Approximating the ground state eigenvalue via the effective potential

Abstract

Abstract In this paper, we study 1D random Schrödinger operators on a finite interval with Dirichlet boundary conditions. We are interested in the approximation of the ground state energy using the minimum of the effective potential. For the 1D continuous Anderson Bernoulli model, we show that the ratio of the ground state energy and the minimum of the effective potential approaches π 2 8 as the domain size approaches infinity. Besides, we will discuss various approximations to the ratio in different situations. There will be numerical experiments supporting our main results for the ground state energy and also supporting approximations for the excited states energies.