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This paper presents a rigorous mathematical framework for solving quantum mechanical systems with quartic potentials using Fredholm integral operators expressed via Airy functions. By establishing an exact dual description through infinite chains and computing specific operator moments, the work advances theoretical understanding of energy spectra and wavefunction asymptotics in higher-dimensional systems. While highly relevant to fundamental physics, this research focuses on analytical methods rather than near-term applications to AI algorithms or quantum hardware engineering.
Analysis of Quartic Potential Problems Using Fredholm Integral Operators with Airy Functions
arXiv:2603.02374v4 Announce Type: replace-cross Abstract: Fredholm integral operators that commute with the Hamiltonians of certain quantum mechanical problems with quartic potentials are introduced in closed analytical form. The operators are expressed in terms of an Airy function, and their eigenvalues fall off exponentially fast. Using these operators we obtain an exact dual description in terms of infinite one-dimensional chains. The systems discussed include the anharmonic quartic oscillator as well as multivariable potentials and higher dimensional systems. We compute a few moments of the Fredholm operators, using matrix models in the higher dimensional case, and we show how those can be used to approximate the energy spectrum and obtain estimates on the asymptotic behavior of the wavefunctions.
Abstract:Fredholm integral operators that commute with the Hamiltonians of certain quantum mechanical problems with quartic potentials are introduced in closed analytical form. The operators are expressed in terms of an Airy function, and their eigenvalues fall off exponentially fast. Using these operators we obtain an exact dual description in terms of infinite one-dimensional chains. The systems discussed include the anharmonic quartic oscillator as well as multivariable potentials and higher dimensional systems. We compute a few moments of the Fredholm operators, using matrix models in the higher dimensional case, and we show how those can be used to approximate the energy spectrum and obtain estimates on the asymptotic behavior of the wavefunctions.
21 pages, 1 table. V4 is an expanded version, including an explicit form for the third moment of the Fredholm operator, explicit forms for the moments with nonzero quadratic terms, additional results for parity-odd states, additional results for multivariable systems including reduction to integrals over eigenvalues with final closed form expression in some cases, and additional references
High Energy Physics - Theory (hep-th); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
arXiv:2603.02374 [hep-th]
(or arXiv:2603.02374v4 [hep-th] for this version)
From: Ori J. Ganor [view email] [v1] Mon, 2 Mar 2026 20:28:57 UTC (17 KB) [v2] Mon, 9 Mar 2026 21:07:46 UTC (17 KB) [v3] Wed, 11 Mar 2026 22:55:15 UTC (15 KB) [v4] Tue, 22 Sep 2026 21:32:20 UTC (31 KB)
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