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Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs
 J. Phys. A: Math. Gen
"... The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one ..."
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Cited by 46 (5 self)
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The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available. A theorem on spectral gap opening for “decorated ” quantum graphs is established (its analog is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions (“scars”). 1
Convergence of spectra of graphlike thin manifolds
 J. Geom. Phys
"... Abstract. We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at th ..."
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Cited by 37 (14 self)
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Abstract. We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices. 1.
Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
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Scattering solutions in a network of thin fibers: small diameter asymptotics, Preprint (mathph/0609021
, 2006
"... Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph Γ. We calculate the Lagrangian gluing conditions at vertices v ∈ Γ for the problems on the limiting graph. ..."
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Cited by 22 (2 self)
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Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph Γ. We calculate the Lagrangian gluing conditions at vertices v ∈ Γ for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network.
Harmonic analysis on metrized graphs
 CANAD. J. MATH
"... This paper studies the Laplacian operator on a metrized graph, and its spectral theory. ..."
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Cited by 20 (4 self)
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This paper studies the Laplacian operator on a metrized graph, and its spectral theory.
Spectra of Schrödinger operators on equilateral quantum graphs
, 2006
"... We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum g ..."
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Cited by 20 (5 self)
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We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum graph is the preimage of the combinatorial spectrum under a certain entire function. Using this correspondence we show that that the number of gaps in the spectrum of the Schrödinger operators admits an estimate from below in terms of the Hill operator independently of the graph structure.
Index theorems for quantum graphs
, 2007
"... Abstract. In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higherorder differential operators as an intermediary. In this ..."
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Cited by 16 (4 self)
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Abstract. In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higherorder differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) onedimensional variety and equipped with a secondorder differential Hamiltonian H (a “Laplacian”) with suitable conditions at vertices. For the case of scaleinvariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heatkernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the indextheory context by factoring the Laplacian into two firstorder operators, H = A ∗ A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is found also that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A ∗.
Selfadjoint extensions of restrictions
, 703
"... Abstract. We provide a simple recipe for obtaining all selfadjoint extensions, together with their resolvent, of the symmetric operator S obtained by restricting the selfadjoint operator A: D(A) ⊆ H → H to the dense, closed with respect to the graph norm, subspace N ⊂ D(A). Neither the knowledge o ..."
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Cited by 15 (1 self)
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Abstract. We provide a simple recipe for obtaining all selfadjoint extensions, together with their resolvent, of the symmetric operator S obtained by restricting the selfadjoint operator A: D(A) ⊆ H → H to the dense, closed with respect to the graph norm, subspace N ⊂ D(A). Neither the knowledge of S ∗ nor of the deficiency spaces of S is required. Typically A is a differential operator and N is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle p: E(h) → P(h), where P(h) denotes the set of orthogonal projections in the Hilbert space h ≃ D(A)/N and p −1 (Π) is the set of selfadjoint operators in the range of Π. The set of selfadjoint operators in h, i.e. p −1 (1), parametrises the relatively prime extensions. Any (Π, Θ) ∈ E(h) determines a boundary condition in the domain of the corresponding extension AΠ,Θ and explicitly appears in the formula for the resolvent (−AΠ,Θ + z) −1. The connection with both von Neumann’s and Boundary Triples theories of selfadjoint extensions is explained. Some examples related to quantum graphs, to Schrödinger operators with point interactions and to elliptic boundary value problems are given. 1. Introduction. On the Hilbert space H with scalar product 〈·, · 〉 we consider the selfadjoint operator
The inverse scattering problem for metric graphs and the traveling salesman problem
, 2006
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Some bound state problems in quantum mechanics
 Proc. of Symposia in Pure Mathematics 76, part 1, Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Gesztesy et al., editors, 463–496, AMS 2007
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