Complex Potentials: Bound States, Quantum Dynamics and Wave Operators

Abstract

Schrödinger operator on half-line with complex potential and the corresponding evolution are studied within perturbation theoretic approach. The total number of eigenvalues and spectral singularities is effectively evaluated. Wave operators are constructed and a criterion is established for the similarity of perturbed and free propagators.

On the number of eigenvalues of Schrödinger operators with complex potentials

“…In dimensions one and three the resolvent kernel is explicit and this is important for the proofs in [31,32]. In contrast, our Theorem 1.2 is valid in arbitrary odd dimensions d ≥ 3, where the resolvent kernel is only given in terms of Bessel functions, which become increasingly more complicated as the dimension increases.…”

Bounds for Schrödinger Operators on the Half-Line Perturbed by Dissipative Barriers

“…Quantitative bounds for the number of eigenvalues of a Schrödinger operator on L 2 (R d ) were proved by Stepin in [32,33] for dimensions d = 1, 3. Bounds for arbitrary odd dimensions were later proved by Frank, Laptev and Safronov in [13], which give better large R estimates when applied to operators H R of the form (1).…”

“…Lemma 7. For any x ∈ [0, ∞) and z ∈ C\{±iγ}, the solution θ to the initial value problem (32) satisfies the inequality…”

Lieb-Thirring and Jensen sums for non-self-adjoint Schrödinger operators on the half-line

“…Proof. A key ingredient of the proof is the following well-known inequality for the Jost function (see, e.g., [37,Lemma 1])…”