DETERMINATION OF A DIFFERENTIAL EQUATION BY TWO OF ITS SPECTRA

Abstract

CONTENTS Introduction Chapter I. Determination of a differential equation by its spectral function § 1. On the spectral function of a differential operator § 2. Derivation of the linear integral equation for the kernel § 3. The inverse problem. Solubility of the integral equation for the kernel § 4. Derivation of the differential equation § 5. Parseval' s equation § 6. The classical Sturm-Liouville problem Chapter II. Determination of a regular Sturm-Liouville equation by two of its spectra § 1. Expression for the normalizing constants in terms of the spectra § 2. Asymptotic formulae for the numbers § 3. The inverse Sturm-Liouville problem Chapter III. Determination of a singular Sturm-Liouville equation by two of its spectra § 1. Formulae for the difference between the traces of two Sturm-Liouville operators under different boundary conditions at the origin § 2. Expression for the numbers in terms of the spectra § 3. On a class of potentials § 4. Solution of the inverse problem for the class Appendix I. Proof of V. A. Ambartsumyan's theorem Appendix II. Derivation of the asymptotic formulae (1.6.6) and (1.6.7) Remarks and notes on the references References