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

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.