Lower bounds on the eigenvalue sums of the Schrödinger operator and the spectral conservation law

2010/01/30 by Oleg Safronov, Safronov, Oleg
Computer Science · Mathematics · Physics and Astronomy · #47A10 #81Q10 #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:47A10 #msc:81Q10

paper · pdf · doi:10.48550/arxiv.1002.0080

openalex publication_date 2010/01/30 · arxiv created 2010/02/12 · arxiv updated 2010/02/26 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

In the first part of the paper we consider the Schrödinger operator -Δ-V(x), V>0. We discuss the relation between the behavior of V at the infinity and the properties of the negative spectrum of H. After that, we consider the case when V changes its sign: V=V₊-V_-, 2V_±=|V|± V. In this case, we treat V and -V symmetrically and study the relation between the behavior of V at the infinity and the negative spectra of the operators H₊=-Δ+V and H_-=-Δ-V.

Related

• On spectral estimates for two-dimensional Schrödinger operators
• On finiteness of the sum of negative eigenvalues of Schroedinger operators
• The rate of accumulation of negative eigenvalues to zero and the absolutely continuous spectrum
• Some spectral bounds for Schroedinger operators with Hardy-type potentials
• On the spectral estimates for the Schrödinger operator on \Z^d, d≥3