Exact Methods for Spectral Problems via Conformal Field Theory, Supersymmetric Gauge Theory, and Topological Strings

In this thesis, we develop and apply methods inspired by two-dimensional conformal field theory, supersymmetric gauge theory, and topological string theory to the study of spectral problems. In the first part, we investigate periodic spectral problems governed by the Lamé and Heun equations, including confluent limits of the latter. These equations arise both as classical limits of BPZ equations satisfied by degenerate Virasoro conformal blocks and as quantized Seiberg--Witten curves of four-dimensional $\mathcal N=2$ $\mathrm{SU}(2)$ gauge theories. The two descriptions are related by the AGT correspondence, under which degenerate classical conformal blocks are mapped to surface-defect partition functions in the Nekrasov--Shatashvili (NS) limit. We solve the connection problem for the Lamé equation and develop a systematic resummation procedure, based on suitable limits of blow-up equations, for bulk and defect NS functions. The resummation resolves their singular instanton expansions at resonant values of the Floquet exponent and reveals the branch structure of the accessory parameters and Floquet solutions, whose branch points determine the edges of spectral bands and gaps. For the Lamé equation, the resonant limit of the resummed accessory parameter agrees with the values obtained independently from isomonodromic and orbifold-defect constructions. The latter also provides the corresponding (anti-)periodic solutions. For the Heun equation, we extend the resummation procedure to include the Floquet solutions, obtaining at resonance both the accessory parameters and the corresponding (anti-)periodic solutions. We further construct their logarithmic companions and analyze the resonant monodromy, distinguishing the mass loci where spectral gaps close from the smaller loci where the resonant monodromy becomes semisimple. In the second part, we study finite-difference equations obtained by quantizing the mirror curves of the $Y^{N,0}$ geometries, which engineer five-dimensional $\mathcal{N}=1$ $\mathrm{SU}(N)$ gauge theories. In the framework of the open topological string/spectral theory (TS/ST) correspondence, we construct exact generalized eigenfunctions that are expected to be entire in the open and closed moduli and to become square-integrable when the spectral moduli belong to the appropriate exact quantization locus. Their four-dimensional limit gives explicit analytic solutions to the eigenvalue problem associated with $\mathsf{H}_N = 2\Lambda^N\cosh(\mathsf{p}) + V_N(\mathsf{x})$, where $V_N(x)$ is an arbitrary monic polynomial of degree $N$. These operators are solvable finite-difference deformations of the Schrödinger Hamiltonians $\mathsf{p}^{2}+V_N(\mathsf{x})$, and our solutions describe both bound states and resonances. We also identify special loci at which the eigenfunctions exhibit enhanced decay. For confining potentials, this enhancement leads to spectral degeneracies, while for potentials unbounded from below, it produces square-integrable states with real energies. We finally discuss the related operators with a $\sinh(\mathsf{p})$ kinetic term, an inverted potential $-V_N(\mathsf{x})$, or both.