Second-order virial expansion for an atomic gas in a harmonic waveguide

Kristensen, Tom; Leyronas, X.; Pricoupenko, Ludovic

doi:10.1103/physreva.93.063636

Cited by 3 publications

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“…Beth and Uhlenbeck, in their famous paper [2], showed that this shift δb 2 is related to the scattering phase shift in 3D. Since then the result has been generalized in lower dimensions as well [3][4][5]. Recently, it was proven that δb 2 is the imprint of the quantum scale anomaly in dilute quantum gases in 1D with three-body and 2D with two-body local interactions [6,7] using a many-body path integral formalism [8].…”

Section: Introductionmentioning

confidence: 98%

Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for the one-dimensional δ -function potential

et al. 2019

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In contrast with the 3D result, the Beth-Uhlenbeck (BU) formula in 1D contains an extra −1/2 term. The origin of this −1/2 term is explained using a spectral density approach. To be explicit, a delta-function potential is used to show that the correction term arises from a pole of the density of states at zero energy. The spectral density method shows that this term is actually an artifact of the non-normalizability of the scattering states and an infrared cutoff regularization scheme has to be used to get the correct result in 1D. The formal derivation of the BU formula would miss this term since it ignores the effects of the boundary terms. While the result is shown for the delta-function potential, the method and result are valid for more general potentials. Additionally, the 1D Levinson's theorem can be extracted from the spectral density method using the asymptotic form of general potentials. The importance of the result lies in the fact that all these correction terms in 1D have a universal source: a pole at zero energy. Similar calculations using quantum field theoretical approaches (without explicit infrared cutoff regularization schemes) also show the same subtleties with the correction term originating from the zero energy scattering states (appendix A). 2

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