Location of zeros of non-trivial positive supersolutions to Schrödinger equations

2019/12/16 by Tomasz Klimsiak, Klimsiak, Tomasz · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1912.07307

openalex publication_date 2019/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Schrödinger operators on L²(E;m) of the form -A+V with singular potentials V. We address the question posed by H. Brezis about the structure of the set \u=0\ for non-negative supersolutions to -Au+Vu=0. The class of operators A we study in the paper includes, in particular, symmetric Levy type operators and symmetric diffusions in divergence form, with strictly positive Green functions. The class of potentials V consists of positive smooth measures, which contains, in particular, Coulomb potentials and harmonic potentials, as well as generalized potentials, i.e. positive Borel measures concentrated on m-negligible sets.

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