Quantum Newtonian cosmology and the biconfluent Heun functions

TL;DR

This paper derives exact solutions for a quantum particle in a Newtonian universe with a cosmological constant, using biconfluent Heun functions, revealing how the cosmological constant influences energy levels and wave functions.

Contribution

It provides the first explicit solutions of the Schrödinger equation in this cosmological context using biconfluent Heun functions, including polynomial solutions and energy spectra.

Findings

Exact wave functions expressed in terms of biconfluent Heun functions.

Explicit first six Heun polynomials for the system.

Energy spectrum analogous to the harmonic oscillator, influenced by the cosmological constant.

Abstract

We obtain the exact solution of the Schr\"odinger equation for a particle (galaxy) moving in a Newtonian universe with a cosmological constant, which is given in terms of the biconfluent Heun functions. The first six Heun polynomials of the biconfluent function are written explicitly. The energy spectrum which resembles the one corresponding to the isotropic harmonic oscillator is also obtained. The wave functions as well as the energy levels codify the role played by the cosmological constant.