Generic Simplicity of the Spectrum of a Schrödinger-type Operator on a Riemannian Manifold

Abstract

Generic simplicity of spectrum of the Schrödinger-type operator, H = \(\Delta\) + V, is investigated in this study. Here, \(\Delta\) is the standard Laplace operator on n-dimensional unit torus and V is the perturbation potential. On the n-dimensional torus, we used Rayleigh- Schrödinger perturbation theory to analyse the splitting behaviour of the spectrum due to infinitesimal perturbation. We proved the existence of a perturbation potential V which guarantees the simplicity of the spectrum of the Schrödinger-type operator \(\Delta\)+V on the n-torus at first order.