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We consider the stationary solutions for a class of Schroedinger equationswith a symmetric double-well potential and a nonlinear perturbation. Here, inthe semiclassical limit we prove that the reduction to a finite-modeapproximation give the stationary solutions, up to an exponentially small term,and that symmetry-breaking bifurcation occurs at a given value for the strengthof the nonlinear term. The kind of bifurcation picture only depends on thenon-linearity power. We then discuss the stability/instability properties ofeach branch of the stationary solutions. Finally, we consider an explicitone-dimensional toy model where the double well potential is given by means ofa couple of attractive Dirac's delta pointwise interactions.

Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit