### 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 language | English |
---|---|

Article number | 205317 |

Journal | Physical Review B - Condensed Matter and Materials Physics |

Volume | 74 |

Issue number | 20 |

DOIs | |

Publication status | Published - 2006 |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

*Physical Review B - Condensed Matter and Materials Physics*,

*74*(20), [205317]. https://doi.org/10.1103/PhysRevB.74.205317

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

Research output: Article

*Physical Review B - Condensed Matter and Materials Physics*, vol. 74, no. 20, 205317. https://doi.org/10.1103/PhysRevB.74.205317

}

TY - JOUR

T1 - Quantum dot potentials

T2 - Symanzik scaling, resurgent expansions, and quantum dynamics

AU - Surzhykov, Andrey

AU - Lubasch, Michael

AU - Zinn-Justin, Jean

AU - Jentschura, U.

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevB.74.205317

DO - 10.1103/PhysRevB.74.205317

M3 - Article

VL - 74

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 0163-1829

IS - 20

M1 - 205317

ER -