# On the discreet spectrum of fractional quantum hydrogen atom in two dimensions

## TL;DR

This paper explores the spectral properties of a fractional 2D hydrogen atom, revealing that bound states exist only for Levy index μ > 1, and introduces analytical and numerical methods to approximate eigenvalues and eigenfunctions.

## Contribution

It provides the first analysis of the fractional 2D hydrogen atom spectrum, showing the existence conditions and lifting degeneracy, with new integral representations and approximation techniques.

## Key findings

- Discrete spectrum exists only for μ > 1.
- Orbital degeneracy is lifted, energy depends on m.
- Analytical approximations match numerical solutions well.

## Abstract

We consider a fractional generalization of two-dimensional (2D) quantum-mechanical Kepler problem corresponding to 2D hydrogen atom. Our main finding is that the solution for discreet spectrum exists only for $\mu>1$ (more specifically $1 < \mu \leq 2$, where $\mu=2$ corresponds to "ordinary" 2D hydrogenic problem), where $\mu$ is the L\'evy index. We show also that in fractional 2D hydrogen atom, the orbital momentum degeneracy is lifted so that its energy starts to depend not only on principal quantum number $n$ but also on orbital $m$. To solve the spectral problem, we pass to the momentum representation, where we apply the variational method. This permits to obtain approximate analytical expressions for eigenvalues and eigenfunctions with very good accuracy. Latter fact has been checked by numerical solution of the problem. We also found the new integral representation (in terms of complete elliptic integrals) of Schr\"odinger equation for fractional hydrogen atom in momentum space. We point to the realistic physical systems like bulk semiconductors as well as their heterostructures, where obtained results can be used.