Coulomb problem in non-commutative quantum mechanics

The aim of this paper is to find out how it would be possible for space non-commutativity (NC) to alter the quantum mechanics (QM) solution of the Coulomb problem. The NC parameter lambda is to be regarded as a measure of the non-commutativity - setting lambda = 0 which means a return to the standard quantum mechanics. As the very first step a rotationally invariant NC space R-lambda(3), an analog of the Coulomb problem configuration space (R-3 with the origin excluded) is introduced. R-lambda(3) is generated by NC coordinates realized as operators acting in an auxiliary (Fock) space F. The properly weighted Hilbert-Schmidt operators in F form H-lambda, a 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 a NC analog of the hamiltonian as a hermitian operator in H-lambda 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 E < 0 and low-energy scattering for E > 0 (both containing 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 lambda -> 0. (C) 2013 AIP Publishing LLC.