The Potential Inversion Theorem

Abstract

Quantum lattice models describe a wide array of physical systems, and are a canonical way to numerically solve the Schrodinger equation. Here we prove the potential inversion theorem, which says that wavefunction probability in these models is preserved under the sign inversion of the potential energy as long as the initial conditions occupy strictly even or odd lattice sites and are real up to a global phase. This implies that electron pairs time evolve like positronium and therefore form bound states. We simulate the dynamics of these paradoxical electronium pairs and show that they are bound together more strongly if their charge is increased. We show how the potential inversion theorem illustrates several seemingly unrelated physical phenomena, including Bloch oscillations, localization, particle-hole symmetry, negative potential scattering, and magnetism.