V. N. FUNTAKOV

EXPANSION IN EIGENFUNCTIONS OF NON-SELF-ADJOINT SINGULAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

(Presented by Academician A. N. Kolmogorov on 12 I 1962)

Let us consider the differential equation

\[ -y''+q(x)y=\lambda^2 y\quad(-\infty<x<\infty), \tag{1} \]

where \(q(x)\) is an arbitrary complex-valued function summable on each finite interval of the axis \(-\infty<x<\infty\), and let us denote by \(\omega_1(\lambda,x)\), \(\omega_2(\lambda,x)\) the solution of this differential equation satisfying the conditions

\[ \omega_1(\lambda,0)=1,\quad \omega_1'(\lambda,0)=0; \]

\[ \omega_2(\lambda,0)=0,\quad \omega_2'(\lambda,0)=1. \tag{2} \]

In the work of V. A. Marchenko \((^3)\), equation (1) was considered on the half-axis \([0,\infty)\) with the boundary condition

\[ y'(0)-hy(0)=0, \tag{3} \]

where \(h\) is an arbitrary complex number. The following result was obtained:

Theorem. Denote by \(Z\) the linear topological space of even entire functions of exponential type, summable on the real axis; by \(T(Z)\) the space conjugate to it. Then to every boundary-value problem (1)—(3) there corresponds a certain generalized function \(R\in T(Z)\) such that

\[ \int_0^\infty f(x)g(x)\,dx=(R,\ E_f(\lambda)E_g(\lambda)), \]

where \(f(x)\) and \(g(x)\) are arbitrary finite functions belonging to \(L^2(0,\infty)\),

\[ E_f(\lambda)=\int_0^\infty f(x)\omega(\lambda,x)\,dx, \]

\((\omega(\lambda,x)\) is the solution of equation (1) satisfying condition (3)).

Further, in the work \((^3)\) necessary and sufficient conditions for the existence of a spectral function \(R\in T(Z)\) were established, and it was proved that to every function \(R\in T(Z)\) satisfying these conditions there corresponds a certain boundary-value problem (1)—(3).

In the present paper the results obtained by V. A. Marchenko are transferred to equation (1), given on the whole interval \((-\infty,\infty)\). The method used in the work is the one first applied by V. A. Marchenko.

Denote by \(K\) the space of infinitely differentiable finite functions \(f(x)\). Consider the functions

\[ C_f(\lambda)=\int_{-\infty}^{\infty} f(x)\cos\lambda x\,dx;\qquad S_f(\lambda)=\int_{-\infty}^{\infty} f(x)\frac{\sin\lambda x}{\lambda}\,dx. \]

As is known (see (2), p. 194), the functions \(C_f(\lambda)\) form a topological space \(Z\) of even entire functions of finite degree \(\psi(\lambda)\), satisfying the inequalities

\[ |\lambda|^q |\psi(\lambda)| \leq C_q e^{a|\tau|}, \qquad q=0,1,\ldots,\qquad \tau=\operatorname{Im}\lambda,\qquad 0<a<\infty . \]

It is easy to show that the functions \(S_f(\lambda)\) also belong to \(Z\). We define generalized functions on \(Z\) as linear continuous functionals \(R[F(\lambda)]\), putting

\[ R[F(\lambda)] = (R,F(\lambda)). \]

The totality of all generalized functions defined in this way will be denoted by \(T(Z)\). A sequence \(R_n\in T(Z)\) converges to \(R\in T(Z)\) if

\[ \lim_{n\to\infty} (R_n,F(\lambda)) = (R,F(\lambda)) \]

for all basic functions \(F(\lambda)\in Z\).

Let \(A(x)\) be an arbitrary locally summable function. Put

\[ A_n(x)= \begin{cases} A(x), & |x|\leq n,\\ 0, & |x|>n, \end{cases} \]

and denote

\[ C_{A_n}(\lambda)=\int_{-\infty}^{\infty} A_n(x)\cos\lambda x\,dx, \qquad S_{A_n}(\lambda)=\int_{-\infty}^{\infty} A_n(x)\frac{\sin\lambda x}{\lambda}\,dx . \]

We define the \(C\)- and \(S\)-Fourier transforms of the function \(A(x)\) by putting

\[ C_A=\lim_{n\to\infty} C_{A_n},\qquad S_A=\lim_{n\to\infty} S_{A_n}, \]

where convergence is understood in the sense of generalized functions.

A fundamental role in proving the main results is played by

Theorem (see (1)). The solutions \(\omega_1(\lambda,x)\), \(\omega_2(\lambda,x)\) of equation (1) can be expressed through \(\cos\lambda x\) and \(\dfrac{\sin\lambda x}{\lambda}\), respectively, with the aid of transformation operators in the form

\[ \omega_1(\lambda,x)=\cos\lambda x+\int_{-|x|}^{|x|} K(x,t)\cos\lambda t\,dt, \]

\[ \omega_2(\lambda,x)=\frac{\sin\lambda x}{\lambda} +\int_{-|x|}^{|x|} K(x,t)\frac{\sin\lambda x}{\lambda}\,dt; \tag{4} \]

\[ \cos\lambda x=\omega_1(\lambda,x)-\int_{-|x|}^{|x|} H(x,t)\omega_1(\lambda,t)\,dt, \]

\[ \frac{\sin\lambda x}{\lambda} =\omega_2(\lambda,x)-\int_{-|x|}^{|x|} H(x,t)\omega_2(\lambda,t)\,dt, \tag{5} \]

where the kernels \(K(x,t)\) and \(H(x,t)\) are absolutely continuous in both variables.

We define the \(\omega\)-Fourier transforms of a function \(f(x)\in K\) by the equalities

\[ E_f(\lambda)=\int_{-\infty}^{\infty} f(x)\omega_1(\lambda,x)\,dx, \qquad G_f(\lambda)=\int_{-\infty}^{\infty} f(x)\omega_2(\lambda,x)\,dx. \]

With the aid of the preceding theorem it is easy to show that the set of \(\omega\)-Fourier transforms of all functions from \(K\) coincides with \(Z\). We can now formulate the principal results obtained; we shall not dwell on their proof.

Theorem. To each differential equation (1) with solutions satisfying condition (2) there corresponds a spectral matrix-function of the second order \(\|R_{ik}\|_{i,k=1,2}\), \(R_{ik}\in T(Z)\), such that

\[ \int_{-\infty}^{\infty} f(x)g(x)\,dx = \bigl(R_{11}, E_f(\lambda)E_g(\lambda)\bigr) + \bigl(R_{12}, E_f(\lambda)G_g(\lambda)\bigr) + \]

\[ + \bigl(R_{21}, G_f(\lambda)E_g(\lambda)\bigr) + \bigl(R_{22}, G_f(\lambda)G_g(\lambda)\bigr), \]

where \(f(x),\,g(x)\in K\); \(E_f(\lambda),\,G_f(\lambda),\,E_g(\lambda),\,G_g(\lambda)\) are their \(\omega\)-Fourier transforms. The elements of the spectral matrix-function \(R_{ik}\) are connected with the kernel \(H(x,t)\) of the transformation operator (5) by the formulas

\[ R_{11}=\frac{1}{2\pi}(1-C_H),\qquad R_{12}=R_{21}=-\frac{\lambda^2}{2\pi}S_H,\qquad R_{22}=\frac{\lambda^2}{2\pi}(1+C_H), \]

where \(C_H\) and \(S_H\) are the \(C\)- and \(S\)-Fourier transforms of the locally summable function \(H(x,0)\).

Corollary. If the function \(f(x)\in K\), then the formula

\[ f(x) = \bigl(R_{11}, E_f(\lambda)\omega_1(\lambda,x)\bigr) + \bigl(R_{12}, E_f(\lambda)\omega_2(\lambda,x)\bigr) + \]

\[ + \bigl(R_{21}, G_f(\lambda)\omega_1(\lambda,y)\bigr) + \bigl(R_{22}, G_f(\lambda)\omega_2(\lambda,x)\bigr) \]

holds.

As is known ([2], p. 180), for every finite continuous function \(\theta(t)\) there exists a sequence of functions \(\theta_\varepsilon(t)\in K\) with the same interval of finiteness \([-a,a]\) such that
\[ \lim_{\varepsilon\to 0}\theta_\varepsilon(t)=\theta(t) \]
uniformly in \(t\). Let \(F(\lambda)\) and \(F_\varepsilon(\lambda)\) be their Fourier transforms. Then it is easy to show that
\[ \lim_{\varepsilon\to 0}F_\varepsilon(\lambda)=F(\lambda) \]
uniformly in \(\lambda\). We extend the functional \(R\in T(Z)\) to the set of all such functions \(F(\lambda)\), putting

\[ (R,F(\lambda))=\lim_{\varepsilon\to 0}(R,F_\varepsilon(\lambda)). \]

Theorem. In order that a generalized matrix-function

\[ \|R_{ik}\|_{i,k=1,2},\qquad R_{ik}\in T(Z),\qquad R_{12}=R_{21},\qquad R_{22}=\frac{\lambda^2}{\pi}(1-\pi R_{11}) \]

be the spectral matrix-function of some problem (1)—(2), it is necessary and sufficient that the following conditions be satisfied:

1) In \(Z\) there do not exist nonzero functions \(E_f(\lambda),\,G_f(\lambda)\) satisfying the equality

\[ \bigl(R_{11}, E_f(\lambda)E_y(\lambda)\bigr) + \bigl(R_{12}, E_f(\lambda)G_y(\lambda)\bigr) + \]

\[ + \bigl(R_{21}, G_f(\lambda)E_y(\lambda)\bigr) + \bigl(R_{22}, G_f(\lambda)G_y(\lambda)\bigr) =0 \]

for all \(E_y(\lambda),\,G_y(\lambda)\in Z\).

2) The functions

\[ \Phi_1(x)=\left(R_{11},\,2\frac{1-\cos \lambda x}{\lambda^2}\right) \qquad(-\infty<\lambda<\infty); \]

\[ \Phi_2(x)=\left(R_{21},\,-\frac{2x\sin \lambda x}{\lambda^3}-\frac{4(1-\cos \lambda x)}{\lambda^4}\right) \qquad(-\infty<\lambda<\infty) \]

have two absolutely continuous derivatives and

\[ \lim_{x\to 0}\frac{\Phi_2''(x)}{x}=0,\qquad \Phi_1''(0)=0. \]

Here the function \(q(x)\) in equation (1) has as many absolutely continuous derivatives as \(\Phi_1''(x)\) and \(\Phi_2''(x)\) have, provided that

\[ \lim_{x\to 0}\frac{\Phi_2^{(k+2)}(x)}{x}\ne \infty,\quad k=1,2,\ldots . \]

The author considers it his duty to express his deep gratitude to his scientific adviser M. A. Naimark, and also thanks V. B. Lidskii for a number of valuable comments.

Moscow Institute of Physics and Technology

Received
19 XII 1961

REFERENCES

¹ A. Sh. Blokh, Dokl. Akad. Nauk SSSR, 92, No. 2 (1953).
² I. M. Gelfand, G. E. Shilov, Generalized Functions and Operations on Them, 1959.
³ V. A. Marchenko, Matem. sbornik, 52 (94), 2, 739 (1960).