Constraint Minimization Problem of the Nonlinear Schr\"{o}dinger Equation with the Anderson Hamiltonian

TL;DR

This paper investigates the minimization problem of the nonlinear Schrödinger equation with a white noise potential, establishing existence, regularity, and tail estimates for the principal eigenvalue within a stochastic PDE framework.

Contribution

It introduces a novel approach to define the energy space using paracontrolled distributions and proves the existence and regularity of least energy solutions under stochastic potentials.

Findings

Existence of a least energy solution for the stochastic nonlinear Schrödinger equation.

Regularity results for the minimizer as a weak solution.

Tail estimates for the principal eigenvalue distribution.

Abstract

We consider the two-dimensional nonlinear Schr\"{o}dinger equation with a white noise potential, described by the Anderson hamiltonian. After define the corresponding energy space via the paracontrolled distribution framework from singular stochastic partial differential equations, we prove the existence of the minimizer as the least energy solution by studying a minimization problem of the corresponding energy functional subject to $L^2$ constraints. Subsequently, we study the regularity of the minimizer, which is a weak solution of the nonlinear Schr\"{o}dinger equation. Finally, we derive a tail estimate for the distribution of the principal eigenvalue corresponding to the least energy solution by energy estimates.