Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation

In this paper we consider Schrodinger operators on $M \times \mathbb{Z}^{d_2}$, with $M=\{M_{1}, \ldots, M_{2}\}^{d_1}$ (`quantum wave guides') with a `$\Gamma$-trimmed' random potential, namely a potential which vanishes outside a subset $\Gamma$ which is periodic with respect to a sub lattice. We prove that (under appropriate assumptions) for strong disorder these operators have \emph{pure point spectrum } outside the set $\Sigma_{0}=\sigma(H_{0,\Gamma^{c}})$ where $H_{0,\Gamma^{c}} $ is the free (discrete) Laplacian on the complement $\Gamma^{c} $ of $\Gamma $. We also prove that the operators have some \emph{absolutely continuous spectrum} in an energy region $\mathcal{E}\subset\Sigma_{0}$. Consequently, there is a mobility edge for such models. We also consider the case $-M_{1}=M_{2}=\infty$, i.~e.~ $\Gamma $-trimmed operators on $\mathbb{Z}^{d}=\mathbb{Z}^{d_1}\times\mathbb{Z}^{d_2}$. Again, we prove localisation outside $\Sigma_{0} $ by showing exponential decay of the Green function $G_{E+i\eta}(x,y) $ uniformly in $\eta>0 $. For \emph{all} energies $E\in\mathcal{E}$ we prove that the Green's function $G_{E+i\eta} $ is \emph{not} (uniformly) in $\ell^{1}$ as $\eta$ approaches $0$. This implies that neither the fractional moment method nor multi scale analysis \emph{can} be applied here.

[1]  W. Kirsch,et al.  Spectral statistics for Anderson models with sporadic potentials , 2017, Journal of Spectral Theory.

[2]  Michael Aizenman,et al.  Random Operators: Disorder Effects on Quantum Spectra and Dynamics , 2015 .

[3]  S. Sodin,et al.  The trimmed Anderson model at strong disorder: localisation and its breakup , 2014, 1409.8009.

[4]  Constanza Rojas-Molina The Anderson model with missing sites , 2013, 1302.3640.

[5]  A. Elgart,et al.  Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models , 2013, 1301.5268.

[6]  M. Aizenman,et al.  Resonant delocalization for random Schr\"odinger operators on tree graphs , 2011, 1104.0969.

[7]  A. Klein,et al.  Absolutely continuous spectrum for random Schrödinger operators on the Bethe strip , 2011, 1101.4328.

[8]  D. Krejčiřík Twisting versus bending in quantum waveguides , 2007, 0712.3371.

[9]  D. Hasler,et al.  Absolutely Continuous Spectrum for the Anderson Model on a Tree: A Geometric Proof of Klein’s Theorem , 2005, math-ph/0511050.

[10]  V. Jaksic CORRUGATED SURFACES AND A.C. SPECTRUM , 2000 .

[11]  W. Kirsch,et al.  Anderson model with decaying randomness: mobility edge , 2000 .

[12]  P. Stollmann,et al.  Multi-scale analysis implies strong dynamical localization , 1999, math-ph/9912002.

[13]  A. Klein Extended States in the Anderson Model on the Bethe Lattice , 1998 .

[14]  W. Kirsch Wegner estimates and Anderson localization for alloy-type potentials , 1996 .

[15]  M. Krishna Anderson model with decaying randomness existence of extended states , 1990 .

[16]  H. Englisch,et al.  Random Hamiltonians ergodic in all but one direction , 1990 .

[17]  A. Klein,et al.  A new proof of localization in the Anderson tight binding model , 1989 .

[18]  Abel Klein,et al.  Random Schrödinger operators , 2008 .

[19]  B. Simon,et al.  Schrödinger Semigroups , 2007 .

[20]  D. Hundertmark,et al.  Spectral Theory of Sparse Potentials , 2000 .

[21]  F. Martinelli,et al.  On the ergodic properties of the specrum of general random operators. , 1982 .