The 3D Perturbed Schrödinger Hamiltonian in a Friedmann Flat Spacetime Testing the Primordial Universe in a Non Commutative Spacetime

In this paper we adapt the mathematical machinery presented in Albeverio et al. (Nanosystem 8(2), 153, 2017) to get, by means of the Laplace-Beltrami operator, the discrete spectrum of the Hamiltonian of the Schrödinger operator perturbed by an attractive 3D delta interaction in a Friedmann flat universe. In particular, as a consequence of the treatment in Albeverio et al. (Nanosystem 8(2), 153, 2017) suitable for a Minkowski spacetime, the discrete spectrum consisting only of the ground state and the first excited state in the above-mentioned cosmic framework can be regained. Thus, the coupling constant λ must be chosen as a function of the cosmic comoving time t as λ/a²(t), with λ being the one of the Hamiltonian studied in the aforementioned article. In this way we can introduce a time dependent delta interaction which is relevant in a primordial universe, where a(t) → 0 and becomes negligible at late times, with a(t) >> 1. We investigate, with such a model, quantum effects provided by point interactions in a strong gravitational regime near the big bang. In particular, as a physically interesting application, we present a method to depict, in a semi-classical approximation, a test particle in a (non commutative) quantum spacetime where, thanks to Planckian effects, the initial classical singularity disappears and, as a consequnce, a ground state with negative energy emerges. Conversely, in a scenario where the scale factor a(t) follows the classical trajectory, this ground state is unstable and thus must be physically ruled out.