The 3D perturbed Schrödinger Hamiltonian in a Friedmann flat spacetime testing the primordial universe in a non commutative spacetime

Abstract: In this paper we adapt the mathematical machinery presented in \cite{P1} to get, by means of the Laplace-Beltrami operator, the discrete spectrum of the Hamiltonian of the Schr\"{o}dinger operator perturbed by an actractive 3D delta interaction in a Friedmann flat universe. In particular, as a consequence of the treatment in \cite{P1} suitable for a Minkowski spacetime, the discrete spectrum of the ground state and the first exited state in the above mentioned cosmic framework can be regained. Thus, the coupling constant $\lambda$ must be choosen as a function of the cosmic comooving time $t$ as ${\lambda}/a^{2}(t)$, with ${\lambda}$ be the one of the static Hamiltonian studied in \cite{P1}. In this way we can introduce a time dependent delta interaction which is relevant in a primordial universe, where $a(t)\rightarrow 0$ and becomes negligible at late times, with $a(t)>>1$. We investigate, with the so obtained 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 instable and thus physically cannot be carried out.