Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for 1D delta-function potential

TL;DR

This paper explains the origin of the extra -1/2 term in the 1D Beth-Uhlenbeck formula using spectral density methods, revealing it as an artifact of non-normalizable states and boundary effects, with implications for general potentials.

Contribution

It introduces a spectral density approach to derive the correction term in the 1D BU formula, highlighting the role of zero-energy poles and boundary conditions, applicable to various potentials.

Findings

The -1/2 term arises from a zero-energy pole in the spectral density.

Infrared cutoff regularization is necessary to obtain correct results in 1D.

The method generalizes to other potentials and relates to Levinson's theorem.

Abstract

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).