The pseudospectrum of an operator with Bessel-type singularities: analytical and computational persp

In this seminar talk we will describe the asymptotic structure of the pseudospectrum of the singular Sturm-Liouville operator L = ∂x(f ∂x)+∂x subject to periodic boundary conditions on a symmetric interval, where the coefficient f is a regular odd function that has only a simple zero at the origin. The operator L is closely related to a remarkable model examined by Davies in 2007, which exhibits surprising spectral properties balancing symmetries and strong non-self-adjointness. In the first part of the presentation, we will derive a concrete construction of classical pseudo-modes for L and give explicit exponential bounds of growth for the resolvent norm in rays away from the spectrum. In the second part, we report on the construction of an adaptive curve-tracing computational method to detect and certify boundary points of the pseudospectrum of L very far from the spectrum. The context of the work and a list of references can be found in the arXiv preprint number 2310.13611. This is research in collaboration with Marco Marletta (Cardiff University) and Catherine Drysdale
(University of Birmingham).