Exceptional points in the Riesz-Feller Hamiltonian with an impenetrable rectangular potential

Abstract

The number of bound states in a standard rectangular potential well depends on the potential depth and width. In an impenetrable one-dimensional rectangular potential well, there are infinite bound states. In this work we study a non-Hermitian Riesz-Feller kinetic energy; i.e., the second-order derivative of the standard kinetic energy operator is replaced by a fractional, $\ensuremath{\alpha}\mathrm{th}$-order derivative. We show that for $\ensuremath{\alpha}<2$ a particle in an impenetrable one-dimensional rectangular potential well contains a finite number of bound states and an infinite number of metastable decaying states. The transitions from bound states to metastable decaying states occur at $\ensuremath{\alpha}$ values that correspond to exceptional points, for which two bound states coalesce. Our findings indicate that one can describe a transition of highly excited bound states to metastable decaying states, for example due to the interactions of atoms and molecules with the environment, by using the Riesz-Feller kinetic energy operator rather than the standard one.