All full-fledged theories in physics boil down to the study of operator equations. They are encompassing differential equations (e.g. equations of motion) that involve operators acting on infinite-dimensional spaces of functions such as Hilbert spaces. From the point of view of spectral theory, it amounts to the eigendecomposition of said operator. In this work, we focus on Schrödinger type spectral problems. For their own sake, but also because they connect to more complicated quantum theories through dualities e.g. supersymmetric Yang-Mills or, via the TS/ST correspondence, topological strings on Calabi-Yau threefolds. The main tool we use in order to extract exact results are exact quantization conditions, functional relations that fully encode the spectrum of a given operator. We explore Hermitian as well as non-Hermitian (resonant) spectral problems through semiclassical WKB analysis, resurgence, dualities, and integral TBA equations, focusing on concrete examples. We develop a method for computing traces of arbitrary functions of an operator exactly.