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.

翻译：在本文中,我们研究的是1D随机Schr\'odinger操作员, 与迪里切特边界条件有一定的间隔。 我们感兴趣的是使用最低有效潜能接近地面状态能源。 对于1D连续的Anderson Bernoulli模型, 我们显示地面状态能源的比例和最小有效潜在方法$\frac\pi ⁇ 2\ ⁇ 8}作为域大小的无限接近点。 此外, 我们将讨论不同情况下与比率的不同近点。 将进行数字实验, 支持我们地面状态能源的主要结果, 同时支持兴奋状态能源的近点。