MATHEMATICS

F. G. MAKSUDOV

EXPANSION IN EIGENFUNCTIONS OF NON-SELFADJOINT SINGULAR SECOND-ORDER DIFFERENTIAL OPERATORS DEPENDING ON A PARAMETER

(Presented by Academician I. N. Vekua on 16 VII 1963)

M. A. Naimark \((^1)\) investigated the spectrum and expansions in eigenfunctions of a non-selfadjoint singular differential operator \(L_\theta\), generated on the half-axis \(R^+ = [0,\infty)\) by the differential expression

\[ l(y)\equiv -y''+p(x)y,\qquad 0\leq x<+\infty, \tag{1} \]

and by the boundary condition

\[ y'(0)-\theta y(0)=0, \tag{2} \]

where \(p(x)\) is a complex-valued function such that \(x^2p(x)\in L(R^+)\), and \(\theta\) is a complex number.

In the present paper, by M. A. Naimark’s method, the expansion in eigenfunctions of the operator \(L_{\theta(\lambda)}\) is studied in the case when, in the boundary conditions (2), \(\theta\) depends on the complex parameter \(\lambda\).

The case in which the differential expression \(l(y)\) is selfadjoint in the sense of Lagrange was considered by A. V. Shtraus \((^2)\). Let \(p(x)\) be a complex-valued function summable on each finite interval \([0,a]\), \(a>0\).

Denote by \(D\) the set of all functions \(y(x)\in L^2(R^+)\) satisfying the conditions: 1) \(y(x)\) and \(y'(x)\) are absolutely continuous on each finite interval \([0,a]\), \(a>0\); 2) \(l(y)\in L^2(R^+)\). Let \(\theta(\lambda)=P(\lambda)/Q(\lambda)\), where \(P(\lambda)\) and \(Q(\lambda)\) are polynomials in powers of the complex parameter \(\lambda\), and moreover

\[ \lim \frac{\theta(\lambda)}{is}\ne 1,\qquad \lambda=s^2. \]

Denote by \(D_{\theta(\lambda_0)}\) the set of all functions \(y(x)\in D\) satisfying, at the fixed point \(\lambda_0\), the condition*

\[ y'(0)-\theta(\lambda_0)y(0)=0. \tag{3} \]

Denote by \(L_{\theta(\lambda_0)}\) the operator with domain \(D_{\theta(\lambda_0)}\) such that, for \(y\in D_{\theta(\lambda_0)}\), \(L_{\theta(\lambda_0)}y=l(y)\).

If \(\lambda\) runs through the set of all points of the \(\lambda\)-plane, then we obtain a family of non-selfadjoint singular operators \(L_{\theta(\lambda)}\), depending on the parameter \(\lambda\). Let \(p(x)\in L(R^+)\).

Theorem 1. The spectrum of the operator \(L_{\theta(\lambda)}\) is continuous on the positive half-axis and discrete in the entire remaining complex \(\lambda\)-plane. The eigenvalues of the operator \(L_{\theta(\lambda)}\) form a bounded set, whose limit points may lie only on a finite interval of the positive half-axis \(\lambda\geq 0\). For values of \(\lambda\) not belonging to the spectrum, the resolvent \(R_\lambda=(L_{\theta(\lambda)}-\lambda I)^{-1}\) of the operator \(L_{\theta(\lambda)}\) is an integral operator with kernel \(K(x,t,\lambda)\), satisfying the conditions

\[ \int_0^\infty |K(x,t,\lambda)|^2\,dx<\infty,\qquad \int_0^\infty |K(x,t,\lambda)|^2\,dt<\infty. \tag{4} \]

\[ \text{* If }\lambda_0\text{ is a pole of }\theta(\lambda),\text{ then condition (3) means }y(0)=0. \]

Suppose that for some \(\varepsilon > 0\) the function \(p(x)\) satisfies the condition

\[ \int_0^\infty e^{\varepsilon x}|p(x)|\,dx < \infty . \tag{5} \]

Then, as M. A. Naimark showed \((^{1})\), the differential equation
\(l(y)=\lambda y\) has solutions \(y_1(x,s)\), \(y_2(x,s)\), where \(y_1\) and \(y_2\) are holomorphic functions of \(s\) in the half-planes \(\operatorname{Im}s > -\frac12\varepsilon\), \(\operatorname{Im}s < \frac12\varepsilon\), respectively. As \(x \to \infty\),

\[ y_1(x,s)=e^{ixs}[1+o(1)], \qquad y_1'(x,s)=e^{ixs}[is+o(1)]; \tag{6a} \]

\[ y_2(x,s)=e^{-ixs}[1+o(1)], \qquad y_2'(x,s)=e^{-ixs}[-is+o(1)]; \tag{6b} \]

as \(s \to \infty\) and \(\operatorname{Im}s \to 0\), and on the positive half-axis \(s \geq 0\), respectively,

\[ y_1(x,s)=e^{ixs}\left[1+O\left(\frac1s\right)\right], \qquad y_1'(x,s)=ise^{ixs}\left[1+O\left(\frac1s\right)\right]; \tag{7a} \]

\[ y_2(x,s)=e^{-ixs}\left[1+O\left(\frac1s\right)\right], \qquad y_2'(x,s)=-ise^{-ixs}\left[1+O\left(\frac1s\right)\right]. \tag{7b} \]

Here the asymptotic formulas (6a) and (6b) hold uniformly with respect to \(s\) in the regions \(\operatorname{Im}s > -\varepsilon_1\), \(\operatorname{Im}s < \varepsilon_1\), \(\varepsilon_1 < \frac12\varepsilon\), while formulas (7a) and (7b) hold for \(\operatorname{Im}s > -\frac12\varepsilon\), \(\operatorname{Im}s < \frac12\varepsilon\), respectively, uniformly with respect to \(x\) from the interval \(R^+\).

Assume that the functions

\[ A(s)=y_1'(0,s)-\theta(\lambda)y(0,s), \]

\[ \widetilde A(s)=y_2'(0,s)-\theta(\lambda)y(0,s) \]

do not vanish on the half-axis \(s \geq 0\); then, from the expression for \(\theta(\lambda)\) and from formulas (7a) and (7b), it follows that \(A(s)\) and \(\widetilde A(s)\) do not vanish in the half-strip \(|\operatorname{Im}s| < \varepsilon_1\), \(\operatorname{Re}s \geq 0\). Therefore, when condition (5) is fulfilled, the spectrum of the operator \(L_{\theta(\lambda)}\) can have only a finite number of eigenvalues. None of the eigenvalues of the operator \(L_{\theta(\lambda)}\) lies on the half-axis \(\lambda \geq 0\). Here \(\lambda=0\) belongs to the continuous spectrum. Suppose that the eigenvalues of the operator \(L_{\theta(\lambda)}\) are simple poles of its resolvent.

Consider the auxiliary boundary-value problem on the interval

\[ -y''+p(x)y=\lambda y, \qquad y'(0)-\theta(\lambda)y(0)=0, \qquad y(b)=0. \tag{8} \]

For sufficiently large \(b\), to each of the eigenvalues \(\lambda_1,\lambda_2,\ldots,\lambda_r\) of the operator \(L_{\theta(\lambda)}\) there corresponds exactly one eigenvalue \(\lambda_1(b),\lambda_2(b),\ldots,\lambda_r(b)\) of the boundary-value problem (7), (8), so that \(\lambda_k(b)\to\lambda_k\) as \(b\to\infty\). All the remaining eigenvalues of the auxiliary boundary-value problem (7), (8) satisfy the relations

\[ \lambda=s^2, \qquad s=\frac{n\pi}{b}+\frac{1}{2ib}\omega\left(\frac{n\pi}{b}\right)+o\left(\frac1b\right) \quad \text{as } b\to\infty \]

uniformly with respect to \(s\) in the half-strip \(|\operatorname{Im}s| < \varepsilon_1\), \(\operatorname{Re}s \geq 0\), where \(\omega(s)=\ln \frac{A(s)}{\widetilde A(s)}\) is a single-valued holomorphic function. Let \(y_{1,b}(x)\), \(y_{2,b}(x),\ldots,y_{r,b}(x)\) be the eigenfunctions of the boundary-value problem (7), (8), corresponding to the eigenvalues \(\lambda_1(b),\lambda_2(b),\ldots,\lambda_r(b)\). Then, as \(b\to\infty\),

\[ \frac{y_{k,b}(x,s)}{\displaystyle \int_0^\infty [y_{k,b}(x)]^2\,dx} = \frac{y_k(x)}{\displaystyle \int_0^\infty [y_k(x)]^2\,dx} +o(1) \]

uniformly with respect to \(x \in [0,c]\), \(c>0\). Moreover, if \(\widetilde{\lambda}=|\widetilde{s}(b)|^2\) are eigenvalues of the problem (7), (8) such that \(\operatorname{Im}s\to 0\) as \(b\to\infty\), then, as \(b\to\infty\),

\[ \frac{1}{b}\int_0^\infty [\widetilde{y}(k,s)]^2\,dx = -2A(\widetilde{s})\widetilde{A}(\widetilde{s}); \qquad \widetilde{y}(x,s)=\widetilde{A}(s)y_1(x,s)-A(s)y_2(x,s) \]

uniformly with respect to \(\widetilde{s}\) in every rectangle
\[ |\operatorname{Im}\widetilde{s}|<\varepsilon_1<\tfrac12\varepsilon,\qquad \alpha\le \operatorname{Re}\widetilde{s}\le \beta,\qquad 0\le \alpha<\beta. \]

Let \(K_b(x,t,\lambda)\) be the resolvent kernel of the auxiliary boundary-value problem (7), (8). If \(\lambda\) is not an eigenvalue of the operator \(L_{\theta(\lambda)}\), then, as \(b\to 0\),

\[ K_b(x,t,\lambda)=K(x,t,\lambda)+o(1) \tag{9} \]

uniformly with respect to \(x,t\) in every finite square \(0\le x,t\le c\), \(c>0\). We shall consider the kernel \(K_b(x,t,\lambda)\) on the curves \(C_{m,q}\) (\(m,q\) are natural numbers), defined as follows: \(C_{m,q}\) is a closed curve composed of two arcs \(C'_{m,q}\), \(C''_{m,q}\), consisting of points \(\lambda\) satisfying the relations

\[ C'_{m,q}:\quad \lambda=s^2,\qquad |s|^2=R_{m,q}^2+i\tau,\qquad -\varepsilon_1\le \tau\le \varepsilon_1,\qquad R_{m,q}=\left(m+\frac{1}{2q}\right)\frac{\tau}{\sqrt{m}}; \]

\[ C''_{m,q}:\quad \lambda=s^2,\qquad |s|^2=R_{m,q}+\varepsilon_1^2,\qquad \operatorname{Im}s>\varepsilon_1. \tag{10} \]

On the curves \(C_{m,q}\), \(|K_b(x,t,\lambda)|\le C/\sqrt{\lambda}\), and, consequently,

\[ \frac{1}{2\pi i}\oint_{C_{m,q}}\frac{K_b(x,t,\lambda)}{\lambda-\lambda_0}\,d\lambda \to 0 \quad\text{as } m\to\infty \tag{11} \]

uniformly with respect to \(q\).

Applying the residue theorem to this integral, passing to the limit as \(m\to\infty\), \(q\to\infty\), and taking into account the relations (9) and (10), we obtain

\[ K(x,t,\lambda) = \sum_{k=1}^{r} \frac{y_k(x)y_k(t)} {(\lambda_k-\lambda)\displaystyle\int_0^\infty [y_k(x)]^2\,dx} - \frac{1}{2\pi}\int_0^\infty \frac{\widetilde{y}(x,s)\widetilde{y}(t,s)} {(s^2-\lambda)A(s)\widetilde{A}(s)} \,ds. \tag{12} \]

Approximating an arbitrary function \(p(x)\) satisfying the condition \(x^2p(x)\in L(R^+)\) by functions \(p(x)\) satisfying condition (4), we arrive at the following result.

Theorem 2. Let \(x^2p(x)\in L(R^+)\), and let the functions \(A(s)\), \(\widetilde{A}(s)\) not vanish for \(s\ge 0\); let the eigenvalues of the operator \(L_{\theta(\lambda)}\) be simple poles of its resolvent, and let
\[ \theta(\lambda)=\frac{P(\lambda)}{Q(\lambda)},\qquad \lim_{s\to\infty}\frac{\theta(\lambda)}{is}\ne 1. \]
Then, for every point \(\lambda\) not belonging to the spectrum of the operator \(L_{\theta(\lambda)}\), equality (12) holds, in which the integral on the right-hand side converges absolutely and uniformly with respect to \(x,t\) in the domain \(0\le x,t<\infty\).

Denote by \(M\) the totality of all functions \(g(x)\in L(R^+)\) satisfying the conditions: 1) \(g'(x)\) exists and is absolutely continuous in every finite interval \([0,a]\), \(a>0\); 2) \(l(g)\in L(R^+)\); 3) \(g(0)=0\), \(g'(0)=0\). Multiplying both sides of (12) by the function \(f(x)=l(g)-\lambda g\) and integrating with respect to \(x\) from \(0\) to \(\infty\), we obtain:

Theorem 3. Under the conditions of Theorem 2, every function \(g(x)\in M\) can be represented in the form

\[ g(x)= \sum_{k=1}^{r} \frac{\alpha_k y_k(x)} {\displaystyle\int_0^\infty [y_k(t)]^2\,dt} - \frac{1}{2\pi}\int_0^\infty \frac{\alpha(s)\widetilde{y}(x,s)} {A(s)\widetilde{A}(s)} \,ds, \tag{13} \]

where

\[ a_k=\int_0^\infty g(t)y_k(t)\,dt,\qquad \alpha(s)=\int_0^\infty g(t)\widetilde y(t,s)\,dt, \]

and the integral in (13) converges absolutely and uniformly with respect to \(x\) in the interval \(0\le x<\infty\).

Theorem 4. Suppose the conditions of Theorem 3 are satisfied and

\[ g(x)\in M,\qquad h(x)\in L(R^+). \]

Then the following analogue of Parseval’s equality holds:

\[ \int_0^\infty g(x)h(x)\,dx = \sum_{k=1}^{r} \frac{\alpha_k\beta_k}{\displaystyle\int_0^\infty [y_k(t)]^2\,dt} - \frac{1}{2\pi}\int_0^\infty \frac{\alpha(s)\beta(s)}{A(s)\widetilde A(s)}\,ds, \tag{14} \]

where

\[ \alpha_k=\int_0^\infty g(t)y_k(t)\,dt,\qquad \alpha(s)=\int_0^\infty g(t)\widetilde y(t,s)\,dt, \]

\[ \beta_k=\int_0^\infty h(t)y_k(t)\,dt,\qquad \beta(s)=\int_0^\infty h(t)\widetilde y(t,s)\,dt. \]

Here the integrals on the left- and right-hand sides of (14) converge absolutely.

Suppose that the function \(A(s)\) has multiple zeros. Let \(\lambda_1,\lambda_2,\ldots,\lambda_r\) be all the eigenvalues of the operator \(L_{\theta(\lambda)}\), and let \(s_1,s_2,\ldots,s_r\) be the zeros of the function

\[ A(s)=y_1'(0,s)-\theta(\lambda)y_1(0,s) \]

of multiplicities, respectively, \(m_1,m_2,\ldots,m_r\). Denote by \(y_{0,k}(x), y_{1,k}(x),\ldots, y_{m_k-1,k}(x)\) chains of eigenfunctions and associated functions corresponding to the eigenvalues \(\lambda_k\), \(k=1,2,\ldots,r\). Then the following holds.

Theorem 5. Suppose \(x^2p(x)\in L(R^+)\), the functions \(A(s)\), \(\widetilde A(s)\) are nonzero on the half-axis \(s\ge0\), and \(\theta(\lambda)\) is the same as in Theorem 2. Then, for any \(\lambda\in G\) not belonging to the spectrum of the operator \(\theta(\lambda)\),

\[ K(x,t,\lambda) = \sum_{k=1}^{r}\sum_{q=1}^{m_k} \frac{\alpha_{q,k}(x,t)}{(\lambda-\lambda_k)^q} - \frac{1}{2\pi} \int_0^\infty \frac{\widetilde y(x,s)\widetilde y(t,s)} {(s^2-\lambda)A(s)\widetilde A(s)}\,ds, \tag{15} \]

where

\[ \alpha_{q,k}(x,t) = \sum_{j=0}^{m_k-q} C_{j+q,k} \sum_{\nu=0}^{j} y_{\nu,k}(x)y_{j-\nu,k}(t), \]

\(C_{q,k}\) are constants, which are determined uniquely, and the integral on the right-hand side of (15) converges absolutely and uniformly with respect to the set of variables \(x,t\) in the domain \(0\le x,t<\infty\).

Let us note that analogous results are also obtained for a certain class of operators of higher order.

In conclusion, the author expresses gratitude to Prof. M. A. Naimark for posing the problem and for valuable advice.

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Moscow State University
named after M. V. Lomonosov

Received
2 VII 1963

CITED LITERATURE

\(^{1}\) M. A. Naimark, Tr. Mosk. Mat. Obshch., 3, 181 (1954).
\(^{2}\) A. V. Shtraus, Izv. AN SSSR, ser. matem., 20, 783 (1956).