Strichartz Estimates for the Schroedinger Equation with Time-Periodic L^{n/2} Potentials

TL;DR

This paper establishes Strichartz estimates for the Schrödinger equation with time-periodic, scaling-critical complex potentials in dimensions three and higher, using resolvent estimates without requiring dispersive bounds.

Contribution

It introduces a novel approach to prove Strichartz estimates directly from resolvent estimates for time-periodic potentials in the critical space.

Findings

Strichartz estimates are valid for potentials in L^{n/2}_x L^_t space.

Eigenvalues can be present at any spectrum location if associated eigenfunctions decay rapidly.

The method avoids the need for dispersive bounds typically used in such analyses.

Abstract

We prove Strichartz estimates for the Schroedinger operator $H = -\Delta + V(t,x)$ with time-periodic complex potentials $V$ belonging to the scaling-critical space $L^{n/2}_x L^\infty_t$ in dimensions $n \ge 3$. This is done directly from estimates on the resolvent rather than using dispersive bounds, as the latter generally require a stronger regularity condition than what is stated above. In typical fashion, we project onto the continuous spectrum of the operator and must assume an absence of resonances. Eigenvalues are permissible at any location in the spectrum, including at threshold energies, provided that the associated eigenfunction decays sufficiently rapidly.