Abstract
Full Text
MATHEMATICS
M. G. GIMADISLAMOV
ON EXPANSION IN EIGENFUNCTIONS OF A NONSELFADJOINT SYSTEM OF DIFFERENTIAL EQUATIONS OF SECOND ORDER
(Presented by Academician P. S. Aleksandrov on 20 IV 1961)
The present work is devoted to the study of expansion in eigenfunctions of a nonselfadjoint system of differential equations in the space of vector-functions. The scalar case was first studied in the fundamental work of M. A. Naimark ((^{1})).
Consider a system of differential expressions of second order, which is written briefly in the form
[
l(y)=-y''+P(x)y,
\tag{1}
]
where (y(x)=(y_1(x),\ldots,y_k(x))) is a vector-function; (P(x)) is a (k)-dimensional complex-valued matrix, summable on the interval ([0,\infty]).
With this differential expression we construct an operator in the space of vector-functions (y(x)\in L_k^2(0,\infty)).
(y(x)\in L_k^2(0,\infty)), if
[
\int_0^\infty \sum_{i=1}^k |y_i(x)|\,dx<\infty .
]
Denote by (D) the set of those vector-functions (y(x)\in L_k^2(0,\infty)) such that: 1) the derivative of the vector-function (y'(x)) exists and is absolutely continuous on every finite interval ([0,b]), (b>0); 2) (l(y)\in L_k^2(0,\infty)). By (D_\Theta) denote the set of vector-functions (y(x)\in D) satisfying the condition
[
y'(0)-\Theta y(0)=0,
\tag{2}
]
where (\Theta) is a fixed (k)-dimensional complex matrix.
Define the operator (L_\Theta) as follows: its domain of definition is (D_\Theta), and for (y(x)\in D_\Theta), (L_\Theta y=l(y)).
Denote by (Y_1(x,s)) and (Y_2(x,s)) ((s^2=\lambda)) linearly independent solutions of the matrix equation
[
-Y''+P(x)Y=\lambda Y .
\tag{3}
]
These solutions are constructed so that they satisfy the following asymptotic formulas:
as (x\to\infty),
[
Y_1(x,s)=e^{isx}[1+o(1)]
]
uniformly with respect to (s), (|s|\ge r>0), (\operatorname{Im}s\ge0);
[
Y_2(x,s)=e^{-isx}[1+o(1)]
]
uniformly with respect to (s), (|s|\ge r>0), (\operatorname{Im}s\le0);
as (s\to\infty),
[
Y_1(x,s)=e^{isx}\left[1+O\left(\frac{1}{s}\right)\right],\qquad
Y_2(x,s)=e^{-isx}\left[1+O\left(\frac{1}{s}\right)\right]
]
uniformly with respect to (x), (0\le x<\infty).
Analogous solutions (Z_1(x,s)) and (Z_2(x,s)) are constructed for the matrix equation
[
-Z''+ZP(x)=\lambda Z.
\tag{4}
]
Denote by (\xi_1(s),\ldots,\xi_k(s)) the eigenvalues of the matrix
[
|A_1(s)-\xi A_2(s)|=0,
]
where
[
A_1(s)=Y_1'(0,s)-\Theta Y_1(0,s),
\tag{5}
]
[
A_2(s)=Y_2'(0,s)-\Theta Y_2(0,s),
\tag{6}
]
and by (\rho_1(x),\ldots,\rho_k(s)) the corresponding eigenvectors. Let (\xi'_1(s),\ldots,\xi'_k(s)) be the eigenvalues, and (\rho'_1(s),\ldots,\rho'_k(s)) the corresponding eigenvectors of the matrix
[
B_1(s)-\xi' B_2(s)=0,
]
where
[
B_1(s)=Z_1'(0,s)-Z_1(0,s)\Theta,
\tag{7}
]
[
B_2(s)=Z_2'(0,s)-Z_2(0,s)\Theta.
\tag{8}
]
If the matrix (P(x)) is summable on the interval ([0,\infty]), then the following holds:
Theorem 1. The spectrum of the operator (L_\Theta) is continuous on the positive half-axis and discrete in the entire remaining complex (\lambda)-plane. The eigenvalues of the operator (L_\Theta) form a bounded set, whose limit points may lie only on the positive half-axis (\lambda\geq 0). For values of (\lambda) not belonging to the spectrum, the resolvent of the operator (L_\Theta) is an integral operator with kernel (K(x,\xi,\lambda)), satisfying the conditions:
[
\int_0^\infty |K(x,\xi,\lambda)|^2\,d\xi<\infty,\qquad
\int_0^\infty |K(x,\xi,\lambda)|^2\,dx<\infty.
]
Theorem 1 follows from the asymptotic behavior of (Y_1(x,s)) and (Y_2(x,s)) for large (s). In what follows we assume that
[
\int_0^\infty x^2|P(x)|\,dx<\infty
]
and that:
1) the eigenvalues of the operator (L_\Theta) are simple poles of its resolvent; 2) the matrices (A_1(s)) and (A_2(s)) are non-singular for (s\geq 0). In this case the discrete part of the spectrum consists of a finite number of points, and the point (\lambda=0) is not a spectral point. Let (\lambda_1,\lambda_2,\ldots,\lambda_r) be the eigenvalues, and (y_1(x),y_2(x),\ldots,y_r(x)) the corresponding eigenvector-functions of the operator (L_\Theta).
Theorem 2. If conditions 1) and 2) are satisfied, then for any point (\lambda) not belonging to the spectrum of the operator (L_\Theta),
[
K(x,\xi,\lambda)=\sum_{j=1}^r
\frac{y_j(x)z_j^(\xi)}
{(\lambda_j-\lambda)\displaystyle\int_0^\infty (y_j,z_j)\,dx}
-\frac{1}{\pi}\int_0^\infty
\sum_{j=1}^k
\frac{[Y_1(x,s)-\xi_jY_2(x,s)]\rho_j\rho_j^{\prime }[z_1(\xi,s)-\xi'_jz_2(\xi,s)]}
{(s^2-\lambda)\xi_j(s)+\xi'_j(s)}\,ds,
\tag{9}
]
where the integral on the right converges absolutely and uniformly with respect to (x,\xi) in the domain (0\leq x,\xi<\infty).
Theorem 3. If conditions 1) and 2) are satisfied, then every vector-function (g(x)\in D_\Theta) can be represented in the form
[
g(x)=\sum_{j=1}^r
\frac{y_j(x)\displaystyle\int_0^\infty (g,z_j)\,dx}
{\displaystyle\int_0^\infty (y_j,z_j)\,dx}
-\frac{1}{\pi}\int_0^\infty
\sum_{j=1}^k
\frac{[Y_1(x,s)-\xi_j(s)Y_2(x,s)]\rho_j\rho_j^{\prime *}F_j(s)}
{\xi_j(s)+\xi'_j(s)}\,ds.
\tag{10}
]
where
[
F_j(s)=\int_0^\infty [z_1(\xi,s)-\xi_j'(s)z_2(\xi,s)]g(\xi)\,d\xi .
]
The integral on the right converges absolutely and uniformly with respect to (x) in the interval (0\leq x<\infty).
There is an analogue of Parseval’s equality.
Let (g(x)\in D_\Theta) and (h(x)\in L_k^2(0,\infty)); then
[
\int_0^\infty (g(x),h(x))\,dx
=
\sum_{j=1}^r
\frac{\alpha_j\beta_j}{\displaystyle\int_0^\infty (y_j,z_j)\,dx}
-\frac{1}{\pi}\int_0^\infty
\sum_{j=1}^k
\frac{(\rho_j\rho_j^*F_j(s),H_j(s))}{\xi_j+\xi_j'}\,ds,
\tag{11}
]
where
[
\alpha_j=\int_0^\infty (y_j,h)\,dx,\qquad
\beta_j=\int_0^\infty (g,z_j)\,dx,\qquad
H_j(s)=\int_0^\infty [Y_1^-\bar{\xi}_jY_2^]h(x)\,dx .
]
Theorem 3 is easily obtained from Theorem 2, and the analogue of Parseval’s equality from Theorem 3.
We outline the proof of Theorem 2. First consider the case when the matrix (P(x)) satisfies the condition
[
\int_0^\infty e^{\varepsilon x}|P(x)|\,dx<\infty
\tag{12}
]
for some (\varepsilon>0). Then we pass to the case when the matrix (P(x)) satisfies the condition
[
\int_0^\infty x^2|P(x)|\,dx<\infty,
]
approximating it by matrices satisfying condition (12).
Consider the auxiliary boundary-value problem on the interval ([0,b]), (b>0),
[
l(y)=\lambda y,\qquad y'(0)-\Theta y(0)=0,\qquad y(b)=0.
\tag{13}
]
Let (K_b(x,\xi,\lambda)) be the resolvent kernel of this boundary-value problem; then, as (b\to\infty),
[
K_b(x,\xi,\lambda)=K(x,\xi,\lambda)+o(1)
\tag{14}
]
uniformly with respect to (x,\xi) in every finite square (0\leq x,\xi\leq c), (c>0). For sufficiently large (\lambda), to each of the eigenvalues (\lambda_1,\ldots,\lambda_r) of the operator (L_\Theta) there corresponds exactly one eigenvalue (\lambda_1(b),\ldots,\lambda_r(b)) of the boundary-value problem (13), so that (\lambda_j(b)\to\lambda_j), (j=1,2,\ldots,r), as (b\to\infty). All remaining eigenvalues of the boundary-value problem, as (b\to\infty), have the following asymptotics:
[
\lambda=s^2,\qquad
s_{nj}^{(j)}=\frac{n\pi}{b}+\frac{1}{2bi}+\ln \xi_j!\left(\frac{n\pi}{b}\right)+\frac{1}{b}o(1)
\tag{15}
]
uniformly with respect to (s) in the domain (|\operatorname{Im}s|\leq\varepsilon_1), (\operatorname{Re}s\geq0). (y(x,s_n^{(j)})=[Y_1(x,s)-\xi_j(s)Y_2(x,s)]\rho_j(s)) are eigenfunctions corresponding to the eigenvalues (15).
If (y_j(x,b)), (j=1,2,\ldots,r), are the eigenfunctions of the boundary-value problem (13), then as (b\to\infty)
[
\frac{y_j(x,b)z_j^(\xi,b)}
{\displaystyle\int_0^b (y_j,z_j)\,dx}
=
\frac{y_j(x)z_j^(\xi)}
{\displaystyle\int_0^\infty (y_j,z_j)\,dx}
+o(1),\qquad
j=1,2,\ldots,r,
\tag{16}
]
uniformly with respect to (x), (0\leq x\leq c), (c>0).
Moreover, as (b\to\infty),
[
\frac{1}{b}\int_{0}^{\infty}(y(x,s_n^{(j)}),\,z(x,s_n^{(j)}))\,dx
=
-\,\xi_j(s)+\xi_j'(s)+o(1)
\tag{17}
]
uniformly with respect to (s) in every rectangle (|\operatorname{Im}s|\leq\varepsilon_1), (0\leq \operatorname{Re}s\leq\beta), (\beta>0).
For the derivation of formula (9), the kernel (K_b(x,\xi,\lambda)) is considered, for (b=m\sqrt{q}), on the contour (C_{m,q}), which is chosen in the same way as in paper (1) ((m,q) are natural numbers). On the contour (C_{m,q}), (|K(x,\xi,\lambda)|\leq c/\sqrt{|\lambda|}), and therefore
[
\frac{1}{2\pi i}\int_{C_{m,q}}\frac{K_b(x,\xi,\lambda)}{(\lambda-\lambda_0)}\,d\lambda
\to 0
\quad\text{as } m\to\infty
]
uniformly with respect to (q). Applying the residue theorem to the last integral and passing to the limit as (m\to\infty), (q\to\infty), taking into account (14), (16), and (17), we obtain formula (9) for (\lambda=\lambda_0). If the eigenvalues (\lambda_1,\ldots,\lambda_r) of the operator (L_\theta) have multiplicities (m_1,\ldots,m_r), respectively, then
[
K(x,\xi,\lambda)
=
\sum_{i=1}^{r}\sum_{p=1}^{m_i}
\frac{G_p^{(i)}(x,\xi)}{(\lambda-\lambda_i)^p}
-
\frac{1}{\pi}\int_{0}^{\infty}
\sum_{j=1}^{k}
\frac{[Y_1-\xi_jY_2]\rho_j\rho_j^{\prime *}[z_1-\xi_j'z_2]}
{(s^2-\lambda)\xi_j+\xi_j'}
\,ds,
\tag{18}
]
where
[
G_p^{(i)}(x,\xi)
=
-
\sum_{j=0}^{m_i-p}
y_j^{(i)}(x)\,z_{m_i-p-j}^{(i)*}(\xi);
]
for all (p) and (i), (G_p^{(i)}(x,\xi)) satisfies the boundary condition (2), and
[
l(y_j^{(i)})-\chi_j y_j^{(i)}=y_{j-1}^{(i)},\qquad
l^*(z_j^{(i)})-\overline{\lambda}j z_j^{(i)}=z,}^{(i)
]
where
[
y_{-1}^{(i)}(x)\equiv 0,\qquad
z_{-1}^{(i)}(\xi)\equiv 0,\qquad
\int_{0}^{\infty}(y_j^{(i)},z_r^{(k)})\,dx
=
\delta_{ik}\delta_{m_i-j-1,r}.
]
The functions (y_0^{(i)}(x),\ldots,y_{m_i-1}^{(i)}(x)) form a chain of eigenfunctions and associated functions corresponding to the eigenvalue (\lambda_i). If (P(x)) satisfies condition (12), then assumption 2) can be dropped; in that case the integration is carried out over a contour obtained in the usual way by deforming the positive half-axis in the complex (\lambda)-plane. Another expansion is considered in paper (4).
In conclusion, the author expresses gratitude to A. G. Kostyuchenko.
Moscow State University
named after M. V. Lomonosov
Received
17 IV 1961
References
- M. A. Naimark, Trans. Moscow Math. Soc., 3, 181 (1954).
- M. A. Naimark, Linear Differential Operators, 1954.
- Z. I. Biglov, Dokl. Akad. Nauk SSSR, 112, No. 5 (1957).
- V. Kabanova, Dokl. Akad. Nauk SSSR, 121, No. 1 (1958).