scholarly journals Linear Wegner estimate for alloy-type Schrödinger operators on metric graphs

2007 ◽  
Vol 48 (9) ◽  
pp. 092107 ◽  
Author(s):  
Mario Helm ◽  
Ivan Veselić
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Metric Graphs ◽  
Wegner Estimate ◽  
Alloy Type

scholarly journals Wegner estimate for discrete Schrödinger operators with Gaussian random potentials

2019 ◽  
Vol 27 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Martin Tautenhahn
Keyword(s):  
Conditional Distribution ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Gaussian Random Process ◽  
Random Potentials ◽  
Wegner Estimate ◽  
Compactly Supported ◽  
Covariance Functions ◽  
Monotonicity Assumption ◽  
Discrete Schrödinger Operators

Abstract We prove a Wegner estimate for discrete Schrödinger operators with a potential given by a Gaussian random process. The only assumption is that the covariance function decays exponentially; no monotonicity assumption is required. This improves earlier results where abstract conditions on the conditional distribution, compactly supported and non-negative, or compactly supported covariance functions with positive mean are considered.

scholarly journals On inverse topology problem for Laplace operators on graphs

2014 ◽  
Vol 6 (2) ◽  
pp. 230-236
Author(s):  
Yu.Yu. Ershova ◽  
I.I. Karpenko ◽  
A.V. Kiselev
Keyword(s):  
Additional Assumption ◽  
Necessary Conditions ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Metric Graphs ◽  
Laplace Operators ◽  
Matching Conditions ◽  
Inverse Topology ◽  
Delta Type ◽  
Laplacian Operators

Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ type. Under one additional assumption, the inverse topology problem is treated. Using the apparatus of boundary triples, we generalize and extend existing results on necessary conditions of isospectrality of two Laplacians defined on different graphs. A result is also given covering the case of Schrodinger operators.

scholarly journals VACUUM ENERGY OF SCHRÖDINGER OPERATORS ON METRIC GRAPHS

2012 ◽  
Vol 14 ◽  
pp. 357-366
Author(s):  
JONATHAN M HARRISON ◽  
KLAUS KIRSTEN
Keyword(s):  
Casimir Force ◽  
Vacuum Energy ◽  
General Scheme ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Classical Dynamics ◽  
Metric Graphs ◽  
Argument Principle ◽  
One Dimensional ◽  
Local Vertex

We present an integral formulation of the vacuum energy of Schrödinger operators on finite metric graphs. Local vertex matching conditions on the graph are classified according to the general scheme of Kostrykin and Schrader. While the vacuum energy of the graph can contain finite ambiguities the Casimir force on a bond with compactly supported potential is well defined. The vacuum energy is determined from the zeta function of the graph Schrödinger operator which is derived from an appropriate secular equation via the argument principle. A quantum graph has an associated probabilistic classical dynamics which is generically both ergodic and mixing. The results therefore present an analytic formulation of the vacuum energy of this quasi-one-dimensional quantum system which is classically chaotic.

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