The pseudospectrum of an operator with Bessel-type singularities

In this paper, we examine 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 our main result, we 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.