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30
Convergence of Spectra of Mesoscopic Systems Collapsing Onto a Graph.
 J. Math. Anal. Appl
, 1999
"... Let M be a finite graph in the plane and M " be a domain that looks like the "fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on M " converges when " ! 0 to the spectrum of an ODE problem on M . Presence of an electromagnetic f ..."
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Cited by 55 (2 self)
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Let M be a finite graph in the plane and M " be a domain that looks like the "fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on M " converges when " ! 0 to the spectrum of an ODE problem on M . Presence of an electromagnetic field is also allowed. Considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry. The results of the paper extend the ones previously obtained by J. Rubinstein and M. Schatzman. 2000 MSC: 35Q40, 35P15, 35J10, 81V99 Key words and phrases: mesoscopic system, Schrodinger operator, spectrum 1 Introduction In recent years one has witnessed growing interest in spectral theory of differential (versus difference) operators on graphs. Although probably one of the first such studies was done in physical chemistry [47], the main thrust 1 in this direction came from the mesoscopic physics [29]. Recent progress in nanotechnology and microelectronics en...
Quantum Wires with Magnetic Fluxes
 Comm. Math. Phys
"... In the present article magnetic Laplacians on a graph are analyzed. We provide a complete description of the set of all operators which can be obtained from a given selfadjoint Laplacian by perturbing it by magnetic fields. In particular, it is shown that generically this set is isomorphic to a tor ..."
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Cited by 20 (5 self)
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In the present article magnetic Laplacians on a graph are analyzed. We provide a complete description of the set of all operators which can be obtained from a given selfadjoint Laplacian by perturbing it by magnetic fields. In particular, it is shown that generically this set is isomorphic to a torus. We also describe the conditions under which the operator is unambiguously (up to unitary equivalence) defined by prescribing the magnetic fluxes through all loops of the graph. 1.
The Generalized Star Product And The Factorization Of Scattering Matrices On Graphs
 J. Math. Phys
, 2000
"... . In this article we continue our analysis of Schrodinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star p ..."
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Cited by 19 (7 self)
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. In this article we continue our analysis of Schrodinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs. We perform a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discussed. 1. INTRODUCTION Potential scattering for one particle Schrodinger operators on the line possesses a remarkable property concerning its (onshell) scattering matrix given as a 2 2 matrixvalued function of the energy. Let the potential V be given as the sum of two potentials V 1 and V 2 with disjoint support. Then the scattering matrix for V at a given energy is obtained from the two scattering matrices for V 1 and V 2 at the same energy by a certain nonlinear, nonco...
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 ∗.
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 ..."
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Cited by 16 (2 self)
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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
"... ..."
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. ..."
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Cited by 10 (6 self)
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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.
Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide
 J. PHYS. A: MATH. THEOR. A
, 2007
"... In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achiev ..."
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Cited by 8 (4 self)
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In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achieve in the limit a nontrivial coupling between the edges including, in particular, the class of δtype boundary conditions. We work out an illustration of this claim in the simplest case when a bent waveguide is squeezed.
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. ..."
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Cited by 8 (5 self)
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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.
Approximations of singular vertex couplings in quantum graphs, mathph/0703051 15 M. Freidlin, A. Wentzell: Diffusion processes on graphs and the averaging principle
, 1993
"... We discuss approximations of the vertex coupling on a starshaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edgepermutation symmetry is present. It is shown that the CheonShigehara technique using δ interactions with nonlinearly s ..."
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Cited by 7 (5 self)
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We discuss approximations of the vertex coupling on a starshaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edgepermutation symmetry is present. It is shown that the CheonShigehara technique using δ interactions with nonlinearly scaled couplings yields a 2nparameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges one can approximate the `n+1 ´parameter family of all 2 timereversal invariant couplings.