Uniqueness for the Schrödinger equation on graphs with potential vanishing at infinity

Abstract We study the uniqueness of solutions to the stationary Schrödinger equation with potential on infinite graphs, within suitable weighted $$\ell ^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> spaces. The potential is allowed to vanish at infinity at a controlled rate. Our results extend those in [20] by considering a broader class of potentials, by removing the assumption that the potential is bounded away from zero.