scholarly journals On the Liouville Property for Divergence Form Operators

1998 ◽  
Vol 50 (3) ◽  
pp. 487-496 ◽  
Author(s):  
Martin T. Barlow

AbstractIn this paper we construct a bounded strictly positive function σ such that the Liouville property fails for the divergence form operator L= ▽ (σ2▽). Since in addition Δσ/σ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schrödinger operators.

1998 ◽  
Vol 312 (4) ◽  
pp. 659-716 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Wolfhard Hansen

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Luca Fresta

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.


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