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15
Resonances from perturbations of quantum graphs with rationally related edges
 J. Phys. A
, 2010
"... rationally related edges ..."
Eigenvalue bracketing for discrete and metric graphs
 J. Math. Anal. Appl
"... Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the m ..."
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Cited by 4 (2 self)
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Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps. 1.
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 ..."
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Cited by 4 (2 self)
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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.
Essential selfadjointness for combinatorial Schrödinger
, 2010
"... We will use in this section the distance dp defined using the weights px,y = cx,y. Let us consider the quadratic form Q(f) = 〈(∆ω,c + 1)f  f〉ω = cxy(f(x) − f(y)) 2 + ∑ We have {x,y}∈E x∈V ω 2 xf(x) 2. Lemma 1.1 For any f: V → R so that Q(f) < ∞ and for any a, b ∈ V, we have f(a) − f(b)  ≤ √ ..."
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Cited by 1 (1 self)
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We will use in this section the distance dp defined using the weights px,y = cx,y. Let us consider the quadratic form Q(f) = 〈(∆ω,c + 1)f  f〉ω = cxy(f(x) − f(y)) 2 + ∑ We have {x,y}∈E x∈V ω 2 xf(x) 2. Lemma 1.1 For any f: V → R so that Q(f) < ∞ and for any a, b ∈ V, we have f(a) − f(b)  ≤ √ Q(f)dp(a, b). Proof.– For any {x, y} ∈ E, f(x) − f(y)  ≤ √ Q(f) / √ cxy. For any path γ from a to b, defined by the vertices x1 = a, x2, · · · , xN = b, we have f(a) − f(b)  ≤ √ Q(f)length(γ). Taking the infimum of the righthandside w.r. to γ we get the result. Lemma 1.1 implies that any function f with Q(f) < ∞ extends to ˆ V as a Lipschitz function ˆ f. We will denote by f ∞ the restriction of ˆ f to V∞. Theorem 1.1 Let f: V → R with Q(f) < ∞, then there exists an unique continuous function F: ˆ V → R which satisfies • (F − f) ∞ = 0 • (∆ω,c + 1)(FV) = 0 Proof.– We will denote by Af the affine space of continuous functions G: ˆV → R which satisfy Q(G) < ∞ and (G − f) ∞ = 0. Q is lower semicontinuous for the pointwise convergence on V as defined by Q = sup Qα with Qα(f) = sum of a finite number of terms in Q. Let Q0: = infG∈Af Q(G) and Gn be a corresponding minimizing sequence. The Gn’s are equicontinuous and pointwise bounded. From Ascoli’s Theorem, this implies the existence of a locally uniformly convergent subsequence Gnk → F. Using semicontinuity, we have Q(F) = Q0. If x ∈ V and δx is the Dirac function at the vertex x, we have d Q(F + tδx) = 2(∆ω,c + 1)F (x)
The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums
, 812
"... Abstract. We present a method which enables one to construct isospectral objects, such as quantum graphs and drums. One aspect of the method is based on representation theory arguments which are shown and proved. The complementary part concerns techniques of assembly which are both stated generally ..."
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Abstract. We present a method which enables one to construct isospectral objects, such as quantum graphs and drums. One aspect of the method is based on representation theory arguments which are shown and proved. The complementary part concerns techniques of assembly which are both stated generally and demonstrated. For that purpose, quantum graphs are grist to the mill. We develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects. We discuss the theoretical implications which include Sunada’s theorem of isospectrality [2] arising as a particular case of this method. A gallery of new isospectral examples is presented and some known examples are shown to result from our theory.
DEPENDENCE OF THE SPECTRUM OF A QUANTUM GRAPH ON VERTEX CONDITIONS AND EDGE LENGTHS
"... Abstract. We study the dependence of the quantum graph Hamiltonian, its resolvent, and its spectrum on the vertex conditions and graph edge lengths. In particular, several results on the interlacing (bracketing) of the spectra of graphs with different vertex conditions are obtained and their applica ..."
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Abstract. We study the dependence of the quantum graph Hamiltonian, its resolvent, and its spectrum on the vertex conditions and graph edge lengths. In particular, several results on the interlacing (bracketing) of the spectra of graphs with different vertex conditions are obtained and their applications are discussed. 1.
VECTORVALUED HEAT EQUATIONS AND NETWORKS WITH COUPLED DYNAMIC BOUNDARY CONDITIONS
, 903
"... setting for interface problems with coupled dynamic boundary conditions belonging to a quite general class. Beside wellposedness, we discuss positivity, L ∞contractivity and further invariance properties. We show that the parabolic problem with dynamic boundary conditions enjoy these properties if ..."
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setting for interface problems with coupled dynamic boundary conditions belonging to a quite general class. Beside wellposedness, we discuss positivity, L ∞contractivity and further invariance properties. We show that the parabolic problem with dynamic boundary conditions enjoy these properties if and only if so does its counterpart with timeindependent boundary conditions. Furthermore, we prove continuous dependence of the solution to the parabolic problem on the boundary conditions in the considered class. 1.
Convergence results for thick graphs Olaf Post
, 2010
"... Many physical systems have branching structure of thin transversal diameter. One can name for instance quantum wire circuits, thin branching waveguides, or carbon nanostructures. In applications, such systems are often approximated by the underlying onedimensional graph structure, a socalled “quant ..."
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Many physical systems have branching structure of thin transversal diameter. One can name for instance quantum wire circuits, thin branching waveguides, or carbon nanostructures. In applications, such systems are often approximated by the underlying onedimensional graph structure, a socalled “quantum graph”. In this way, many properties of the system like conductance can be calculated easier (sometimes even explicitly). We give an overview of convergence results obtained so far, such as convergence of Schrödinger operators, Laplacians and their spectra. 1