Counterexample to the Laptev–Safronov Conjecture

Abstract

AbstractLaptev 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 $$\mathbb {R}^d$$ R d , $$d\ge 2$$ d ≥ 2 , and an $$L^q$$ L q norm of the potential, for any $$q\in [d/2,d]$$ q ∈ [ d / 2 , d ] . Frank (Bull Lond Math Soc 43(4):745–750, 2011) proved that the conjecture is true for $$q\in [d/2,(d+1)/2]$$ q ∈ [ d / 2 , ( d + 1 ) / 2 ] . We construct a counterexample that disproves the conjecture in the remaining range $$q\in ((d+1)/2,d]$$ q ∈ ( ( d + 1 ) / 2 , d ] . As a corollary of our main result we show that, for any $$q>(d+1)/2$$ q > ( d + 1 ) / 2 , there is a complex potential in $$L^q\cap L^{\infty }$$ L q ∩ L ∞ such that the discrete eigenvalues of the corresponding Schrödinger operator accumulate at every point in $$[0,\infty )$$ [ 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.