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Nonlinear Schrödinger equations with Stark potential. (English) Zbl 1069.35083

Summary: We study the nonlinear Schrödinger equations with a linear potential \[ i\varepsilon\partial_tu+\frac 12\varepsilon^2 \Delta u=V(x)u+\lambda |u|^{2\sigma}u,\quad u|_{t=0}= u_0. \] A change of variables makes it possible to deduce results concerning finite time blow up and scattering theory from the case with no potential.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81U05 \(2\)-body potential quantum scattering theory
35P25 Scattering theory for PDEs