Existence of complex Schrödinger operators in higher dimensions with substantially more eigenvalues than real-valued counterparts

Determine whether, in dimensions d ≥ 2, there exist Schrödinger operators H = −Δ + V on L^2(R^d) with complex-valued potentials V whose discrete spectrum contains significantly more eigenvalues than that of Schrödinger operators with real-valued potentials in the same integrability class (e.g., V ∈ L^q(R^d) for finite q).

Background

The paper reviews known bounds for sums of eigenvalues of non-selfadjoint Schrödinger operators with complex potentials and contrasts them with classical Lieb–Thirring bounds for real-valued potentials. In one dimension, examples exist that show unexpectedly many eigenvalues compared to resonances for complex potentials.

Whether a similar phenomenon can occur in higher dimensions is unknown, and resolving this would clarify how non-selfadjointness affects the discrete spectrum size relative to the real-valued case.