Quantum mechanics on one-dimentional Cayley-Klein geometries

Two exactly-solvable quantum mechanical problems, namely harmonic oscillator and Coulomb systems in one-dimensional spaces of constant curvature, are dis- cussed in the context of the unified description of Caley-Klein geometries. Gen- eral expressions for the discrete eigenvalues and corresponding eigenfunctions of Schrödinger operator are obtained by factorization method. In both cases the space curvature appears in these expressions as a parameter. The energy levels of Coulomb particle are shifted in curved space as compared with flat space up or down, depending on curvature sign: positive or negative. The same is held for the oscillator.