Spectrum and eigenfunctions in the momentum representation for 1D Coulomb-like potential with deformed Heisenberg algebra leading to minimal length are found exactly. It is shown that correction due to the deformation is proportional to square root of the deformation parameter. We obtain the same spectrum using Bohr-Sommerfeld quantization condition.

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Exact solution of Dirac equation for a particle whose potential energy and mass are inversely proportional to the distance from the force centre has been found. The bound states exist provided the length scale a which appears in the expression for the mass is smaller than the classical electron radius e 2 /mc 2 . Furthermore, bound states also exist for negative values of a even in the absence of the Coulomb interaction. Quasirelativistic expansion of the energy has been carried out, and a modified expression for the fine structure of energy levels has been obtained. The problem of kinetic energy operator in the Schrödinger equation is discussed for the case of positiondependent mass. In particular, we have found that for highly excited states the mutual ordering of the inverse mass and momentum operator in the non-relativistic theory is not important.

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