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arXiv · 2401.07262

Eigenfunctions and Quantum Transport with Applications to Trimmed Schrodinger Operators

Peter D> Hislop·Werner Kirsch·M. Krishna
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Abstract

We provide a simple proof of dynamical delocalization, that is, time-increasing lower bounds on quantum transport for discrete, one-particle Schrodinger operators on $\ell^2 (\mathbb{Z}^d)$, provided solutions to the Schrodinger equation satisfy certain growth conditions. The proof is based on basic resolvent identities and the Combes-Thomas estimate on the exponential decay of the Green's function. As a consequence, we prove that generalized eigenfunctions for energies outside the spectrum of $H$ must grow exponentially in some directions. We also prove that if $H$ has any absolutely continuous spectrum, then the Schrodinger operator exhibits dynamical delocalization. We apply the general result to $\Gamma$-trimmed Schrodinger operators, with periodic $\Gamma$, and prove dynamical delocalization for these operators. These results also apply to the $\Gamma$-trimmed Anderson model, providing a random, ergodic model exhibiting both dynamical localization in an energy interval and dynamical delocalization.

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Peter D> Hislop, Werner Kirsch, M. Krishna. 2024-01-14. Eigenfunctions and Quantum Transport with Applications to Trimmed Schrodinger Operators. https://arxiv.org/abs/2401.07262

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