Boundary conditions associated with the general left-definite theory for differential operators

Abstract

In the early 2000s, Littlejohn and Wellman developed so-called nth left-definite theory. Namely, they fully determined the ‘left-definite domains’ and spectral properties of powers of self-adjoint Sturm–Liouville operators associated with classical orthogonal polynomials. We study how these left-definite domains relate with explicit classical Glazman–Krein–Naimark (GKN) boundary conditions.When n is small, we significantly simplify previously challenging analysis by introducing an explicit method for checking whether a given set of functions yields GKN conditions. This reduces to computing the rank of a relatively small matrix. We include explicit computations for n=2,…,5. Further, for arbitrary powers n of Sturm–Liouville operators with a complete system of orthogonal eigenfunctions, we show that these left-definite domains are given by GKN boundary conditions involving some of the polynomial eigenfunctions. We also study and extend a conjecture by Littlejohn–Wicks regarding the equality of four different formulations for these domains.