  The aim of this paper is to find out how would possible space
non-commutativity (NC) alter the QM solution of the Coulomb problem. The NC
parameter lambda is to be regarded as a measure of the non-commutativity -
setting lambda = 0 means a return to the standard quantum mechanics. As the
very first step a rotationaly invariant non-commutative 3D space, an analog of
the Coulomb problem configuration space (3D space with origin extracted), is
introduced. The non-commutative space in question is generated by NC
coordinates realized as operators acting in an auxiliary Fock space. The
properly weighted Hilbert-Schmidt operators in this Fock space form an NC
analog 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 the
hamiltonian as a hermitian operator is one of the key parts of this paper. The
resulting problem is exactly solvable. The full solution is provided, including
formulas for the bound states for negative energy E and low-energy scattering
for positive E (both containig NC corrections analytic in lambda) and also
formulas for high-energy scattering and unexpected bound states at ultra-high
energy (both containing NC corrections singular in lambda). All the NC
contributions to the known QM solutions either vanish or disappear in the limit
of vanishing lambda.
