Resolvent estimates for non-self-adjoint differential operators in one dimension

Abstract

We consider non-self-adjoint differential operators H in one dimension and study the norm of the resolvent operator, ‖R(H,λ)‖, near or inside the numerical range when the spectral parameter λ is large. More precisely, we derive asymptotic estimates for three classes of operators: generalised Airy operators, Schrödinger operators with complex potential and the generator of the wave equation with unbounded damping. In each case, we determine the leading order term, including explicit constants, and then show that the estimate is optimal. For Schrödinger operators, we develop a methodology to find an upper bound for the norm of the resolvent based on the separate local/non-local analyses of the action of H- λ on functions in the operator domain. We apply this technique to estimate an upper bound for the resolvent norm of the quadratic operator family T(λ) associated with the generator G of the damped wave equation and subsequently leverage that estimate to derive a result for the resolvent of the generator itself. We show that, in both cases, the norm of the resolvent operator depends on the resolvent norm of an appropriate generalised Airy operator which we also investigate using Schur's test and an extension of Laplace's method. We illustrate our results with a range of examples and explore several applications and extensions. In particular, we compute the asymptotic level curves for the norm of the resolvent of Schrödinger operators H and the quadraticfamily T(λ). Furthermore we study consequences for the long-time behaviour of the norm of associated strongly continuous semigroups for each operator class, leading to statements regarding the stability of solutions of the corresponding abstract Cauchy problems.

Date of Award	Jul 2023
Original language	English
Awarding Institution	Queen's University Belfast
Supervisor	Petr Siegl (Supervisor), Hannah Mitchell (Supervisor) & Anna Zhigun (Supervisor)