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 eigenvalues 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 Schrö dinger equation and, hence, for studying the dynamical evolution of the wave packets in the double-well-like potentials.
A. S. Surzhykov et al., "Quantum Dot Potentials: Symanzik Scaling, Resurgent Expansions, and Quantum Dynamics," Physical review B: Condensed matter and materials physics, vol. 74, no. 20, pp. 205317-1-205317-13, American Physical Society (APS), Nov 2006.
The definitive version is available at https://doi.org/10.1103/PhysRevB.74.205317
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