Some remarks on a differential expression with an indefinite weight function

Abstract

This chapter discusses the differential expression with an indefinite weight function. The chapter focuses on with the generation of a self-adjoint operator, in an integrable-square Hilbert function space (H may be shown to be a Hilbert space), from a boundary value problem associated with the differential equation on a half-line. A classical boundary value problem (B α ) may be determined by requiring solutions of the differential equation to satisfy the boundary condition at 0, and the boundary condition at ∞ —that is, the solution should belong to H. In the usual terminology, X is an eigenvaue of the problem B α non-trivial solution of the differential equation satisfying the boundary conditions this solution is then an eigenfunction associated with the eigenvalue. When α ∈ P, with some additional conditions on the coefficients p, q and r, the problem B α can be characterized by means of a uniquely determined unbounded self-adjoint operator T α in the Hilbert function space H 1 α . The eigenvalues and eigenfunctions of B α are equivalent to the eigenvalues and eigenvectors of T α in H 1 α . These results, however, are not valid when α ∈ N.