Eigenfunctions of deformed Schr\"odinger equations

TL;DR

This paper constructs exact eigenfunctions for a class of deformed Schr"odinger operators related to supersymmetric gauge theories, revealing special spectral properties and degeneracies at specific parameter values.

Contribution

It introduces a method to obtain explicit, analytic eigenfunctions for polynomial potentials in deformed Schr"odinger equations, connecting spectral theory with topological string theory.

Findings

Eigenfunctions are entire in x and valid for arbitrary energies.

Spectral degeneracies occur at Toda points in parameter space.

Eigenfunctions exhibit enhanced decay at special parameter loci.

Abstract

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.