Quantitative Bounds Versus Existence of Weakly Coupled Bound States for Schrödinger Type Operators

2016/10/31 by Vu Hoang, Dirk Hundertmark, Hoang, Vu +5 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1610.09891

openalex publication_date 2016/10/31 · openalex created_date 2022/09/10 · openalex updated_date 2026/08/01

Abstract

It is well-known that for usual Schroedinger operators weakly coupled bound states exist in dimensions one and two, whereas in higher dimensions the famous Cwikel-Lieb-Rozenblum bound holds. We show for a large class of Schrödinger-type operators with general kinetic energies that these two phenomena are complementary. In particular, we explicitly get a natural semi-classical type bound on the number of bound states precisely in the situation when weakly coupled bound states exist not.

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