Dispersive Estimates for Four Dimensional Schrödinger and Wave Equations with Obstructions at Zero Energy
نویسندگان
چکیده
We investigate L(R) → L∞(R4) dispersive estimates for the Schrödinger operator H = −∆ + V when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator Ft satisfying ‖Ft‖L1→L∞ . 1/ log t for t > 2 such that ‖ePac − Ft‖L1→L∞ . t , for t > 2. We also show that the operator Ft = 0 if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential.
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