Two rotating particles interacting via two-body Gaussian potential harmonically confined in two spatial dimensions

TL;DR

This paper investigates the energy spectrum of two rotating bosons in two dimensions interacting via Gaussian potential, analyzing effects of interaction parameters, angular momentum, and potential range on ground state energies.

Contribution

It provides a detailed analysis of how Gaussian interaction range and strength influence the energy spectrum and ground state properties of rotating two-boson systems, comparing with delta-function interactions.

Findings

Ground state energy becomes independent of interaction range for non-interacting case.

Energy diverges negatively for strong attractive interactions in the s-wave state.

Gaussian potential allows convergence with smaller Hilbert space compared to delta-function potential.

Abstract

We study two spinless bosons interacting via two-body Gaussian potential subjected to an externally impressed rotation about an axis confined in a harmonic trap in two-spatial dimensions. We obtain a transcendental equation for the relative angular momentum $|m|$ state with various values of the two-body interaction range $\sigma$ and the two-body interaction strength $g_{2}$ to study the resulting energy spectrum and analyze the role of Hilbert space dimensions $\widetilde{N}$. We compare results for both attractive and repulsive interaction for $\delta$-function potential and Gaussian potential for various values of interaction range. We study the effects of interaction parameters and relative angular momentum on the ground state energy and its various components, namely, kinetic energy, trap potential and interaction potential. For a given $|m|$ and non-interacting case, we observe that the ground state energy becomes independent of interaction range. However, for a given relative angular momentum and interaction strength $g_{2}>0$, there is an increase in ground state energy with an increase in interaction range. Below the interaction strength $g_{2}V(r)\leq -1$, ground state energy diverges to physically unacceptable negative-infinity for $|m|=0$ state. Further, for $|m|=1$, the ground state energy becomes independent of the interaction strength. For a $|m|$, we present a comparative study between the Gaussian interaction potential and the $\delta$-function potential. Further, we observe that for a given $g_{2}$ and $|m|$, for $\delta$-function potential {\it i.e.} $\sigma\to 0$, to achieve the convergence of ground state energy, we require a considerably large critical Hilbert space. Whereas, in the case of Gaussian interaction potential with $\sigma\to 1$, the ground state energy converges for a considerably small critical Hilbert space.