Counterexample to the Laptev–Safronov conjecture

• Main Authors: Sabine Bögli, Jean-Claude Cuenin
• Format: Other Non-Article Part of Journal/Newspaper
• Language: unknown
• Published: 2022
• Subjects: Mathematical physics; Pure mathematics; Quantum physics; Science & Technology; Physical Sciences; Physics; Mathematical; laptev
• Online Access: https://figshare.com/articles/journal_contribution/Counterexample_to_the_Laptev_Safronov_conjecture/22094606

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 R d , d≥ 2 , and an L q 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 L q ∩ 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.