  In this paper we give a streamlined derivation of the exact quantization
condition (EQC) on the quantum periods of the Schr\"odinger problem in one
dimension with a general polynomial potential, based on Wronskian relations. We
further generalize the EQC to potentials with a regular singularity, describing
spherical symmetric quantum mechanical systems in a given angular momentum
sector. We show that the thermodynamic Bethe ansatz (TBA) equations that govern
the quantum periods undergo nontrivial monodromies as the angular momentum is
analytically continued between integer values in the complex plane. The TBA
equations together with the EQC are checked numerically against Hamiltonian
truncation at real angular momenta and couplings, and are used to explore the
analytic continuation of the spectrum on the complex angular momentum plane in
examples.
