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In this paper, we study 1-d random Schr\"odinger operators on a finiteinterval with Dirichlet boundary conditions. We are interested in theapproximation of the ground state energy using the minimum of the effectivepotential. For the 1-d continuous Anderson Bernoulli model, we show that theratio of the ground state energy and the minimum of the effective potentialapproaches $\frac{\pi^2}{8}$ 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 groundstate energy and also supporting approximations for the excited statesenergies.

Approximating the ground state eigenvalue via the effective potential