A Regularized $(XP)^2$ Model

TL;DR

This paper explores a regularized $(XP)^2$ Hamiltonian model across classical, semi-classical, and quantum regimes, revealing discrete spectra, classical solutions via elliptic functions, and novel quantization forms with numerical spectral analysis.

Contribution

It introduces a regularized $(XP)^2$ model with multiple quantization forms, analyzes its spectral properties, and connects classical solutions to elliptic functions and Mathieu functions.

Findings

Discrete spectrum with logarithmic state density increase

Classical solutions described by elliptic functions

Numerical spectra show minor differences among quantization forms

Abstract

We investigate a dynamic model described by the classical Hamiltonian $H(x,p)=(x^2+a^2)(p^2+a^2)$, where $a^2>0$, in classical, semi-classical, and quantum mechanics. In the high-energy $E$ limit, the phase path resembles that of the $(XP)^2$ model. However, the non-zero value of $a$ acts as a regulator, removing the singularities that appear in the region where $x, p \sim 0$, resulting in a discrete spectrum characterized by a logarithmic increase in state density. Classical solutions are described by elliptic functions, with the period being determined by elliptic integrals. In semi-classical approximation, we speculate that the asymptotic Riemann-Siegel formula may be interpreted as summing over contributions from multiply phase paths. We present three different forms of quantized Hamiltonians, and reformulate them into the standard Schr\" odinger equation with $\cosh 2x$-like potentials. Numerical evaluations of the spectra for these forms are carried out and reveal minor differences in energy levels. Among them, one interesting form possesses Hamiltonian in the Schr\" odinger equation that is identical to its classical version. In such scenarios, the eigenvalue equations can be expressed as the vanishing of the Mathieu functions' value at $i\infty$ points, and furthermore, the Mathieu functions can be represented as the wave functions.