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 Schrödinger 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, generalized eigenfunctions of $H_N$, valid for arbitrary polynomial potentials and describing both bound and resonant states.

We also comment on the case with a $\sinh(p)$ kinetic term.

Our solutions are entire in $x$ for all generalized eigenvalues, and become square-integrable for a discrete subset of those.

An interesting feature is the existence of special loci in the parameter space of the potential, where the eigenfunctions exhibit enhanced decay, 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.