Inverse problems for second order integral and integro-differential operators

Abstract

Inverse spectral problems are studied for second order integral and integro-differential operators. Uniqueness results are obtained, and algorithms for the solutions are provided along with necessary and sufficient conditions for the solvability of these nonlinear inverse problems.

Appendix

Let the function $y(x)=S(x,\lambda )$ be the solution of the equation

$$\begin{aligned} y''(x)-q(x)y(x)-\int _0^x B(x,t)y(t)\,dt=\lambda y(x),\quad x\in [0,\pi ], \quad \lambda =-\rho ^2, \end{aligned}$$

under the initial conditions $S(0,\lambda )=0,\; S'(0,\lambda )=1$. Then $S(x,\lambda )$ is the solution of the integral equation

$$\begin{aligned} S(x,\lambda )= & {} \frac{\sin \rho x}{\rho }+\int _0^x \frac{\sin \rho (x-\tau )}{\rho } \nonumber \\&\times \Big (q(\tau )S(\tau ,\lambda )+\int _0^\tau B(\tau ,s)S(s,\lambda )\,ds\Big )\,d\tau . \end{aligned}$$

(4.1)

Since

$$\begin{aligned} \frac{\sin \rho (x-\tau )}{\rho }=\int _\tau ^x \cos \rho (t-\tau )\,dt, \end{aligned}$$

then (4.1) can be written as follows:

$$\begin{aligned} S(x,\lambda )= & {} \frac{\sin \rho x}{\rho } \\&+\int _0^x dt \int _0^t\Big (q(\tau )S(\tau ,\lambda )+\int _0^\tau B(\tau ,s)S(s,\lambda )\,ds\Big ) \cos \rho (t-\tau )\,d\tau . \end{aligned}$$

Solving this equation by the method of successive approximations, we get

$$\begin{aligned} S(x,\lambda )= & {} \sum _{n=0}^\infty S_n(x,\lambda ),\quad S_0(x,\lambda )=\frac{\sin \rho x}{\rho }, \\ S_{n+1}(x,\lambda )= & {} \int _0^x dt \int _0^t \Big (q(\tau )S_n(\tau ,\lambda )+\int _0^\tau B(\tau ,s)S_n(s,\lambda )\,ds\Big ) \\&\times \cos \rho (t-\tau )\,d\tau ,\quad n\ge 0, \end{aligned}$$

and consequently,

$$\begin{aligned} S_n(x,\lambda )=\int _0^x K_n(x,\xi )\frac{\sin \rho \xi }{\rho }\,d\xi ,\quad n\ge 1, \end{aligned}$$

where

$$\begin{aligned} K_1(x,\xi )= & {} \frac{1}{2}\int _{0}^{\xi } q(\tau )\,d\tau +\frac{1}{4}\int _{\xi }^{x}\left( q\Big (\frac{t+\xi }{2}\right) -q\Big (\frac{t-\xi }{2}\Big )\Big )\,dt \\&+\,\frac{1}{2}\int _{\xi }^{x}dt \int _{t-\xi }^{t} B(\tau ,\xi -t+\tau )\,d\tau \\&+\,\frac{1}{2}\int _{\xi }^{x}dt \int _{(t+\xi )/2}^{t} B(\tau ,\xi +t-\tau )\,d\tau \\&-\,\frac{1}{2}\int _{\xi }^{x}dt \int _{(t-\xi )/2}^{t-\xi } B(\tau ,t-\tau -\xi )\,d\tau \\ K_{n+1}(x,\xi )= & {} \frac{1}{2}\int _{\xi }^{x}\left( \int _{t-\xi }^{t} q(\tau )K_n(\tau ,\xi +\tau -t)\,d\tau \right. \\&+\,\int _{(t+\xi )/2}^{t} q(\tau )K_n(\tau ,\xi -\tau +t)\,d\tau \\&-\,\int _{(t-\xi )/2}^{t-\xi } q(\tau )K_n(\tau ,-\,\xi -\tau +t)\,d\tau \\&+\,\int _{t-\xi }^{t} d\tau \int _{\xi -t+\tau }^{\tau } B(\tau ,s) K_n(s,\xi +\tau -t)\,ds \\&+\,\int _{(t+\xi )/2}^{t} d\tau \int _{\xi +t-\tau }^{\tau } B(\tau ,s)K_n(s,\xi -\tau +t)\,ds \\&-\,\left. \int _{(t-\xi )/2}^{t-\xi } d\tau \int _{-\xi +t-\tau }^{\tau } B(\tau ,s) K_n(s,-\,\xi -\tau +t)\,ds\right) \,dt. \end{aligned}$$

This yields

$$\begin{aligned} S(x,\lambda )= \frac{\sin \rho x}{\rho }+\int _0^x K(x,\xi )\frac{\sin \rho \xi }{\rho }\,d\xi , \end{aligned}$$

where

$$\begin{aligned} K(x,\xi )=\sum _{n=1}^\infty K_n(x,\xi ),\quad K(x,0)=0, \end{aligned}$$

and the series converges absolutely and uniformly for $0\le \xi \le x\le \pi$.