2019/04/22 by Valmir Bucaj, David Damanik, Jake Fillman +4 · 6 citations
Mathematics · Physics and Astronomy · #Spectral Theory in Mathematical Physics #Quantum chaos and dynamical systems #Quantum many-body systems
paper · pdf · doi:10.1090/tran/7832
We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg’s theorem. That is, a Schrödinger operator in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l squared left-parenthesis double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">ℓ 2(\mathbb Z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose potential is given by independent, identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions, and its unitary group exhibits exponential off-diagonal decay, uniformly in time. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogues of these models.