Quantum dot potentials

Symanzik scaling, resurgent expansions, and quantum dynamics

Andrey Surzhykov, Michael Lubasch, Jean Zinn-Justin, U. Jentschura

Research output: Article

6 Citations (Scopus)

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 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 Schrödinger equation and, hence, for studying the dynamical evolution of the wave packets in the double-well-like potentials.

Original languageEnglish
Article number205317
JournalPhysical Review B - Condensed Matter and Materials Physics
Volume74
Issue number20
DOIs
Publication statusPublished - 2006

Fingerprint

Semiconductor quantum dots
quantum dots
scaling
expansion
instantons
Eigenvalues and eigenfunctions
Ground state
eigenvectors
Hamiltonians
Wave packets
ground state
energy
numerical integration
wave packets
energy spectra
eigenvalues
perturbation
coefficients
configurations

ASJC Scopus subject areas

  • Condensed Matter Physics

Cite this

Quantum dot potentials : Symanzik scaling, resurgent expansions, and quantum dynamics. / Surzhykov, Andrey; Lubasch, Michael; Zinn-Justin, Jean; Jentschura, U.

In: Physical Review B - Condensed Matter and Materials Physics, Vol. 74, No. 20, 205317, 2006.

Research output: Article

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