Calogero type bounds in two dimensions

For a Schrödinger operator on the plane R2 with electric potential V and an Aharonov–Bohm magnetic field, we obtain an upper bound on the number of its negative eigenvalues in terms of the L1(R2) -norm of V. Similar to Calogero’s bound in one dimension, the result is true under monotonicity assumpti...

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Bibliographic Details
Main Authors: Ari Laptev, Larry Read, Lukas Schimmer
Format: Default Article
Published: 2022
Subjects:
Applied mathematics
Pure mathematics
Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Mathematics
General Physics
Calogero's bound
Aharanov-Bohm magnetic field
Cwikel-Lieb-Rozenblum inequality
Hardy inequality
Online Access:https://hdl.handle.net/2134/24591960.v1
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Summary:For a Schrödinger operator on the plane R2 with electric potential V and an Aharonov–Bohm magnetic field, we obtain an upper bound on the number of its negative eigenvalues in terms of the L1(R2) -norm of V. Similar to Calogero’s bound in one dimension, the result is true under monotonicity assumptions on V. Our method of proof relies on a generalisation of Calogero’s bound to operator-valued potentials. We also establish a similar bound for the Schrödinger operator (without magnetic field) on the half-plane when a Dirichlet boundary condition is imposed and on the whole plane when restricted to antisymmetric functions.

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