q_0(\lambda)\).

For any complex \(\lambda\), let us consider the system of functions

\[ y_1(x;\lambda),\ y_2(x;\lambda),\ldots,\ y_{2n}(x;\lambda), \tag{1} \]

which are solutions of the equation

\[ l[y]-\lambda y=0 \]

and satisfy the conditions

\[ y_k^{[j-1]}(x_0;\lambda)=\delta_{jk}\qquad (j,k=1,2,\ldots,2n), \]

where the expression on the left denotes the quasiderivative of order \(j-1\).

By \(y(x;\lambda)\) we shall denote the column matrix composed of the functions (1), and by \(y'(x;\lambda)\) the corresponding transposed matrix.

Let \(E_t\) \((-\infty<t<+\infty)\) be an arbitrary spectral function of the operator \(L\) (in general nonorthogonal), and let \(R_\lambda\) be the corresponding generalized resolvent. Applying the Stieltjes inversion formula to the generalized resolvent \(R_\lambda\), we obtain the formula for all spectral functions of the operator \(L\).

Theorem 2. For any real \(\alpha\) and \(\beta\), the operator

\[ E_{\alpha,\beta}=\frac{E_\beta+E_{\beta+0}}{2}-\frac{E_\alpha+E_{\alpha+0}}{2} \]

is integral. Its kernel is determined by the formula

\[ K(x,s;\alpha,\beta)=\int_\alpha^\beta y'(x;\lambda)\,dT(\lambda)\,y(s;\lambda), \]

where

\[ T(\lambda)=\frac{1}{\pi}\lim_{\tau\to +0}\int_0^\lambda \operatorname{Im} M(\sigma+i\tau)\,d\sigma. \]

Moreover,

\[ \int_a^b |K(x,s;\alpha,\beta)|^2\,ds<\infty;\qquad \int_a^b |K(x,s;\alpha,\beta)|^2\,dx<\infty \]

\[ (a<x,\ s<b). \]

The following assertion is closely connected with Theorem 2.

Theorem 3. For any function \(f(x)\in L^2(a,b)\) the expansion

\[ f(x)=\int_{-\infty}^{+\infty} y'(x;\lambda)\,dT(\lambda)\,\eta(f;\lambda), \tag{2} \]

holds, where

\[ \eta(f;\lambda)=\int_b^b f(s)y(s;\lambda)\,ds. \tag{3} \]

Moreover, the integrals in the right-hand sides of formulas (2), (3) converge correspond—

respectively in the sense of the metrics of the spaces \(L^{2}(a,b)\) and \(L_T^2(-\infty,+\infty)\), and the equality

\[ \int_a^b |f(x)|^2\,dx = \int_{-\infty}^{+\infty} \eta^*(f;\lambda)\,dT(\lambda)\,\eta(f;\lambda) \]

holds.

1. Suppose now that both endpoints of the interval \((a,b)\) are regular. Let \(A(\lambda)\) and \(B(\lambda)\) be square matrices of order \(2n\), depending on the complex parameter \(\lambda\) and satisfying the following conditions:

a) the rank of the rectangular matrix \(\|A(\lambda)\ B(\lambda)\|\) is equal to \(2n\) for every \(\lambda\);

b) \(A(\lambda)\) and \(B(\lambda)\) are entire matrix functions of the parameter \(\lambda\);

c) for every non-real \(\lambda\),

\[ \frac{1}{\lambda-\bar\lambda}\,[B(\lambda)JB^*(\lambda)-A(\lambda)JA^*(\lambda)]\ge 0, \]

where the skew-symmetric matrix \(J\) of order \(2n\) has the form

\[ J= \begin{Vmatrix} & & & -1\\ & & -1 & \\ & \iddots& & \\ -1& & & \\ 1 & & & \\ & \ddots & & \\ & & 1 & \\ & & & 1 \end{Vmatrix}, \]

and all unmarked positions contain zeros.

Every function \(y(x)\) for which \(l[y]\) has meaning possesses quasi-derivatives \(y^{[j]}(x)\) \((j=0,1,\ldots,2n-1)\) that are absolutely continuous on the interval \([a,b]\). By \(\hat y(x)\) we denote the one-column matrix function composed of these quasi-derivatives:
\[ \hat y(x)=(y^{[0]}(x),y^{[1]}(x),\ldots,y^{[2n-1]}(x)). \]
Consider the following boundary-value problem:

\[ l[y]-\lambda y=0, \tag{4} \]

\[ A(\lambda)\hat y(a)+B(\lambda)\hat y(b)=0^*. \tag{5} \]

To the boundary condition (5) there corresponds a certain generalized resolvent \(R_\lambda\) of the operator \(L\); for any function \(f(x)\in L^2(a,b)\), \(R_\lambda f\), for every non-real \(\lambda\), is defined as the solution of the equation

\[ l[y]-\lambda y=f, \]

satisfying the boundary condition (5) (cf. (4)). Owing to this circumstance, propositions on generalized resolvents and the corresponding spectral functions prove useful in the study of the boundary-value problem (4), (5).

Theorem 4. All eigenvalues of the boundary-value problem (4), (5) are real, and their set is countable and coincides with the set of poles of the meromorphic matrix function

\[ M(\lambda)=\frac12[A(\lambda)Y(a;\lambda)+B(\lambda)Y(b;\lambda)]^{-1}\times \]
\[ \times [A(\lambda)Y(a;\lambda)-B(\lambda)Y(b;\lambda)]J^{-1}, \]

\[ \text{* Boundary-value problems of this type occur in applications (see, for example, (³), p. 144).} \]

where \(Y(x;\lambda)\) is the matrix of quasi-derivatives corresponding to the system (1), i.e.

\[ Y(x;\lambda)=\left\|\,\hat y_1(x;\lambda);\ \hat y_2(x;\lambda)\ldots \hat y_{2n}(x;\lambda)\,\right\|. \]

All these poles are simple.

Let

\[ \lambda_1,\lambda_2,\ldots,\lambda_k,\ldots \]

be the sequence of all eigenvalues of the boundary-value problem (4), (5).

Denote by \(T_k\) the residue of the matrix function \(-M(\lambda)\) at the point \(\lambda_k\). Put

\[ H_k(x;s)=y'(x;\lambda_k)T_k y(s;\lambda_k)\qquad (a\leq x,s\leq b;\quad k=1,2,\ldots) \]

and introduce into consideration the integral operators \(Q_k\) in \(L^2(a,b)\):

\[ Q_k f=\int_a^b H_k(x;s)f(s)\,ds\qquad (k=1,2,\ldots). \]

For any \(k\), the operator \(Q_k\) is, obviously, nonnegative.

Denote by \(\mathfrak M_k\) the subspace of eigenfunctions of the boundary-value problem (4), (5) corresponding to the eigenvalue \(\lambda_k\) \((k=1,2,\ldots)\).

Theorem 5. For any \(k\) \((k=1,2,\ldots)\),

\[ Q_k L^2(a,b)=Q_k\mathfrak M_k=\mathfrak M_k. \]

Theorem 6. Every function \(f(x)\) from \(L^2(a,b)\) can be expanded in a series, convergent in the metric of this space, in the eigenfunctions of the boundary-value problem (4), (5):

\[ f=\sum_{k=1}^{\infty} Q_k f. \]

Ulyanovsk State Pedagogical Institute
named after I. N. Ulyanov

Received
10 V 1956

CITED LITERATURE

\(^{1}\) N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, 1950.
\(^{2}\) M. A. Naimark, Linear Differential Operators, 1954.
\(^{3}\) A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, 1951.
\(^{4}\) A. V. Shtraus, DAN, 111, No. 4, 773 (1956).