Spectra of PT-symmetric fractional Schrödinger equations with multiple quantum wells

Abstract

We evaluate the eigenvalues of a type of one-dimensional PT-symmetric fractional Schrodinger equation with multiple quantum wells potential profile. By using a finite-difference scheme, we solve the fractional Schrodinger equation and present the algorithm. We study the effects of different parameters on the pairwise coalescence of eigenvalues. We show that by using the mentioned parameters, we can tune the position of the pairwise coalescence of the eigenvalues and the surface area between the two eigenvalues that intersect. An interesting phenomenon is that a small value of the fractionality as much as 0.15 can destroy the pairwise coalescence of eigenvalues and produce a single energy level. We also, consider the Hofstadter butterfly of a PT-symmetric one-dimensional system and show that by increasing the intensity of the potential imaginary part, we can kill the butterfly.