Bifurcation and Stability for Nonlinear SchrödingerEquations with DoubleWell Potential in the SemiclassicalLimit

We consider the stationary solutions for a class of Schrödinger equations witha symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassicallimit we prove that the reduction to a finite-mode approximation give the stationary solutions,up to an exponentially small term, and that symmetry-breaking bifurcation occurs ata given value for the strength of the nonlinear term. The kind of bifurcation picture onlydepends on the nonlinearity power. We then discuss the stability/instability properties ofeach branch of the stationary solutions. Finally, we consider an explicit one-dimensional toymodel where the double well potential is given by means of a couple of attractive Dirac’sdelta pointwise interactions.