Eigenvalue Splitting of Polynomial Order for a System of Schrödinger Operators with Energy-Level Crossing

2021 ◽  
Author(s):  
Marouane Assal ◽  
Setsuro Fujiié
Keyword(s):  
Energy Level ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Level Crossing

scholarly journals Resonance free domain for a system of Schrödinger operators with energy-level crossings

2020 ◽  
pp. 2150007
Author(s):  
Kenta Higuchi
Keyword(s):  
Energy Level ◽  
Differential Operators ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Level Crossings ◽  
Diagonal Part ◽  
Free Domain ◽  
Small Interactions ◽  
Directed Cycles

We consider a [Formula: see text] system of 1D semiclassical differential operators with two Schrödinger operators in the diagonal part and small interactions of order [Formula: see text] in the off-diagonal part, where [Formula: see text] is a semiclassical parameter and [Formula: see text] is a constant larger than [Formula: see text]. We study the absence of resonance near a non-trapping energy for both Schrödinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by [Formula: see text] and the coefficient [Formula: see text] is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hamiltonians.

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.

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