Severin Schraven
Biography
Bio: Severin Schraven, Technical University of Munich, interested in Spectral Theory and Quantum Mechanics.
Title: Counting eigenvalues of Schr¨odinger operators using the landscape function
Abstract: We prove an upper and a lower bound on the rank of the spectral projections of the Schr¨odinger operator −∆ + V in terms of the volume of the sublevel sets of an effective potential 1 u . Here, u is the ‘landscape function’ of David-Filoche-Mayboroda (2021), namely a solution of (−∆ + V )u = 1 in R d . We prove the result for non-negative potentials satisfying a Kato-type and a doubling condition, in all spatial dimensions, in infinite volume, and show that no coarse graining is required. Our result yields in particular a necessary and sufficient condition for discreteness of the spectrum. This is joint work with S. Bachmann and R. Froese.
