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Solution of Quantum Quartic Potential Problems with Airy Fredholm Operators

Ori J. Ganor

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.

Solution of Quantum Quartic Potential Problems with Airy Fredholm Operators

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.
Paper Structure (7 sections, 49 equations, 1 figure, 1 table)

This paper contains 7 sections, 49 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and setup
  2. Operators in the chain
  3. Steepest descent approximation
  4. Parity odd states
  5. Asymptotic behavior
  6. Other systems
  7. Discussion and outlook

Figures (1)

  • Figure 1: Comparison of steepest-descent and perturbation theory approximations to the ground state energy. The graphs depict the ground state energy ("Actual $E_0$") of the Hamiltonian \ref{['eqn:H']} with $\gl=1$ as a function of $\al$ in the range $0\le\al\le5$ together with the $0^{th}$ order steepest-descent approximation $\ApproxE_0$ from \ref{['eqn:ApproxE0']}. Also shown is the perturbative approximation including the $0^{th}$ and first order corrections ($E_0^{(0)}$ and $E_0^{(0)}+E_0^{(1)}$). Note that $\ApproxE_0$ agrees with first-order perturbation theory in the perturbative regime ($\al\gg 1$) and is much better in the nonperturbative regime $\al\lesssim1$. (For the purpose of graphing, each function is approximated by $25$ Bezier curves in LaTeX.)