Quantitative Bounds Versus Existence of Weakly Coupled Bound States for Schrödinger Type Operators
Vu HoangDepartment of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249, USADirk HundertmarkDepartment of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street (MC-382), Urbana, IL, 61801, USAJohanna RichterHausdorff Center for Mathematics, Endenicher Allee 62, 53115, Bonn, GermanySemjon VugalterDepartment of Mathematics, Institute for Analysis, Karlsruhe Institute of Technology, 76131, Karlsruhe, Germany
2022en
ABI
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Abstract It is well-known that for usual Schrödinger 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. 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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