Semiclassical nonlinear Schrödinger equations with potential and focusing initial data
The authors study the semiclassical limit for solutions \(u^{\varepsilon}\) to a Schrödinger equation with repulsive subcritical nonlinearity and potential, with concentrating initial data. The potential \(V\) is assumed to be polynomial of degree 2, and the eigenvalues of the quadratic part of \(V\) are allowed to be negative. We stress that in such a case the energy of \(u^{\varepsilon}\) contains negative terms which are not a priori controlled by the positive ones, so global existence in \(H^1\) is not trivial. For any \(T>0\), the authors show that \(u^{\varepsilon}\) cannot blow up for \(0<| t| < T\). Moreover they prove the asymptotics for the semiclassical limit: the behavior near \(t=0\) is governed by the nonlinear term where the potential is negligible, while for \(| t| \) large we have the reversed situation. The transition between these two regimes is measured by the scattering operator associated to the potential \(V\).
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