Calogero model with Yukawa-like interaction

We study an extension of one-dimensional Calogero model involving strongly coupled and electrically charged particles. Besides Calogero term g2x2, 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) factorized as xɛΦ(x) with ɛ(ɛ−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″(x) at the singular point x=0 of the potential. Convergence of ∫dx|Ψ(x)|2 requires ɛ>−12 and is shown to be sensitive to the zero mode of Φ(x) at x=0.