Constraint Minimization Problem of the Nonlinear Schrödinger Equation with the Anderson Hamiltonian

We consider the two-dimensional nonlinear Schrödinger equation with the Anderson hamiltonian, which given by the Laplacian plus a white noise potential. After establishing the energy space through the paracontrolled distribution framework, we prove the existence of the minimizer as the least energy solution through studying the minimization problem of the corresponding energy functional subject to L2 constraints. Subsequently, we study the regularity of the minimizer, which is a ground state solution of the nonlinear Schrödinger equation. Finally, we derive a tail estimate for the distribution of the principal eigenvalue corresponding to the ground state solution by energy estimates.