Results 1  10
of
38
Heat kernels on metric graphs and a trace formula
, 2007
"... We study heat semigroups generated by selfadjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kerne ..."
Abstract

Cited by 31 (4 self)
 Add to MetaCart
We study heat semigroups generated by selfadjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.
The inverse scattering problem for metric graphs and the traveling salesman problem
, 2006
"... ..."
(Show Context)
Sch’nol’s theorem for strongly local forms
, 2009
"... We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions. ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions.
Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376
"... Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
Gaussian estimates for a heat equation on a network
 Netw. Heter. Media
, 2007
"... ar ..."
(Show Context)
Semigroup methods for evolution equations on networks
"... 2These notes derive from the lecture series “Evolution equations on networks”, held in March 2007 at the Mathematical Department at Lousiana State University. The lectures were conceived with the aim of presenting known results obtained over the last few years by a group of few researchers with a de ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
(Show Context)
2These notes derive from the lecture series “Evolution equations on networks”, held in March 2007 at the Mathematical Department at Lousiana State University. The lectures were conceived with the aim of presenting known results obtained over the last few years by a group of few researchers with a definite operator theoretical background, most of whom currently are and/or have recently been based at the universities of Tübingen and Ulm. While elaborating these notes, I have however tried to homogenize the discussion and to provide the reader with several examples and applications, in particular to neuronal modelling. Ideally, the presented method relies upon tools coming from operator semigroups as well as graph theory. Most proofs have only been sketched, and in general I have tried to keep the presentation as fluid and selfcontained as possible. Though, I have tried to mention most relevan recent developments in the theory, including approximation problems, quantum graphs, and nonstandard boundary conditions. Mistakes and typos may well have slipped in the text. I will be glad to receive feedbacks, suggestions, and criticisms: please do not hesitate and send me an email. The cover picture has been taken in Berlin during Der Berg (The Mountain), an art installation organized in sommer 2005 inside the Palast der Republik (Palace of the Republic), the former House of Parliament of the German Democratic Republic. The palast has been subsequently demolished.
NONWEYL RESONANCE ASYMPTOTICS FOR QUANTUM GRAPHS
"... Abstract. We consider the resonances of a quantum graph G that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of G in a disc of a large radius. We call G a Weyl graph if the coefficient in front of ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Abstract. We consider the resonances of a quantum graph G that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of G in a disc of a large radius. We call G a Weyl graph if the coefficient in front of this leading term coincides with the volume of the compact part of G. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the nonWeyl case occurs. 1.
Contraction semigroups on metric graphs
 Analysis on Graphs and its Applications, volume 77 of Proceedings of Symposia in Pure Mathematics
, 2008
"... Dedicated to Volker Enss on the occasion of his 65th birthday ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Dedicated to Volker Enss on the occasion of his 65th birthday
FINITE PROPAGATION SPEED AND CAUSAL FREE QUANTUM FIELDS ON NETWORKS
, 907
"... ABSTRACT. Laplace operators on metric graphs give rise to KleinGordon and wave operators. Solutions of the KleinGordon equation and the wave equation are studied and finite propagation speed is established. Massive, free quantum fields are then constructed, whose commutator function is just the Kl ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
ABSTRACT. Laplace operators on metric graphs give rise to KleinGordon and wave operators. Solutions of the KleinGordon equation and the wave equation are studied and finite propagation speed is established. Massive, free quantum fields are then constructed, whose commutator function is just the KleinGordon kernel. As a consequence of finite propagation speed Einstein causality (local commutativity) holds. Comparison is made with an alternative construction of free fields involving RTalgebras. PACS: 03.65.Nk, 03.70.+k, 73.21.Hb 1.
KREIN FORMULA AND SMATRIX FOR EUCLIDEAN SURFACES WITH CONICAL SINGULARITIES
"... Abstract. Using the Krein formula for the difference of the resolvents of two selfadjoint extensions of a symmetric operator with finite deficiency indices, we establish a comparison formula for ζregularized determinants of two selfadjoint extensions of the Laplace operator on a Euclidean surface ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Using the Krein formula for the difference of the resolvents of two selfadjoint extensions of a symmetric operator with finite deficiency indices, we establish a comparison formula for ζregularized determinants of two selfadjoint extensions of the Laplace operator on a Euclidean surface with conical singularities (E. s. c. s.). The ratio of two determinants is expressed through the value S(0) of the Smatrix, S(λ), of the surface. We study the asymptotic behavior of the Smatrix, give an explicit expression for S(0) relating it to the Bergman projective connection on the underlying compact Riemann surface and derive variational formulas for S(λ) with respect to coordinates on the moduli space of E. s. c. s. with trivial holonomy. 1.