Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces

We generalize the $L^p$ spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators.Comment: 30 page

The Stein-Tomas inequality in trace ideals

The goal of this review is to explain some recent results regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).Comment: Proceedings of the Laurent Schwartz semina

Maximizers for the Stein-Tomas inequality

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein-Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein-Tomas inequality. Our result is valid in any dimension.Comment: 37 page

Galactic Plane Hα\alpha Surveys: IPHAS & VPHAS+

The optical Galactic Plane H$\alpha$ surveys IPHAS and VPHAS+ are dramatically improving our understanding of Galactic stellar populations and stellar evolution by providing large samples of stars in short lived, but important, evolutionary phases, and high quality homogeneous photometry and images over the entire Galactic Plane. Here I summarise some of the contributions these surveys have already made to our understanding of a number of key areas of stellar and Galactic astronomy.Comment: 5 pages, 2 figures, refereed proceeding of the "The Universe of Digital Sky Surveys" conference, November 2014, to be published in the Astrophysics and Space Science Proceeding

Atomic swelling upon compression

The hydrogen atom under the pressure of a spherical penetrable confinement potential of a decreasing radius $r_{0}$ is explored, as a case study. A novel counter-intuitive effect of atomic swelling rather than shrinking with decreasing $r_{0}$ is unraveled, when $r_{0}$ reaches, and remains smaller than, a certain critical value. Upon swelling, the size of the atom is shown to increase by an order of magnitude, or more, compared to the size of the free atom. Examples of changes of photoabsorption properties of confined hydrogen atom upon its swelling are uncovered and demonstrated.Comment: 5 pages, 4 figure

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

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

Extremizers for the Airy–Strichartz inequality

We identify the compactness threshold for optimizing sequences of the Airy–Strichartz inequality as an explicit multiple of the sharp constant in the Strichartz inequality. In particular, if the sharp constant in the Airy–Strichartz inequality is strictly smaller than this multiple of the sharp constant in the Strichartz inequality, then there is an optimizer for the former inequality. Our result is valid for the full range of Airy–Strichartz inequalities (except the endpoints) both in the diagonal and off-diagonal cases