  It has been recently realized that, in the case of polynomial potentials, the
exact WKB method can be reformulated in terms of a system of TBA equations. In
this paper we study this method in various examples. We develop a graphical
procedure due to Toledo, which provides a fast and simple way to study the
wall-crossing behavior of the TBA equations. When complemented with exact
quantization conditions, the TBA equations can be used to solve spectral
problems exactly in Quantum Mechanics. We compute the quantum corrections to
the all-order WKB periods in many examples, as well as the exact spectrum for
many potentials. In particular, we show how this method can be used to
determine resonances in unbounded potentials.
