We investigate consequences of space non-commutativity in quantum mechanics of the hydrogen atom. We introduce rotationally invariant noncommutative spaceR 3 0 -an analog of the hydrogen atom (H-atom) configuration space R 3 0 = R 3 \ {0}. The spaceR 3 0 is generated by noncommutative coordinates realized as operators in an auxiliary (Fock) space F. We introduce the Hilbert spaceĤ of wave functionsψ formed by properly weighted Hilbert-Schmidt operators in F. Finally, we define an analog of the H-atom Hamiltonian inR 3 0 and explicitly determine the bound state energies E λ n and the corresponding eigenstatesψ λ njm . The Coulomb scattering problem inR 3 0 is under study.

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The main point of this paper is to examine a "hidden" dynamical symmetry connected with the conservation of Laplace-Runge-Lenz vector (LRL) in the hydrogen atom problem solved by means of noncommutative quantum mechanics (NCQM). The basic features of NCQM will be introduced to the reader, the key one being the fact that the notion of a point, or a zero distance in the considered configuration space, is abandoned and replaced with a "fuzzy" structure in such a way that the rotational invariance is preserved.The main facts about the conservation of LRL vector in both classical and quantum theory will be reviewed. Finally we will search for an analogy in the NCQM, provide our results and their comparison with the QM predictions. The key notions we are going to deal with are non-commutative space, Coulomb-Kepler problem and symmetry.

Recently we formulated the Coulomb problem in a rotationally invariant NC configuration space specified by NC coordinates <em>x_i, i</em> = 1, 2, 3, satisfying commutation relations<em> [x_i, x_j ] = 2iλε_ijkx_k</em> (<em>λ</em> being our NC parameter). We found that the problem is exactly solvable: first we gave an exact simple formula for the energies of the negative bound states <em>E^λ_n</em> &lt; 0 (n being the principal quantum number), and later we found the full solution of the NC Coulomb problem. In this paper we present an exact calculation of the NC Coulomb scattering matrix <em>S^λ_j (E)</em> in the <em>j</em>-th partial wave. As the calculations are exact, we can recognize remarkable non-perturbative aspects of the model: 1) energy cut-off — the scattering is restricted to the energy interval 0 &lt; <em>E</em> &lt; <em>E</em>_crit = 2/<em>λ</em>²; 2) the presence of two sets of poles of the S-matrix in the complex energy plane — as expected, the poles at negative energy <em>E</em>^I_<em>λ</em>n = <em>E</em>^<em>λ</em>_n for the Coulomb attractive potential, and the poles at ultra-high energies <em>E</em>^II_<em>λ</em>n = <em>E</em>_crit − <em>E^λ</em>_n for the Coulomb <em>repulsive</em> potential. The poles at ultra-high energies disappear in the commutative limit <em>λ</em>→0.

The object under scrutiny is the dynamical symmetry connected with conservation of the Laplace-Runge-Lenz vector (LRL) in the hydrogen atom problem solved by means of noncommutative quantum mechanics (NCQM). The considered noncommutative configuration space has such a “fuzzy”<br />structure that the rotational invariance is not spoilt. An analogy with the LRL vector in the NCQM is brought to provide our results and also a comparison with the standard QM predictions.