We study Schrödinger operators H=−Δ+V in L2(Ω) where Ω is Rd or the half-space Rd+, 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.
Boegli, S. (2017). Schrödinger Operator with Non-Zero Accumulation Points of Complex Eigenvalues. Communications in Mathematical Physics, 352(2), 629-639. https://doi.org/10.1007/s00220-016-2806-5