MATHEMATICS

Z. I. BIGLOV

EXPANSION IN EIGENFUNCTIONS OF A SYSTEM OF SECOND-ORDER DIFFERENTIAL EQUATIONS

(Presented by Academician I. G. Petrovskii on 23 IX 1956)

The spectral expansion is studied for operators generated by a system of second-order differential expressions

\[ l(y)=-y''+P(x)y \qquad (0\le x<\infty), \tag{1} \]

in the case where the real symmetric matrix of order \(n\), \(P(x)\), is summable on the interval \((0,\infty)\). This note is a continuation of the author’s preceding note \((^1)\).

We introduce the following notation. Let \(L_0\) denote the operator in the space \(L_n^2(0,\infty)\) of vector-functions
\[ y(x)=\{y_1(x),y_2(x),\ldots,y_n(x)\}, \]
summable with the square of the norm, generated by the differential expression (1) and the boundary conditions at zero

\[ y'(0)=\theta y(0) \]

(\(\theta\) is a Hermitian matrix). Denote by \(\Omega_1(x,s)\), \(\Omega_2(x,s)\) \((s^2=\lambda)\) linearly independent solutions of the matrix equation

\[ l(Y)-\lambda Y=0. \tag{2} \]

Further, denote by \(\xi_1(s),\xi_2(s),\ldots,\xi_n(s)\) the eigenvalues of the problem:

\[ [A_1(s)-\xi A_2(s)]\rho=0, \tag{3} \]

where

\[ A_1(s)=\Omega_1'(0,s)-\theta\Omega_1(0,s), \]

\[ A_2(s)=\Omega_2'(0,s)-\theta\Omega_2(0,s). \]

Finally, let

\[ \Omega_i(x,s)=-\Omega_1(x,s)\xi_i(s)+\Omega_2(x,s), \qquad i=1,2,\ldots,n. \]

Under the condition that the eigenvalue problem (3) has a complete system of orthogonal eigenvectors \(\rho_1(s),\rho_2(s),\ldots,\rho_n(s)\), the following theorems are valid:

Theorem 1 (Parseval equality). For every vector-function \(f(x)\) belonging to \(L_n^2(0,\infty)\), the equality

\[ \int_0^\infty \|f(x)\|^2\,dx = \]

\[ = \sum_{k=1}^{\infty} \frac{|a_k|^2}{\displaystyle \int_0^\infty \|y_k(x)\|^2\,dx} - \frac{1}{\pi}\int_0^\infty \frac{ \displaystyle \sum_{i,j=1}^{n} F_i^*(s)\rho_i(s)\rho_j^*(s)\overline{c_i}c_jF_j(s)\,ds }{ \displaystyle \sum_{i=1}^{n} (|\xi_i(s)|^2+1)\|\rho_i\|^2|c_i|^2 }, \]

where

\[ \alpha_k=\int_0^\infty (f,y_k)\,dx; \]

\(y_k\) is an eigenfunction of \(L_\theta\); \(c_1,c_2,\ldots,c_n\) are constant numbers,

\[ F_i(s)=\operatorname{l.i.m}_{\,n\to\infty}\int_0^n \Omega_i(x,s)f(x)\,dx. \]

(The symbol l.i.m. denotes the limit in the sense of the norm in the Hilbert space generated by the spectral matrix of the operator \(L_\theta\). See, in this connection, \((^3)\).)

Theorem 2. For the kernel \(K(x,\xi,\mu)\) of the resolvent of the operator \(L_\theta\) the following integral representation holds \((\operatorname{Im}\mu\ne 0)\):

\[ K(x,\xi,\mu)= \sum_{k=1}^{\infty} \frac{y_k(x)y_k^*(\xi)} {(\lambda_k-\mu)\displaystyle\int_0^\infty \|y_k\|^2\,dx} - \frac{1}{\pi}\int_0^\infty \frac{ \displaystyle\sum_{i,j=1}^{n} \Omega_i(x,s)\rho_i(s)\rho_j^*(s)c_i\overline{c_j}\Omega_j^*(\xi,s) }{ (s^2-\mu)\left[\displaystyle\sum_{i=1}^{n}\left(|\xi_i(s)|^2+1\right)\|\rho_i\|^2|c_i|^2\right] }\,ds. \]

The integral on the right-hand side of this equality converges absolutely and uniformly with respect to \(x,\xi\) in the region \(0\le x,\xi<\infty\).

In proving these theorems, the method proposed by M. A. Naimark in his paper \((^2)\) is used. We shall give the principal points of the proof.

Linearly independent matrix solutions \(\Omega_1(x,s)\), \(\Omega_2(x,s)\) of equation (2) are constructed so that they satisfy the asymptotic formulas: as \(x\to\infty\),

\[ \Omega_1(x,s)=e^{isx}[1+o(1)] \]

uniformly with respect to \(s\), \(|s|\ge r>0\), \(\operatorname{Im}s\ge 0\);

\[ \Omega_2(x,s)=e^{-isx}[1+o(1)] \]

uniformly with respect to \(s\), \(|s|\ge r>0\), \(\operatorname{Im}s\le 0\); as \(s\to\infty\),

\[ \Omega_1(x,s)=e^{isx}[1+O(1/s)],\quad \Omega_2(x,s)=e^{-isx}[1+O(1/s)] \]

uniformly with respect to \(x\), \(0\le x<\infty\).

With the aid of these formulas we find asymptotic formulas for the eigenfunctions of the boundary-value problem on the finite interval \([0,b]\)

\[ l(y)-\lambda y=0,\quad y'(0)-\theta y(0)=0,\quad y(b)=0. \tag{4} \]

The eigenvalues of this boundary-value problem form \(n\) infinite sequences
\(\{\lambda_k^{(i)}\}\), \(i=1,2,\ldots,n\), \(k=1,2,3,\ldots\), such that

\[ \lambda_k^{(i)}=[s_k^{(i)}]^2,\quad s_k^{(i)}=-\,\frac{k\pi}{b}-\frac{1}{2bi}\ln \xi_i\!\left(\frac{k\pi}{b}\right) +\frac{1}{b}o_j(1) \]

as \(b\to\infty\).

Suppose that problem (3) has a complete system of eigenvectors
\(\rho_1(s),\rho_2(s),\ldots,\rho_n(s)\); then the eigenfunctions of the boundary-value problem (4) are determined up to a constant vector
\(c=(c_1,c_2,\ldots,c_n)\) and, as \(b\to\infty\), satisfy the asymptotic formulas

\[ y_k(x)=\sum_{i=1}^{n}\Omega_i(x,s_k)\rho_i(s_k)c_i+o(1), \quad k=1,2,3,\ldots . \]

Further, for the eigenfunction \(y(x,s)\) of the boundary-value problem (4), corresponding to the eigenvalue \(\lambda=s^2,\ s>0\), as \(b\to\infty\) the asymptotic formula

\[ \frac{1}{b}\int_0^\infty \|y(x,s)\|^2\,dx = \sum_{i=1}^n \left(|\xi_i(s)|^2+1\right)\|\rho_i(s)\|^2\,|c_i|^2+o(1) \tag{5} \]

holds uniformly with respect to \(s,\ 0\le s<\infty\).

Further, the operator \(L_\theta\) is self-adjoint, and its spectrum on the negative half-axis \(\lambda>0\) is discrete and bounded below, while on the positive half-axis \(\lambda>0\) it is continuous \({}^{1}\). Taking this into account and using the asymptotic formulas for the eigenvalues and eigenfunctions of the boundary-value problem (4) and formula (5), from the corresponding spectral representations for the boundary-value problem (4), by passing to the limit as \(b\to\infty\), we arrive at Theorems 1 and 2.

Bashkir State
Pedagogical Institute
named after K. A. Timiryazev

Received
10 XII 1955

REFERENCES

\({}^{1}\) Z. I. Biglov, DAN, 99, No. 4 (1954).
\({}^{2}\) M. A. Naimark, Tr. Moscow Math. Soc., 3 (1954).
\({}^{3}\) I. Kats, Zap. Scientific-Research Institute of Mathematics and Mechanics and Kharkov Math. Soc., 22 (1950).