Tail Bounds for the Dyson Series of Random Schrödinger Equations

Abstract We study Schrödinger equations on $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> and $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , $$d\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> with random potentials of strength $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> . Our main result gives tail bounds for the terms of the Dyson series that are effective at time scales on the order of $$\lambda ^{-2+\varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msup> </mml:math> . As corollaries, we obtain estimates on the frequency localization and spatial delocalization of approximate eigenfunctions in the spirit of Schlag et al. (J Anal Math 88(1):173–220, 2002) and Chen (J Stat Phys 120(1):279–337, 2005). These estimates also apply to Floquet states associated to time-periodic potentials. Our proof is elementary in that we use neither sophisticated harmonic analysis as in Schlag et al. (2002) nor diagrammatic arguments as in Chen (2005). Instead, we use only the noncommutative Khintchine inequality from random matrix theory combined with pointwise dispersive estimates for the free Schrödinger equation.