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.
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