Phase transitions of energy and wave functions and bound states in the continuum

Abstract

This paper studies a particle subjected to an infinite potential well in the circumstance of a fractional dimensional L\'evy path. To obtain analytic expression for the wave functions and energy levels, we introduce the fractional corresponding operator and a generalized de Moivre's theorem. Phase transitions of the energy and wave functions are found when the L\'evy path dimension changes from integer to noninteger in nature. More importantly, we demonstrate the existence of stable bound states in the continuum in a simple potential. The results predict a phenomenon in which all bound states energy levels of the particle are continuous and the particle remains in bound states. This phenomenon can be demonstrated that this is a characteristic phenomenon of a fractional system. This phenomenon provides both an a priori criterion for theoretically describing an unknown quantum system with fractional derivatives and a sufficient condition for verifying the preparation of a fractional quantum system in experiment. Finally, we compare our results for fractional quantum systems with the existing results and explain the cause of the reported phenomenon.