Eigenfunctions of deformed Schr\"odinger equations

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the standard Schr\"odinger operators $p^2 + V_N(x)$, and they arise naturally from the quantization of the Seiberg-Witten curve of four-dimensional, $\mathcal{N} = 2$, SU($N$) supersymmetric Yang-Mills theory. Using the open topological string/spectral theory correspondence, we construct exact, analytic eigenfunctions of $H_N$, valid for arbitrary polynomial potentials and describing both bound and resonant states. Our solutions are entire in $x$ for generic values of the energy, and become $L^2$-normalizable only at a discrete set of energies. An interesting feature of these Hamiltonians is the existence of special loci in the parameter space of the potential, the so-called Toda points. The eigenfunctions exhibit enhanced decay at these points, leading to spectral degeneracies for confining potentials and to a real energy spectrum for unbounded ones. Our results provide a rare example of a quantum-mechanical spectral problem that is exactly solvable, admitting explicit, analytic eigenfunctions for both bound and resonant states.