ON RECONSTRUCTION OF THE POTENTIAL IN THE STURM-LIOUVILLE EQUATION FROM INCOMPLETE SPECTRAL DATA

Abstract

During past few years methods were developed to reconstruct negative potentials in the Sturm-Liouville equation through characteristics of a discrete spectrum. Several numerical tests clarify limitations of these methods. The tests show: (1) only the negative part of a potential having negative and positive parts can be reconstructed through characteristics of the negative spectrum of the Sturm-Liouville equation (the potential is multiplied by a suitable factor $\omega$); (2) spectral data for eigenvalues not larger than $-B \omega^2,$ $B>0$, can be used to reconstruct the potential where it is less than $-B \omega^2$; (3) discontinuities of the potential result in deterioration of approximations with the growth of $\omega$.

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Computational Seismology, Vol. 3.