Direct scaling analysis of localization in single-particle quantum systems on graphs with diagonal disorder
The author considers random discrete Schrödinger operators (Anderson-type Hamiltonians) \(H(\omega)\) on a graph \(\mathcal Z\), i.e., operators acting in the Hilbert space \(\ell^2({\mathcal Z})\), of the form \[ H(\omega)=-\Delta+gV(x,\omega), \] where \(g \in {\mathbb R}, V : {\mathcal Z} \times \Omega \to {\mathbb R}\) is a random field relative to some probability space \((\Omega,{\mathcal F}, {\mathbb P})\), and \(\Delta\) is the so-called canonical graph Laplacian on \(\mathcal Z\). The passage from the traditional Anderson model on a lattice to the framework of graphs is motivated by the analysis of \(N\)-particle systems with quantum-mechanical symmetries (bosonic and fermionic systems). The aim of the author is to obtain spectral localization (pure point spectrum and exponential decay of eigenfunctions) and dynamical localization (non spreading of the wave packets) in arbitrarily large finite subsets of the graph, which imply the spectral and strong dynamical localization in the entire graph. The traditional tool in the study of these problem is the Multi-Scale Analysis proposed by \textit{J. Fröhlich} and \textit{T. Spencer} [Commun. Math. Phys. 124, No. 2, 285--299 (1989; Zbl 0698.60051)] and \textit{H. von Dreifus} and \textit{A. Klein} [Commun. Math. Phys. 88, 151--184 (1983; Zbl 0519.60066)]. In the paper under review, the author introduces a simplified version of the Multi-Scale Analysis of Anderson models on a lattice and, more generally, on a countable graph with polynomially bounded growth of balls. The author demonstrates applications of this new scaling procedure to a particular case where the external potential is generated by an independent and identically distributed random field. He also considers weakly disordered media, where localization can be established only in a specific energy band, and models with correlated, but strongly mixing random potentials. These results are used to prove the strong dynamical localization (dynamical localization not only with probability one, but in expectation). The proof is based on the bootstrap Multi-Scale Analysis developed by \textit{F. Germinet} and \textit{A. Klein} [Commun. Math. Phys. 222, No. 2, 415--448 (2001; Zbl 0982.82030)].
scientific article; zbMATH DE number 6121940
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