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The aim of this paper is to find out how would possible spacenon-commutativity (NC) alter the QM solution of the Coulomb problem. The NCparameter lambda is to be regarded as a measure of the non-commutativity -setting lambda = 0 means a return to the standard quantum mechanics. As thevery first step a rotationaly invariant non-commutative 3D space, an analog ofthe Coulomb problem configuration space (3D space with origin extracted), isintroduced. The non-commutative space in question is generated by NCcoordinates realized as operators acting in an auxiliary Fock space. Theproperly weighted Hilbert-Schmidt operators in this Fock space form an NCanalog of the Hilbert space of the wave functions. We will refer to them as"wave functions" also in the NC case. The definition of an NC analog of thehamiltonian as a hermitian operator is one of the key parts of this paper. Theresulting problem is exactly solvable. The full solution is provided, includingformulas for the bound states for negative energy E and low-energy scatteringfor positive E (both containig NC corrections analytic in lambda) and alsoformulas for high-energy scattering and unexpected bound states at ultra-highenergy (both containing NC corrections singular in lambda). All the NCcontributions to the known QM solutions either vanish or disappear in the limitof vanishing lambda.

Coulomb problem in NC quantum mechanics: Exact solution and non-perturbative aspects