Calogero model with Yukawa like interaction

Abstract

We study an extension of one dimensional Calogero model involving strongly coupled and electrically charged particles. Besides Calogero term $\frac{g}{% 2x^{2}}$, there is an extra factor described by a Yukawa like coupling modeling short distance interactions. Mimicking Calogero analysis and using developments in formal series of the wave function $Ψ(x) $ factorised as $x^εΦ(x) $ with $ε(ε-1) =g$, we develop a technique to approach the spectrum of the generalized system and show that information on full spectrum is captured by $Φ(x) $ and $Φ^{\prime \prime}(x) $ at the singular point $x=0$ of the potential. Convergence of $% \int dx| Ψ(x) | ^{2}$ requires $ε>-{1/2}$ and is shown to be sensitive to the zero mode of $Φ(x) $ at $x=0$. \textbf{Key words}: \textit{Hamitonian systems, quantum integrability, Calogero model, Yukawa like potential.}