Nonlinear double well Schrödinger equations in the semiclassical limit (Q2574161)

This paper deals with time-dependent Schrödinger equations with a symmetric double-well potential and nonlinear perturbations. Both local and nonlocal terms are considered as the perturbations. (Spatial dimension \(d=1\) or \(2\) are assumed for the case of the local perturbation.) If the nonlinear term is absent, the linear Hamiltonian has even parity and odd parity eigenstates and a time dependent state generically performs a beating motion between the two wells. It is shown that under some generic assumptions on the double-well potential, a new asymmetric stationary state appears and the beating motion gradually disappears for increasing nonlinearity. Main apparatus of the proof is the two level approximation which considers the projection onto the two-dimensional subspace spanned by eigenfunctions associated to two lowest eigenvalues of the linear Schrödinger operator. The dynamical system obtained by the reduction is exactly soluble and its solution approximately describes the solution of the original nonlinear Schrödinger equation. A rigorous semiclassical estimate of the error is given.