A Solvable Deformation of Quantum Mechanics

A Solvable Deformation of Quantum Mechanics

The conventional Hamiltonian H=p²+VN(x), where the potential VN(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 t...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2019
Main Authors: Grassi, A., Mariño, M.
Format: Article
Language:English
Published: Інститут математики НАН України 2019
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/210050
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:A Solvable Deformation of Quantum Mechanics / A. Grassi, M. Mariño // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 111 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-210050
record_format dspace
spelling Grassi, A.
Mariño, M.
2025-12-02T09:27:43Z
2019
A Solvable Deformation of Quantum Mechanics / A. Grassi, M. Mariño // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 111 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 14N35; 58C40; 51P05; 81T13; 81Q60; 82B23; 81Q80
arXiv: 1806.01407
https://nasplib.isofts.kiev.ua/handle/123456789/210050
https://doi.org/10.3842/SIGMA.2019.025
The conventional Hamiltonian H=p²+VN(x), where the potential VN(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 = 2cosh(p) + VN(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 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.
We would like to thank Yoan Emery, Giovanni Felder, Matthias Gaberdiel, Jie Gu, Nikita Nekrasov, Massimiliano Ronzani, and Szabolcs Zakany for useful discussions. We are particularly thankful to Jie Gu for extending the Bender-Wu package to Hamiltonians like the one we study in this paper. The work of M.M. is supported in part by the Fonds National Suisse, subsidies 200021-156995 and 200020-141329, and by the NCCR 51NF40-141869 "The Mathematics of Physics" (SwissMAP).
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Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
A Solvable Deformation of Quantum Mechanics
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title A Solvable Deformation of Quantum Mechanics
spellingShingle A Solvable Deformation of Quantum Mechanics
Grassi, A.
Mariño, M.
title_short A Solvable Deformation of Quantum Mechanics
title_full A Solvable Deformation of Quantum Mechanics
title_fullStr A Solvable Deformation of Quantum Mechanics
title_full_unstemmed A Solvable Deformation of Quantum Mechanics
title_sort solvable deformation of quantum mechanics
author Grassi, A.
Mariño, M.
author_facet Grassi, A.
Mariño, M.
publishDate 2019
language English
container_title Symmetry, Integrability and Geometry: Methods and Applications
publisher Інститут математики НАН України
format Article
description The conventional Hamiltonian H=p²+VN(x), where the potential VN(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 = 2cosh(p) + VN(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 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.
issn 1815-0659
url https://nasplib.isofts.kiev.ua/handle/123456789/210050
citation_txt A Solvable Deformation of Quantum Mechanics / A. Grassi, M. Mariño // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 111 назв. — англ.
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AT marinom asolvabledeformationofquantummechanics
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AT marinom solvabledeformationofquantummechanics
first_indexed 2025-12-07T21:24:12Z
last_indexed 2025-12-07T21:24:12Z
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