Calogero model with Yukawa like interaction

TL;DR

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Contribution

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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 $\Psi (x) $ factorised as $x^{\epsilon}\Phi (x) $ with $\epsilon (\epsilon -1) =g$, we develop a technique to approach the spectrum of the generalized system and show that information on full spectrum is captured by $\Phi (x) $ and $\Phi ^{\prime \prime}(x) $ at the singular point $x=0$ of the potential. Convergence of $% \int dx| \Psi (x) | ^{2}$ requires $\epsilon >-{1/2}$ and is shown to be sensitive to the zero mode of $\Phi (x) $ at $x=0$. \textbf{Key words}: \textit{Hamitonian systems, quantum integrability, Calogero model, Yukawa like potential.}