Counterexample to the Laptev–Safronov conjecture

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.