Approximating the ground state eigenvalue via the effective potential

TL;DR

This paper investigates how well the minimum of the effective potential approximates the ground state energy in 1D random Schrödinger operators, revealing a specific asymptotic ratio and supporting findings with numerical experiments.

Contribution

It establishes the asymptotic ratio of ground state energy to effective potential minimum in the 1D Anderson Bernoulli model and explores various approximation methods.

Findings

Ratio of ground state energy to effective potential minimum approaches π²/8 as domain size increases

Numerical experiments confirm theoretical predictions for ground and excited state energies

Various approximation techniques are analyzed and supported by numerical results

Abstract

In this paper, we study 1-d random Schr\"odinger 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 1-d continuous Anderson Bernoulli model, we show that the ratio of the ground state energy and the minimum of the effective potential approaches $\frac{\pi^2}{8}$ 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.