Strichartz estimates for the wave and Schroedinger equations with potentials of critical decay

Abstract

We prove weighted L 2 estimates for the solutions of linear Schrödinger and wave equation with potentials that decay like |x| −2 for large x, by deducing them from estimates on the resolvent of the associated elliptic operator. We then deduce Strichartz estimates for these equations.

Kato Smoothing and Strichartz Estimates for Wave Equations with Magnetic Potentials

“…Potentials of critical decay. Our second example is a simplification of a proof in [2], where Strichartz estimates were obtained for Schrödinger and wave equations of the form…”

“…In order to apply the same procedure to the wave equation, in [2] the following estimate for the wave flow is proved:…”

“…[19], [6] or [2]). The first goal of this note is to deduce smoothing estimates for abstract wave equations within the framework of Kato's theory: given a non negative selfadjoint operator H and a closed operator A on the Hilbert space H, we prove that A is H-(super)smooth =⇒ AH It is clear that the range of applications is quite wide.…”

“…In Section 3 we picked three. The first two are mostly known results: smoothing estimates for the flows generated by powers of the Laplacian; and the smoothing estimates for wave equations with potentials of critical decay which were obtained in [2].…”

Sobolev spaces adapted to the Schrödinger operator with inverse-square potential

“…The operator L a arises frequently in mathematics and physics, commonly as a scaling limit of more complicated problems. Several instances where this occurs in physics are discussed in the mathematical papers [7,8,15,28,29]; they range from combustion theory to the Dirac equation with Coulomb potential, and to the study of perturbations of classic space-time metrics such as Schwarzschild and Reissner-Nordström.…”