Basic Sturm–Liouville problems

Abstract

This paper is devoted to studying a q-analogue of Sturm–Liouville eigenvalue problems. We formulate a self-adjoint q-difference operator in a Hilbert space. Some of the properties of the eigenvalues and the eigenfunctions are discussed. Green's function is constructed and the problem in question is inverted into a q-type Fredholm integral operator with a symmetric kernel. The set of eigenfunctions is shown to be a complete orthogonal set in the Hilbert space. Examples involving basic trigonometric functions are involved.