Search

Let H be a separable Hilbert space and let H1 be a Hilbert space whose elements are vector functions f(x) (0≤x

We prove the existence of a transformation operator with a condition at infinity that sends a solution of the matrix equation−y″ + My=λ2y (M is a constant Hermitian matrix) into a solution of the matrix equation−y″+Q(x)y+My=λ2y (the matrix function Q(x) is continuously differentiable for 0 ≤ x

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 § 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 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 § 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 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 theoremAppendix II. Derivation of the asymptotic formulae (1.6.6) and (1.6.7)Remarks and notes on the referencesReferences