Typical Macroscopic Long-Time Behavior for Random Hamiltonians

Abstract We consider a closed macroscopic quantum system in a pure state $$\psi _t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ψ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces $${\mathcal {H}}_\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> </mml:math> (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of $$\psi _t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ψ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> looks like macroscopically, specifically on how much of $$\psi _t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ψ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> lies in each $${\mathcal {H}}_\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> </mml:math> . Previous bounds concerned the absolute error for typical $$\psi _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ψ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and/or t and are valid for arbitrary Hamiltonians H ; now, we provide bounds on the relative error, which means much tighter bounds, with probability close to 1 by modeling H as a random matrix, more precisely as a random band matrix (i.e., where only entries near the main diagonal are significantly nonzero) in a basis aligned with the macro spaces. We exploit particularly that the eigenvectors of H are delocalized in this basis. Our main mathematical results confirm the two phenomena of generalized normal typicality (a type of long-time behavior) and dynamical typicality (a type of similarity within the ensemble of $$\psi _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ψ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> from an initial macro space). They are based on an extension we prove of a no-gaps delocalization result for random matrices by Rudelson and Vershynin (Geom Funct Anal 26:1716–1776, 2016).