Solution of Quantum Quartic Potential Problems with Airy Fredholm Operators

TL;DR

This paper introduces Airy Fredholm operators that commute with quantum quartic Hamiltonians, providing new tools for high-precision numerical analysis and dual descriptions of complex quantum systems.

Contribution

It presents a novel class of Fredholm operators expressed via Airy functions that commute with quartic potential Hamiltonians, expanding analytical and numerical approaches.

Findings

Eigenvalues decay exponentially fast

Operators enable high-accuracy numerical methods

Dual descriptions relate to infinite chains and nonlocal quantum field theories

Abstract

Fredholm integral operators that commute with the Hamiltonians of certain quantum mechanical problems with quartic potentials are introduced. The operators are expressed in terms of an Airy function, and their eigenvalues fall off exponentially fast. They may help with high-accuracy numerical analysis, and their existence leads to dual descriptions in terms of infinite one-dimensional chains with variables on nodes, and weights on nodes and links. The systems discussed include the anharmonic quartic oscillator as well as multivariable potentials and higher dimensional systems, including certain quantum field theories with nonlocal interactions.