Resolvent estimates for one-dimensional Schrödinger operators with complex potentials

Abstract

We study one-dimensional Schrödinger operators H = −∂ 2 x + 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.

Resolvent estimates for the one-dimensional damped wave equation with unbounded damping

“…The following result will be used later on and we include it here for convenience. We refer to [4,Lem. 4.4] for a proof.…”

“…[7,11,27,5]) and non-semi-classical (e.g. [24,22,3,12,4]) methods. One approach, pioneered in [7] and subsequently developed non-semi-classically in [24,3,22,12], relies on the construction of pseudomodes (or approximate eigenfunctions) for the operator at hand (Schrödinger, damped wave equation, Dirac, biharmonic) inside the numerical range thereby finding lower bounds on (H − λ) −1 .…”

“…One approach, pioneered in [7] and subsequently developed non-semi-classically in [24,3,22,12], relies on the construction of pseudomodes (or approximate eigenfunctions) for the operator at hand (Schrödinger, damped wave equation, Dirac, biharmonic) inside the numerical range thereby finding lower bounds on (H − λ) −1 . For Schrödinger operators with complex potentials, lower and upper bounds have recently been found in [4] using different (non-semi-classical) methods.…”

“…The aim of the work presented in this paper is to apply the new techniques developed in [4] to the study of the resolvent of the NSA generator G for the onedimensional damped wave equation (DWE) described by…”

“…Although we shall defer a more rigorous definition of this operator, along with an explanation of how it relates to G, until Sub-section 2.4, we observe here that its structure is that of a λ-dependent Schrödinger operator with the complex potential q(x) + 2λa(x) + λ 2 . Whereas the fact that q is "no stronger" than a discourages us from (for example) attempting to recast the problem as a (relatively bounded) perturbation of a self-adjoint operator, it does on the other hand broadly fit into the framework used to prove [4,Thm. 4.2], where the asymptotic behaviour of (H − λ) −1 along the real axis for a one-dimensional Schrödinger operator with a complex potential H was determined.…”