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).