Effective Schrodinger equation for one-dimensional systems with rapidly oscillating boundary conditions

The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.