Unitary quantum matter-bounce in a universe with a positive cosmological constant

We analyze the Wheeler-DeWitt quantization of a spatially flat Friedmann–Lemaïtre–Robertson–Walker universe containing pressureless dust and a positive cosmological constant ( Λ > 0 Λ > 0 ). Following relational time framework, we establish a direct mathematical correspondence between the cosmological Hamiltonian and the radial Schrödinger equation for the scattering states of the non-relativistic hydrogen atom. This exact solvability allows us to rigorously construct the physical Hilbert space and ensure the self-adjointness of the Hamiltonian. As a concrete result, we show that the wave packets unitarily evolve depicting a non-singular quantum bounce, systematically replacing the classical Big Bang singularity. Finally, we discuss the physical relevance of this exact solution within the matter-bounce scenario. We demonstrate that this framework provides a robust quantum origin for a bounce during a dust-dominated contracting phase—a necessary prerequisite for generating a scale-invariant spectrum of primordial perturbations—derived from the unitary dynamics of the quantized background.