Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 025, 42 pages      arXiv:1806.01407      https://doi.org/10.3842/SIGMA.2019.025

A Solvable Deformation of Quantum Mechanics

Alba Grassi a and Marcos Mariño b
a) Simons Center for Geometry and Physics, SUNY, Stony Brook, NY, 1194-3636, USA
b) Département de Physique Théorique et Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland

Received October 15, 2018, in final form March 23, 2019; Published online March 31, 2019

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.

Key words: topological string theory; supersymmetric gauge theory; quantum mechanics; spectral theory.

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