An Inverse Sturm‐Liouville Problem by Three Spectra

Abstract It is well known that the equation of small transverse vibrations of a smooth string may be reduced by means of Liouville transform [1] to the Sturm‐Liouville equation. Then the inverse problem, i. e. determination of the potential, may be solved by given spectra of two boundary problems [2, 3, 4]. We consider here the inverse Sturm‐Liouville problem with the following given data: the spectrum of a boundary problem (I) on the interval [0, a] and the spectra of two‐boundary problems (II and III) on the intervals [0, 1/2a] and [1/2a, a], correspondingly. The potentials of the problems I and II (I and III) coincide on the interval [0, 1/2a] (on the interval [1/2a, a]). The conditions are found which are sufficient for three sequences of real numbers to be the spectra of the three boundary problems generated by real potential from L2 (0, a). These conditions are not much different from the necessary ones. The physical meaning of the problems is as follows. The first sequence is the spectrum of a smooth string with fixed ends. The two other sequences are the spectra of the same string clamped at the point of equilibrium.