p-Adic quantum mechanics, infinite potential wells, and continuous-time quantum walks

This paper discusses a [Formula: see text]-adic version of the infinite potential well in quantum mechanics (QM). This model describes the confinement of a particle in a [Formula: see text]-adic ball. We rigorously solve the Cauchy problem for the Schrödinger equation and determine the stationary solutions. The [Formula: see text]-adic balls are fractal objects. By dividing a [Formula: see text]-adic ball into a finite number of sub-balls and using the wavefunctions of the infinite potential well, we construct a continuous-time quantum walk (CTQW) on a fully connected graph, where each vertex corresponds to a sub-ball in the partition of the original ball. In this way, we establish a connection between [Formula: see text]-adic QM and quantum computing.