Dispersive Estimates for Schrödinger Operators with Measure-valued Potentials in R

Dispersive Estimates for Matrix Schrödinger Operators in Dimension Two

We consider the non-selfadjoint operator H = [ −∆ + μ− V1 −V2 V2 ∆− μ+ V1 ] where μ > 0 and V1, V2 are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain L(R)× L(R)→ L∞(R2)× L∞(R2) dispersive decay estimates for the evolution ePac. We also obtain the following weighted estimate ‖wePa...

Wave Operator Bounds for 1-dimensional Schrödinger Operators with Singular Potentials and Applications

Boundedness of wave operators for Schrödinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.

Decay Estimates for Four Dimensional Schrödinger, Klein-gordon and Wave Equations with Obstructions at Zero Energy

We investigate dispersive estimates for the Schrödinger operator H = −∆+V with V is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion eχ(H)Pac(H) = O(1/(log t))A0 +O(1/t)A1 +O((t log t) )A2 +O(t (log t))A3. Here A0, A1 : L (R) → L∞(Rn), while A2, A3 are o...

Dispersive Estimates for Schrödinger Operators: a Survey

Schrödinger Operators with Complex-valued Potentials and No Resonances

In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on R. In odd dimensions d ≥ 3 we study the fundament...