scholarly journals Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

2020 ◽  
pp. 1
Author(s):  
Rostislav Grigorchuk ◽  
Brian Simanek
Keyword(s):  
Cayley Graphs ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
Lamplighter Group

scholarly journals Supersymmetric Cluster Expansions and Applications to Random Schrödinger Operators

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Luca Fresta
Keyword(s):  
Density Of States ◽  
Local Density ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
Local Density Of States ◽  
Weak Disorder ◽  
Cluster Expansions ◽  
Strong Disorder ◽  
Lifshitz Tail

AbstractWe study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy. As an application, we establish Lifshitz-tail-type estimates for the local density of states and thus localization at weak disorder.

scholarly journals LOCALIZATION FOR ONE DIMENSIONAL LONG RANGE RANDOM HAMILTONIANS

1999 ◽  
Vol 11 (01) ◽  
pp. 103-135 ◽  
Author(s):  
VOJKAN JAKŠIĆ ◽  
STANISLAV MOLCHANOV
Keyword(s):  
Long Range ◽  
Spectral Properties ◽  
Schrödinger Operators ◽  
Pure Point ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Free Part ◽  
Random Hamiltonians

We study spectral properties of random Schrödinger operators hω=h0+vω(n) on l2(Z) whose free part h0 is long range. We prove that the spectrum of hω is pure point for typical ω whenever the off-diagonal terms of h0 decay as |i-j|-γ for some γ>8.

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