Self-adjoint Sturm-Liouville problems with an infinite number of boundary conditions

Abstract

We study Sturm–Liouville (SL) problems on an infinite number of intervals, adjacent endpoints are linked by means of boundary conditions, and characterize the conditions which determine self-adjoint operators in a Hilbert space which is the direct sum of the spaces for each interval. These conditions can be regular or singular, separated or coupled. Furthermore, the inner products of the summand spaces may be multiples of the usual inner products with different spaces having different multiples. We also extend the GKN Theorem to cover the infinite number of intervals theory with modified inner products and discuss the connection between our characterization and the classical one with the usual inner products. Our results include the finite number of intervals case.