A Solvable Deformation of Quantum Mechanics

TL;DR

This paper introduces an exactly solvable deformation of quantum mechanics derived from Seiberg-Witten theory, providing a closed-form quantization condition that encompasses bound states and resonances without resummation.

Contribution

It presents a novel, exactly solvable Hamiltonian deformation linked to topological string theory, expanding the understanding of quantum spectra and symmetry phenomena.

Findings

Exact quantization condition derived for the deformed Hamiltonian

Spectrum determination without resummation techniques

Observation of spontaneous parity symmetry breaking

Abstract

The conventional Hamiltonian $H= p^2+ V_N(x)$, where the potential $V_N(x)$ is a polynomial of degree $N$, has been studied intensively since the birth of quantum mechanics. In some cases, its spectrum can be determined by combining the WKB method with resummation techniques. In this paper we point out that the deformed Hamiltonian $H=2 \cosh(p)+ V_N(x)$ is exactly solvable for any potential: a conjectural exact quantization condition, involving well-defined functions, can be written down in closed form, and determines the spectrum of bound states and resonances. In particular, no resummation techniques are needed. This Hamiltonian is obtained by quantizing the Seiberg-Witten curve of $\mathcal{N}=2$ Yang-Mills theory, and the exact quantization condition follows from the correspondence between spectral theory and topological strings, after taking a suitable four-dimensional limit. In this formulation, conventional quantum mechanics emerges in a scaling limit near the Argyres-Douglas superconformal point in moduli space. Although our deformed version of quantum mechanics is in many respects similar to the conventional version, it also displays new phenomena, like spontaneous parity symmetry breaking.