Quantum Dot Potentials: Symanzik Scaling, Resurgent Expansions and Quantum Dynamics

TL;DR

This paper explores the energy spectrum and eigenfunctions of a special class of double-well-like quantum potentials, focusing on the Fokker-Planck potential, by comparing resurgent weak-coupling expansions with strong-coupling expansions across a wide parameter range.

Contribution

It introduces a detailed comparison between resurgent weak-coupling series and strong-coupling expansions for the Fokker-Planck potential, extending analysis into the strong-coupling regime.

Findings

Overlap between weak- and strong-coupling regimes identified

First coefficients of strong-coupling expansion determined

Eigenfunctions computed for dynamical wavepacket analysis

Abstract

This article is concerned with a special class of the ``double-well-like'' potentials that occur naturally in the analysis of finite quantum systems. Special attention is paid, in particular, to the so-called Fokker-Planck potential, which has a particular property: the perturbation series for the ground-state energy vanishes to all orders in the coupling parameter, but the actual ground-state energy is positive and dominated by instanton configurations of the form exp(-a/g), where a is the instanton action. The instanton effects are most naturally taken into account within the modified Bohr-Sommerfeld quantization conditions whose expansion leads to the generalized perturbative expansions (so-called resurgent expansions) for the energy values of the Fokker-Planck potential. Until now, these resurgent expansions have been mainly applied for small values of coupling parameter g, while much less attention has been paid to the strong-coupling regime. In this contribution, we compare the energy values, obtained by directly resumming generalized Bohr-Sommerfeld quantization conditions, to the strong-coupling expansion, for which we determine the first few expansion coefficients in powers of g^(-2/3). Detailed calculations are performed for a wide range of coupling parameters g and indicate a considerable overlap between the regions of validity of the weak-coupling resurgent series and of the strong-coupling expansion. Apart from the analysis of the energy spectrum of the Fokker-Planck Hamiltonian, we also briefly discuss the computation of its eigenfunctions. These eigenfunctions may be utilized for the numerical integration of the (single-particle) time-dependent Schroedinger equation and, hence, for studying the dynamical evolution of the wavepackets in the double-well-like potentials.