Sturm–Liouville Operators

Abstract

Chapter 15 deals with the Hilbert space theory of Sturm–Louville operators $-\frac{d^{2}}{dx^{2}}+ q(x)$ on intervals. First, we study the case of regular end points. Then we develop the fundamental results of H. Weyl’s classical limit point–limit circle theory. Some general limit point and limit circle criteria are proved. Next, we define boundary triplets in the various cases (regular end points, limit point case, limit circle case), determine their gamma fields and Weyl functions, and describe all self-adjoint extensions. In the final section, we derive formulas for the resolvents of some self-adjoint extensions.