Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential

Abstract

In this paper, we show the that the ground state energy of the one dimensional Discrete Random Schroedinger Operator with Bernoulli Potential is controlled asymptotically as the system size N goes to infinity by the random variable \ell_N, the length the longest consecutive sequence of sites on the lattice with potential equal to zero. Specifically, we will show that for almost every realization of the potential the ground state energy behaves asymptotically as $\frac{{\pi }^2}{{\ell }_N+1{)}^2}$ in the sense that the ratio of the quantities goes to one.

References in corpus (4)

• Direct observation of Anderson localization of matter-waves in a controlled disorder
• Three-Dimensional Anderson Localization of Ultracold Matter
• An Invitation to Random Schroedinger operators
• Glass to superfluid transition in dirty bosons on a lattice

Cited by in corpus (6)

• From Extreme Values of I.I.D. Random Fields to Extreme Eigenvalues of Finite-volume Anderson Hamiltonian
• Approximating the ground state eigenvalue via the effective potential
• Lifschitz Tails for Random Schrödinger Operator in Bernoulli Distributed Potentials
• Ground State Energy of Mean-field Model of Interacting Bosons in Bernoulli Potential
• Full distribution of the ground-state energy of potentials with weak disorder
• Landscape approximation of the ground state eigenvalue for graphs and random hopping models