Harmonic and anharmonic quantum-mechanical oscillators in noninteger dimensions

We present new results for time-independent solutions for a Schrödinger equation with noninteger dimension by considering different, harmonic and anharmonic, forms for the potential energy. The solutions obtained for these potentials are exact and expressed in terms of the special functions such as Laguerre and Gegenbauer polynomials, associated Legendre functions, and hypergeometric functions. Graphical comparison of the probability density function with the ones for two-dimensional and three-dimensional case is given. We derive the mean values rβsinδθ¯ for the harmonic oscillator in noninteger dimensions, which may be of interest in the perturbation theory for calculation of energy corrections. We consider anharmonic Kratzer potential energy function and we obtain bound and scattering states. Exact results in case of different forms of θ-dependent potentials are presented. In addition, they can be connected to rich variety of situations which enable us to model anisotropic interactions in real space.