Classical and quantum mechanics in the Snyder space

The Snyder model is an example of noncommutative spacetime admitting a fundamental length scale and invariant under Lorentz transformations. Here, we consider its nonrelativistic counterpart, i.e. the Snyder model restricted to three-dimensional Euclidean space. We discuss the classical and the quantum mechanics of a free particle in this framework, and show that they strongly depend on the sign of a coupling constant λ, appearing in the fundamental commutators. If λ is negative, momenta are bounded, while for λ > 0 a minimal localization length arises. We also give the exact solution of the harmonic oscillator equations both in the classical and the quantum case, and show that its frequency is energy dependent.