scholarly journals Heat-kernel and Resolvent Asymptotics for Schrodinger Operators on Metric Graphs

2014 ◽  
Author(s):  
J. Bolte ◽  
S. Egger ◽  
R. Rueckriemen
Keyword(s):  
Heat Kernel ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Metric Graphs

scholarly journals Heat Kernel Coefficients for Two-Dimensional Schrödinger Operators

2008 ◽  
Vol 283 (3) ◽  
pp. 853-860 ◽  
Author(s):  
Yuri Berest ◽  
Tim Cramer ◽  
Farkhod Eshmatov
Keyword(s):  
Heat Kernel ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Two Dimensional

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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