Bohr-Sommerfeld quantization rules in the semiclassical limit

Hagedorn, George A.; Robinson, Sam L.

doi:10.1088/0305-4470/31/50/009

Cited by 11 publications

(4 citation statements)

References 5 publications

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“…It is the simplicity of the Bohr Sommerfeld quantization scheme that has kept it popular [18,19] even now. We have shown that this condition can be effectively used for a wide variety of potentials where it has not traditionally been applied.…”

Section: Resultsmentioning

confidence: 99%

Bohr Sommerfeld quantisation and molecular potentials

2011

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We combine, within the Bohr Sommerfeld quantization rule, a systematic perturbation with asymptotic analysis of the action integral for potentials which support a finite number of bound states with E < 0 to obtain an interpolation formula for the energy eigenvalues. We find interpolation formulae for the Morse potential as well as potentials of the form V = V 0 a x 2k − a x k . For k = 6 i.e. the well known Lennard Jones potential this yields results within 1 per cent of the highly accurate numerical values. For the Morse potential this procedure yields the exact answer. We find that the result for the Morse potential which approaches zero exponentially is the k → ∞ limit of the Lennard Jones class of potentials.

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Section: Resultsmentioning

confidence: 99%

Bohr Sommerfeld quantisation and molecular potentials

2011

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“…In the semiclassical spectral analysis of non-selfadjoint operators in dimension 2, the work [25] elucidated the rôle played by certain flow invariant Lagrangian tori in the complexified phase space, and showed how the latter could be used to obtain complete asymptotic expansions for all eigenvalues in suitable domains in the complex spectral plane, in the spirit of the classical Bohr-Sommerfeld quantization condition, well known in dimension one [9], [4], [8]. In the context of small non-selfadjoint perturbations of selfadjoint operators, the work [25] has been pursued in a series of papers [15], [16], [17], [31], under the additional assumption that the classical flow of the unperturbed part should be periodic in a fixed energy shell.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Neglecting the Floquet conditions for the time being, we should have, 8) where the weight Φ is such that…”

Section: Microlocal Hilbert Spaces Outside Of a Tiny Neighborhood Of mentioning

confidence: 99%

Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension $2$

Hitrik¹,

Sjöstrand²

2008

Ann. Sci. École Norm. Sup.

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We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the strength of the non-selfadjoint perturbation ≫ h (or sometimes ≫ h 2 ) is not too large.

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“…We started from the analysis of the pair electronelectron distribution functions and related potentials of the mean force [13]. Our analysis, based on the semiclassical Bohr-Sommerfeld quantization condition [14,15], allows us to estimate the energy levels averaged over the canonical ensemble. We also demonstrate the stochastic nature of the energy levels disturbing the Maxwell distribution.…”

Section: Introductionmentioning

confidence: 99%

Bound States of the Exchange—Correlation Excitons in the Uniform Electron Gas by the Monte Carlo Simulations

2022

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The modified path integral representation of Wigner functions and the new Monte Carlo approach has been suggested to account for the impact of the interparticle interaction on the Pauli exclusion principle of fermions. This approach also allows to calculate the momentum distribution functions and to reduce the “sign problem” that is inaccessible to the standard path integral Monte Carlo methods. The obtained pair electron–electron distribution functions for the “uniform electron gas” demonstrate the short-range quantum ordering of electrons associated with exchange–correlation excitons. The exchange–correlation exciton is caused by the interaction of electrons with positively charged exchange holes and the excluded volume effect. The developed approach allows one to study the density–temperature range of the exciton arising, existence, and decay. Using the potential of the mean force and semiclassical Bohr–Sommerfeld quantization condition, we have demonstrated the existence of bound states disturbing the Maxwellian distribution and estimated their average energy levels. The exchange–correlation excitons have not been observed earlier in the standard path integral Monte Carlo simulations.

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Bohr-Sommerfeld quantization rules in the semiclassical limit

Cited by 11 publications

References 5 publications

Bohr Sommerfeld quantisation and molecular potentials

Bohr Sommerfeld quantisation and molecular potentials

Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension $2$

Bound States of the Exchange—Correlation Excitons in the Uniform Electron Gas by the Monte Carlo Simulations

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