Bound and scattering states of extended Calogero model with an additional PT invariant interaction

TL;DR

This paper explores PT-invariant extensions of the Calogero model, revealing real spectra, bound states with generalized exclusion statistics, and scattering states with phase shifts, highlighting differences in statistics parameters due to nonhermitian interactions.

Contribution

It introduces PT-invariant nonhermitian interactions into the Calogero model, analyzing their effects on spectra, bound states, and scattering behavior, which was not previously studied.

Findings

Energy eigenvalues remain real despite nonhermitian PT-invariant interactions.

Bound states exhibit generalized exclusion statistics.

Scattering states have phase shifts related to exchange statistics, with parameters differing from exclusion statistics.

Abstract

Here we discuss two many-particle quantum systems, which are obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the Calogero model with and without confining potential. It is shown that the energy eigenvalues are real for both of these quantum systems. For the case of extended Calogero model with confining potential, we obtain discrete bound states satisfying generalised exclusion statistics. On the other hand, the extended Calogero model without confining term gives rise to scattering states with continuous spectrum. The scattering phase shift for this case is determined through the exchange statistics parameter. We find that, unlike the case of usual Calogero model, the exclusion and exchange statistics parameter differ from each other in the presence of PT invariant interaction.