The discrete spectrum of Schrödinger operators with $\delta$-type conditions on regular metric trees

Abstract

This paper deals with the spectral properties of the self-adjoint Schrödinger operators $L_Q=-D^2+Q$ with $\delta$-type conditions on regular metric trees. Firstly, we prove that the operator ${\mathcal{L}}_{\delta ,Q}$ given in this paper is self-adjoint if it is lower semibounded. Then a necessary and sufficient condition is given for the spectrum of the operator ${\mathcal{L}}_{\delta ,Q}$ to be discrete. The condition is an analog of Molchanov's discreteness criteria. Finally, using the theory of deficiency indices we get the necessary and sufficient condition which ensures the spectra of the self-adjoint Schrödinger operators with general boundary conditions to be discrete.