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

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.