INVERSE PROBLEM FOR INTERIOR SPECTRAL DATA OF THE STURM-LIOUVILLE OPERATOR

KIYOSHI MOCHIZUKI AND IGOR TROOSHIN

We consider the Sturm-Liouville operator L generated by the differential expression l(y) = −y′′ + q(x)y (0 ≤ x ≤ 1) subject to the boundary conditions y(0) cos α + y′(0) sin α = 0, 0 ≤ α < π, and y(1) cos β + y′(1) sin β = 0, 0 ≤ β < π, where q(x) ∈ L(0, 1) is a real-valued function.

As is well known, the operator L has a discrete spectrum consisting of simple eigenvalues λ_{n}, n = 0, 1, . . .. We denote by y_{n}(x), n = 0, 1, … the corresponding eigenfunctions. The problem of reconstructing a potential q(x) from the spectra of its boundary value problem is known as the inverse Sturm-Liouville problem. It was first posed by Ambarzumian (1929). Later Borg (1946) showed that one spectra does not define the potential uniquely , to prove the uniqueness of the potential q(x), one needs additional spectral data. Borg proved that the spectra of two boundary value problems for an operator with different boundary conditions at one end (and identical conditions repeated at the other end) uniquely determined the potential q(x), i.e. he took the second spectrum as an additional spectral data. Later Marchenko (1950) substituted the second spectrum by normalising coefficients α_{n} = ∫_{0}^{1} y_{n}^{2}(x)dx, n = 0, 1, 2, … and proved that the eigenvalues λ_{n}, n = 0, 1, 2, … and normalising coefficients α_{n}, n = 0, 1, 2, … uniquely determined the potential q(x).

The aim of our present work is to investigate a possibility to recover potential from the known eigenvalues and some information on eigenfunctions in the internal point b ∈ (0, 1). We proved, for example, the following theorem.

Let us introduce a second Sturm-Liouville operator L̃ generated

by differential expression with a real-valued potential q̃(x) ∈ L(0, 1): l̃(y) = −y′′ + q̃(x)y (0 ≤ x ≤ 1) subject to the same boundary conditions . We denote by λ̃_{n}, n = 0, 1, … and ỹ_{n}(x), n = 0, 1, … eigenvalues and corresponding eigenfunctions of operator L̃.

Theorem If for any n = 0, 1, …, λ_{n} = λ̃_{n}, \frac{y′_{n}( \frac{1}{2} )}{y_{n}( \frac{1}{2} )} = \frac{ỹ′_{n}( \frac{1}{2} )}{ỹ_{n}( \frac{1}{2} )}, then q(x) = q̃(x) a.e. on the [0, 1].

In the case b ≠ 1/2, the uniqueness of q(x) can be proven if we require the knowledge of part of the second spectrum.

DEPARTMENT OF MATHEMATICS, TOKYO METROPOLITAN UNIVERSITY, HA-CHIOJI, TOKYO 192-0397, JAPAN.E-MAIL: MOCHIZUK@COMP.METRO-U.AC.JP

PRECISION MECHANICS AND CONTROL INSTITUTE OF RUSSIAN ACADEMY OF SCI-ENCES, 24, RABOCHAYA, SARATOV,410028, RUSSIA