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

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.