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Localisation and Delocalisation for a Simple Quantum Wave Guide with Randomness

Published in Birkhauser

2022

In this paper, we consider Schrödinger operators on M×Zd2$M\times {\mathbb{Z}}^{{d}_{2}}$, with M={M1,…,M2}d1$M=\{{M}_{1},\dots ,{M}_{2}{\}}^{{d}_{1}}$ (‘quantum wave guides’) with a ‘Γ$\mathrm{\Gamma}$-trimmed’ random potential, namely a potential which vanishes outside a subset Γ$\mathrm{\Gamma}$ which is periodic with respect to a sub-lattice. We prove that (under appropriate assumptions) for strong disorder these operators have *pure point spectrum *outside the set Σ0=σ(H0,Γc)${\text{\Sigma}}_{0}=\text{\sigma}({H}_{0,{\mathrm{\Gamma}}^{c}})$ where H0,Γc${H}_{0,{\mathrm{\Gamma}}^{c}}$ is the free (discrete) Laplacian on the complement Γc${\mathrm{\Gamma}}^{c}$ of Γ$\mathrm{\Gamma}$. We also prove that the operators have some *absolutely continuous spectrum* in an energy region E⊂Σ0$\mathcal{E}\subset {\text{\Sigma}}_{0}$. Consequently, there is a mobility edge for such models. We also consider the case −M1=M2=∞$-{M}_{1}={M}_{2}=\mathrm{\infty}$, i.e. Γ$\mathrm{\Gamma}$-trimmed operators on Zd=Zd1×Zd2${\mathbb{Z}}^{d}={\mathbb{Z}}^{{d}_{1}}\times {\mathbb{Z}}^{{d}_{2}}$. Again, we prove localisation outside Σ0${\text{\Sigma}}_{0}$ by showing exponential decay of the Green function GE+iη(x,y)${G}_{E+i\eta}(x,y)$ uniformly in η>0$\eta >0$. For *all* energies E∈E$E\in \mathcal{E}$ we prove that the Green’s function GE+iη${G}_{E+i\eta}$ is *not* (uniformly) in ℓ1${\ell}^{1}$ as η$\eta $ approaches 0. This implies that neither the fractional moment method nor multi-scale analysis *can* be applied here.

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About the journal

Journal | Annales Henri Poincare |
---|---|

Publisher | Birkhauser |

Open Access | Yes |