Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials

Abstract

Classical and quantum walks on some finite paths are introduced. It is shown that these walks have explicit solutions given in terms of exceptional Krawtchouk polynomials and their properties are explored. In particular, fractional revival is shown to take place in the corresponding quantum walks.

The single-indexed exceptional Krawtchouk polynomials

“…Here we give the explicit form of the properties of the X-Krawtchouk polynomials { K(j,d) n (x)} with j = 2 and d = 2, that were used in [14]. For x = −1, 0, .…”

“…The ordinary Krawtchouk polynomials have applications in many areas including signal processing, coding theory and so on [12,17]. The authors recently found that the exceptional Krawtchouk (X-Krawtchouk) polynomials lead to interesting continuous-time classical and quantum walks [14]. With further applications in mind, this study motivates the examination of the properties of these X-Krawtchouk polynomials.…”