Local energy decay and smoothing effect for the damped Schrödinger equation

Local decay for the damped wave equation in the energy space

Energy Decay of Damped Systems

We present a new and simple bound for the exponential decay of second order systems using the spectral shift. This result is applied to finite matrices as well as to partial differential equations of Mathematical Physics. The type of the generated semigroup is shown to be bounded by the upper real part of the numerical range of the underlying quadratic operator pencil.

Scale invariant energy smoothing estimates for the Schrödinger equation with small magnetic potential

We consider some scale invariant generalizations of the smoothing estimates for the free Schrödnger equation obtained by Kenig, Ponce and Vega in [21], [22]. Applying these estimates and using appropriate commutator estimates, we obtain similar scale invariant smoothing estimates for perturbed Schrödnger equation with small magnetic potential.

Self-Focusing in the Damped Nonlinear Schrödinger Equation

We analyze the effect of damping (absorption) on critical self-focusing. We identify a threshold value δth for the damping parameter δ such that when δ > δth damping arrests blowup. When δ < δth, the solution blows up at the same asymptotic rate as the undamped nonlinear Schrödinger equation.

A remark on the Schrödinger smoothing effect

— We prove the equivalence between the smoothing effect for a Schrödinger operator and the decay of the associate spectral projectors. We give two applications to the Schrödinger operator in dimension one. Résumé. — On donne une caractérisation de l’effet régularisant pour un opérateur de Schrödinger par la décroissance de ses projecteurs spectraux. On en déduit deux applications à l’opérateur ...