Schrödinger Operator with Non-Zero Accumulation Points of Complex Eigenvalues

Abstract

We study Schrödinger operators H=-Δ+V in L2(Ω) where Ω is Rd or the half-space R+d, subject to (real) Robin boundary conditions in the latter case. For p> d we construct a non-real potential V∈Lp(Ω)∩L∞(Ω) that decays at infinity so that H has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum σess(H) = [ 0 , ∞). This demonstrates that the Lieb–Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.