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 $L={\mathrm{\partial }}_x(f{\mathrm{\partial }}_x)+{\mathrm{\partial }}_x$ 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.