Number of Complex Eigenvalues for a Class of Dissipative Schrödinger Operators

For a class of dissipative Schrödinger operators H = −∆ + V (x) with a complex-valued potential V = V1−iV2 with V2 ≥ 0 and |V (x)| = O(|x| ) as |x| tends to infinity, we prove that the complex eigenvalues of H can not accumulate to zero. In the perturbation regime where V2 is sufficiently small, we show under some conditions that N(V ) = N(V1)+k, where k is the multiplicity of the zero resonance of −∆ + V1 and N(V ) (resp., N(V1) ) is the number of eigenvalues of −∆+ V (resp., −∆+ V1).