New tools for extremal problems in semiclassical spectral theory

An important area of research in spectral theory is the study of semiclassical approximations — describing the limiting behavior of complex spectral quantities in terms of simpler objects.

Such approximations are crucial in understanding, for instance, quantum mechanical systems as the number of particles becomes large.

The proposed project seeks to develop new tools to understand in what situations and to what degree these approximations are accurate. The specific focus is in the realm of Lieb–Thirring inequalities.

These inequalities bound sums of negative eigenvalues of Schrödinger operators and are central tools in semiclassical spectral theory and quantum mechanics.

The proposal takes a non-standard perspective on this classical topic by developing a unified strategy to investigate operators which almost saturate the inequalities in the semiclassical limit.

The strategy takes a well-developed method for proving semiclassical asymptotics under weak assumptions, and turns it on its head — obtaining information about operators from knowledge of semiclassical expansions instead of the reverse.

The proposal pushes the research frontiers of spectral theory in multiple directions — including the study of universal estimates and semiclassical asymptotics — and brings breakthrough developments in the field of spectral extremal problems into a semiclassical setting, where the quantities of interest depend on an infinite number of eigenvalues instead of a fixed and finite number.