Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates

Abstract

We generalize the theorems of Stein-Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schrödinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-posedness of the Hartree equation in Schatten spaces, to Lieb-Thirring bounds for eigenvalues of Schrödinger operators with complex potentials, and to Schatten properties of the scattering matrix.

Improved Eigenvalue Bounds for Schrödinger Operators with Slowly Decaying Potentials

“…Finally we should also mention that there are versions of (1.1) concerning sums of eigenvalues (e.g. [13,6,7,15,14]), but these will not be discussed here. Several works also deal with a class of potentials outside the L q -scale (e.g.…”

“…where |z| = 1 and 0 ≤ Re ζ ≤ (d + 1)/2 (see also [15,3,12,22] where the same family is considered). It suffices to prove the bounds…”

Eigenvalue bounds for Stark operators with complex potentials

“…The latter observation allows one to turn the analysis of eigenvalues into the study of zeros of analytic functions. Similar ideas were successfully used in the paper [8] by Frank and Sabin for the study of the eigenvalues of the Schrödinger operator perturbed by a decaying potential. Among other related papers are the articles [5], [7].…”

Eigenvalue Estimates for Bilayer Graphene

“…The latter implies that (A.1) holds with r = (d − 1)/2. The Schatten space estimates for the resolvent of the Laplacian were first proved by Frank-Sabin [10]. By a duality argument, (A.3) implies…”