Spectral analysis of the sixth-order Krall differential expression

Abstract

In this paper, we construct a self-adjoint operator $$\widehat{T}_{A,B}$$ generated by the sixth-order Krall differential expression in the extended Hilbert space $$L^{2}(-1,1)\oplus \mathbb {C}^{2}.$$ To obtain $$\widehat{T} _{A,B},$$ we apply a new general theory, the so-called GKN-EM theory, developed recently by Littlejohn and Wellman that extends the classical Glazman–Krein–Naimark theory using a complex symplectic geometric approach developed by Everitt and Markus. This work extends earlier studies of the Krall expression by both Littlejohn and Loveland.