Determination of a Differential Equation by Two of its Spectra

CONTENTSIntroductionChapter 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 K(x, t) § 3. The inverse problem. Solubility of the integral equation for the kernel K(x, t) § 4. Derivation of the differential equation § 5. Parseval' s equation § 6. The classical Sturm-Liouville problemChapter 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 α_n § 3. The inverse Sturm-Liouville problemChapter 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 α_n(h_1) in terms of the spectra § 3. On a class of potentials § 4. Solution of the inverse problem for the class Ω_1/4Appendix I. Proof of V. A. Ambartsumyan's theoremAppendix II. Derivation of the asymptotic formulae (1.6.6) and (1.6.7)Remarks and notes on the referencesReferences