Uniqueness for discrete Schrodinger evolutions

TL;DR

This paper establishes a uniqueness result for solutions to the discrete Schrödinger equation, showing that rapid decay at two times implies the solution is trivial, and extends Hardy uncertainty principles to discrete settings.

Contribution

It introduces new uniqueness theorems for discrete Schrödinger evolutions, including sharp Hardy uncertainty principles for specific operators and methods for general potentials.

Findings

Solutions with rapid decay at two times are trivial.

Sharp Hardy uncertainty principles are established for free and compactly supported potentials.

Logarithmic convexity methods are used for general bounded potentials.

Abstract

We prove that if a solution of the discrete time-dependent Schr\"odinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shr\"odinger operator and for operators with compactly supported time-independent potentials a sharp analog of the Hardy uncertainty principle is obtained, using an argument based on the theory of entire functions. Logarithmic convexity of weighted norms is employed in the case of general real-valued time-dependent bounded potentials. In the latter case the result is not optimal.