Counting eigenvalues of Schrödinger operators using the landscape function

2023/06/06 by Sven Bachmann, Bachmann, Sven, Richard Froese +3 · 1 citation
Computer Science · Materials Science · Mathematics · #35J10 #35P20 #47B93 #81Q10 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quasicrystal Structures and Properties #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2306.03936

openalex publication_date 2023/06/06 · openalex created_date 2023/06/09 · openalex updated_date 2026/07/28

Abstract

We prove an upper and a lower bound on the rank of the spectral projections of the Schrödinger operator -Δ+ V in terms of the volume of the sublevel sets of an effective potential (1)/(u). Here, u is the `landscape function' of [(David, G., Filoche, M., & Mayboroda, S. (2021) Advances in Mathematics, 390, 107946)], namely a solution of (-Δ+ V)u = 1 in \bbR^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. In the case of polynomial potentials, we prove that the spectrum is discrete if and only if no directional derivative vanishes identically.

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