Quantum Systems in Bernoulli Potentials

Abstract

Quantum mechanics is a theory developed to explain both particle and wave-like properties of small matter such as light and electrons. The consequences of the theory can be counter-intuitive but lead to mathematical and physical theory rich in fascinating phenomena and challenging questions. This dissertation investigates the nature of quantum systems in Bernoulli distributed random potentials for systems on the one dimensional lattice {0, 1, ..., L, L+1} ⊂ Z in the large system limit L → ∞. For single particle systems, the behavior of the low energy states is shown to be approximated by systems where positive potential is replaced by infinite potential. The approximate shape of these states is described, the asymptotics of their eigenvalues are calculated in the large system limit L → ∞, and a Lifschitz tail estimate on the sparsity of low energy states is proven. For interacting multi-particle systems, a Lieb-Liniger model with Bernoulli distributed potential is studied in the Gross-Pitaevskii approximation. First, to investigate localization in these settings, a general inequality is proven to bound from below the support of the mean-field. The bound depends on the per particle energy, number of particles, and interaction strength. Then, the ground state for the one-dimensional lattice with Bernoulli potential is studied in the large system limit. Specifically, the case where the product of interaction strength and particle density is near zero is considered to investigate whether localization can be recovered.