Counterexample to the Laptev-Safronov Conjecture

Boegli, Sabine; Cuenin, Jean-Claude

doi:10.1007/s00220-022-04546-z

Counterexample to the Laptev-Safronov Conjecture

Boegli, Sabine; Cuenin, Jean-Claude

Authors

Dr Sabine Boegli sabine.boegli@durham.ac.uk
Associate Professor

Jean-Claude Cuenin

Abstract

Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009) conjectured an inequality between the magnitude of eigenvalues of a non-self-adjoint Schrödinger operator on Rd, d≥2, and an Lq norm of the potential, for any q∈[d/2,d]. Frank (Bull Lond Math Soc 43(4):745–750, 2011) proved that the conjecture is true for q∈[d/2,(d+1)/2]. We construct a counterexample that disproves the conjecture in the remaining range q∈((d+1)/2,d]. As a corollary of our main result we show that, for any q>(d+1)/2, there is a complex potential in Lq∩L∞ such that the discrete eigenvalues of the corresponding Schrödinger operator accumulate at every point in [0,∞). In some sense, our counterexample is the Schrödinger operator analogue of the ubiquitous Knapp example in Harmonic Analysis. We also show that it is adaptable to a larger class of Schrödinger type (pseudodifferential) operators, and we prove corresponding sharp upper bounds.

Citation

Boegli, S., & Cuenin, J. (2023). Counterexample to the Laptev-Safronov Conjecture. Communications in Mathematical Physics, 398(3), 1349-1370. https://doi.org/10.1007/s00220-022-04546-z

Journal Article Type	Article
Acceptance Date	Oct 3, 2022
Online Publication Date	Nov 17, 2022
Publication Date	2023-03
Deposit Date	Oct 27, 2022
Publicly Available Date	May 23, 2023
Journal	Communications in Mathematical Physics
Print ISSN	0010-3616
Electronic ISSN	1432-0916
Publisher	Springer
Peer Reviewed	Peer Reviewed
Volume	398
Issue	3
Pages	1349-1370
DOI	https://doi.org/10.1007/s00220-022-04546-z
Related Public URLs	https://arxiv.org/abs/2109.06135

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