Two trapped particles interacting by a finite-ranged two-body potential in two spatial dimensions

We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-ranged Gaussian-shaped potential in two spatial dimensions. We derive an approximate transcendental equation and solved it using exact diagonalization to study the resulting energy spectrum as a function of the inter-particle interaction strength, relative angular momenta and analyse the role of Hilbert space. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. Complementary, we also explicitly verify by a variational treatment that in the zero-range limit the positive delta potential in two dimensions only reproduces the non-interacting results, if the Hilbert space in not truncated. Finally, we established and discuss the connection between our finite-range treatment and regularized zero-range results from the literature.