A regularized (XP)2 model
Year: 2024
Type: article
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, respectively. 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 ∼ 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 Schrödinger equation with cosh 2 x -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 Schrödinger 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 ∞ points, and furthermore, the Mathieu functions can be represented as the wave functions.
Source: Physica Scripta
Authors: Yu-Qi Chen, Zhao-Feng Ge
Institutions: Chinese Academy of Sciences, Institute of Theoretical Physics
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,285
OpenAlex ID: W4391464336