  In this paper we examine the asymptotic structure of the pseudospectrum of
the singular Sturm-Liouville operator $L=\partial_x(f\partial_x)+\partial_x$
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.
