SOME PROBLEMS IN THE THEORY OF A STURM-LIOUVILLE EQUATION

Abstract

CONTENTS Introduction Chapter One. The solution of the Cauchy problem for the one dimensional wave equation § 1. The application of the method of successive approximations § 2. Reduction to the Goursat problem § 3. The solution of the mixed problem on the half line § 4. The solution of a mixed problem on a finite interval Chapter Two. Eigenfunction expansions § 1. Derivation of auxiliary formulae § 2. Preliminary estimate of the spectral function. The case of the whole line § 3. The asymptotic behaviour of the spectral function. The case of the whole line § 4. The asymptotic behaviour of the spectral function. The case of the half line § 5. Riesz summability of the spectral function § 6. Proof of the uniform convergence theorem § 7. Convergence and summability of eigenfunction expansions Chapter Three. Differentiation of eigenfunction expansions § 1. A preliminary estimate of the derivatives of the spectral function § 2. The asymptotic behaviour of the derivatives of the spectral function § 3. Uniform summability of differentiated eigenfunction expansions § 4. The summability of differentiated expansions in an ordinary and generalised Fourier integral § 5. Convergence of the differentiated eigenfunction expansion § 6. Establishing the Fourier method for the one-dimensional wave equation Chapter Four. Eigenfunction expansions for unboundedly increasing potential § 1. Derivation of auxiliary identities § 2. Some estimates for Green's function § 3. The asymptotic behaviour of the trace of Green's function § 4. The asymptotic distribution of the eigenvalues § 5. Summability of expansions and of differentiated expansions in eigenfunctions § 6. Convergence of expansions and differentiated expansions in eigenfunctions § 7. An example Appendix Notes and references to the literature Literature cited