# The $x_i$-eigenvalue problem on some new fuzzy spheres

## TL;DR

This paper investigates the eigenvalue problem for coordinate observables on fuzzy spheres, showing their spectra and eigenvectors approximate those of quantum particles constrained on the classical sphere.

## Contribution

It analyzes the eigenvalues and eigenvectors of coordinate operators on new fuzzy spheres, demonstrating their properties align with classical quantum particle behavior on spheres.

## Key findings

- Eigenvalues and eigenvectors approximate classical coordinate operators.
- Spectral properties support fuzzy spheres as quantum geometric models.
- Results validate fuzzy spheres as effective approximations of quantum particles on spheres.

## Abstract

We study the eigenvalue equation for the 'Cartesian coordinates' observables $x_i$ on the fully $O(2)$-covariant fuzzy circle $\{S^1_\Lambda\}_{\Lambda\in\mathbb{N}}$ ($i=1,2$) and on the fully $O(3)$-covariant fuzzy 2-sphere $\{S^2_\Lambda\}_{\Lambda\in\mathbb{N}}$ ($i=1,2,3$) introduced in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451]. We show that the spectrum and eigenvectors of $x_i$ fulfill a number of properties which are expected for $x_i$ to approximate well the corresponding coordinate operator of a quantum particle forced to stay on the unit sphere.