Narrow Results

We show that the time-dependent Schrödinger equation (TDSE) for a potential of the form V(x,t)=A(t)x²+B(t)x+C(t) and time-dependent mass can be transformed into the same TDSE with constant mass. We obtain an explicit formula relating solutions of the TDSE for time-dependent mass and for constant mass to each other.

Fractional calculus of variations (FCV) has recently attracted considerable attention as it is deeply related to the fractional quantization procedure. In this work, the FCV from extended Erdélyi-Kober fractional integral is constructed. Our main goal is to exhibit a general treatment for dissipative systems, in particular the harmonic oscillator (HO) that has time-dependent mass and time-dependent frequency. The general linear equation of damped Erdélyi-Kober harmonic oscillator is constructed from which a time-dependent mass generalized law was derived exhibiting different types of behavior. This relatively new time-dependent mass law permits us to point out several possible cases simultaneously in contrast to many models discussed in the literature and without making use of any types of fractional derivatives. Some results on Hamiltonian part, namely Hamilton equations for the damped HO were obtained and discussed in detail.

The problem of a spineless charged particle with a time-dependent decaying mass interacting with a Coulomb and an inverse quadratic potentials is considered. The Green’s function is explicitly evaluated. The energy levels as well as the wave functions for the bound states are exactly determined.