Resolvent estimates for one-dimensional Schrödinger operators with complex potentials

Abstract

We study one-dimensional Schrödinger operators H=−∂x2+V with unbounded complex potentials V and derive asymptotic estimates for the norm of the resolvent, Ψ(λ):=‖(H−λ)−1‖, as |λ|→+∞, separately considering λ∈RanV and λ∈R+. In each case, our analysis yields an exact leading order term and an explicit remainder for Ψ(λ) and we show these estimates to be optimal. We also discuss several extensions of the main results, their interrelation with some aspects of semigroup theory and illustrate them with examples.