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APPROXIMATION OF QUANTUM GRAPH VERTEX COUPLINGS BY SCALED SCHRÖDINGER OPERATORS ON THIN BRANCHED MANIFOLDS
"... Abstract. We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrödinger operators can approximate nontrivial vertex couplings. The lat ..."
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Abstract. We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrödinger operators can approximate nontrivial vertex couplings. The latter include not only the δcouplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric δ ′couplings and conjecture that the same method can be applied to all couplings invariant with respect to the time reversal. 1.
Monotone Unitary Families
"... A unitary family is a family of unitary operators U(x) acting on a finite dimensional hermitian vector space, depending analytically on a real parameter x. It is monotone if 1 i U ′ (x)U(x) −1 is a positive operator for each x. ..."
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Cited by 3 (2 self)
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A unitary family is a family of unitary operators U(x) acting on a finite dimensional hermitian vector space, depending analytically on a real parameter x. It is monotone if 1 i U ′ (x)U(x) −1 is a positive operator for each x.
Asymptotics of eigenfunctions on plane domains
, 2007
"... Abstract. We consider a family of domains (ΩN)N>0 obtained by attaching an N ×1 rectangle to a fixed set Ω0 = {(x, y) : 0 < y < 1, −φ(y) < x < 0}, for a Lipschitz function φ ≥ 0. We derive full asymptotic expansions, as N → ∞, for the mth Dirichlet eigenvalue (for any fixed m ∈ N) and for the associ ..."
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Cited by 2 (2 self)
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Abstract. We consider a family of domains (ΩN)N>0 obtained by attaching an N ×1 rectangle to a fixed set Ω0 = {(x, y) : 0 < y < 1, −φ(y) < x < 0}, for a Lipschitz function φ ≥ 0. We derive full asymptotic expansions, as N → ∞, for the mth Dirichlet eigenvalue (for any fixed m ∈ N) and for the associated eigenfunction on ΩN. The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain Ω∞. We determine the first variation of this scattering phase, with respect to φ, at φ ≡ 0. This is then used to prove sharpness of results, obtained previously by the same authors, about the location of extrema and nodal line of eigenfunctions on convex domains. Contents
SPECTRAL ANALYSIS OF METRIC GRAPHS AND RELATED SPACES
"... Abstract. The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the g ..."
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Cited by 1 (0 self)
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Abstract. The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs. Moreover, we describe a connection of metric graphs with manifolds. Finally, we comment on Cheeger’s inequality and trace formulas for metric and discrete (generalised) Laplacians. 1.
Propagation of Waves in Networks of Thin Fibers
, 902
"... The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can desc ..."
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Cited by 1 (0 self)
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The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can describe the propagating waves, the spectrum and the resolvent in terms of solutions of ordinary differential equations on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODE on the graph uniquely, one needs to know the gluing conditions (GC) on the vertices of the graph. Unlike other publications on this topic, we consider the situation when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. We show that the GC in this case can be expressed in terms of the scattering matrices related to individual junctions. The results are extended to the values of the spectral parameter below the threshold and around it. 1
CONVERGENCE OF SECTORIAL OPERATORS ON VARYING HILBERT SPACES
"... Abstract. Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on varying spaces is natural. However, it seems that the first results in this direction have been obtained only recen ..."
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Abstract. Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on varying spaces is natural. However, it seems that the first results in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on scales of Hilbert spaces. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces and show that this kind of convergence is compatible with the functional calculus of the operator and moreover implies convergence of the spectrum. Finally, we present examples for which this convergence can be checked, including convergence of coefficients of parabolic problems. Convergence of a manifold (roughly speaking consisting of thin tubes) towards the manifold’s skeleton graph plays a prominent role, being our main application. 1.
A GENERAL APPROXIMATION OF QUANTUM GRAPH VERTEX COUPLINGS BY SCALED SCHRÖDINGER OPERATORS ON THIN BRANCHED MANIFOLDS
"... Abstract. We demonstrate that any selfadjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schrödinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local chang ..."
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Cited by 1 (1 self)
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Abstract. We demonstrate that any selfadjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schrödinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently ‘lifted ’ to the manifold. For the corresponding operator a normresolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero. 1.
unknown title
, 902
"... solvable model for scattering on a junction and a modified analytic perturbation procedure. ..."
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solvable model for scattering on a junction and a modified analytic perturbation procedure.
SPECTRAL ANALYSIS OF METRIC GRAPHS AND RELATED SPACES OLAF POST
, 712
"... Abstract. The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the g ..."
Abstract
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Abstract. The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs. Moreover, we describe a connection of metric graphs with manifolds. Finally, we comment on Cheeger’s inequality and trace formulas for metric and discrete (generalised) Laplacians. 1.
Then
, 711
"... Abstract. A unitary family is a family of unitary operators U(x) acting on a finite dimensional hermitian vector space, depending analytically on a real parameter x. It is monotone if 1 i U ′(x)U(x) −1 is a positive operator for each x. We prove a number of results generalizing standard theorems on ..."
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Abstract. A unitary family is a family of unitary operators U(x) acting on a finite dimensional hermitian vector space, depending analytically on a real parameter x. It is monotone if 1 i U ′(x)U(x) −1 is a positive operator for each x. We prove a number of results generalizing standard theorems on the spectral theory of a single unitary operator U0, which correspond to the ’commutative’ case U(x) = eixU0. Also, for a twoparameter unitary family – for which there is no analytic perturbation theory – we prove an implicit function type theorem for the spectral data under the assumption that the family is monotone in one argument. 1.