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98
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.
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
 Journal of Physics A: Mathematical and General
"... Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) ..."
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Cited by 27 (7 self)
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Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature. 1.
DUNKL OPERATORS: THEORY AND APPLICATIONS
, 2002
"... These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the D ..."
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Cited by 21 (1 self)
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These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable particle systems of CalogeroMoserSutherland type, and discuss some systems of orthogonal polynomials associated with them. A major part is devoted to positivity results for the intertwining operator and the Dunkl kernel, the Dunkltype heat semigroup, and related probabilistic aspects. The notes conclude with recent results on the asymptotics of the Dunkl kernel.
Equality of bulk and edge Hall conductance revisited
 Comm. Math. Phys
, 2002
"... The integral quantum Hall eect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to dierent de nitions of Hall conductance, which agree under appropriate hypotheses, as shown by SchulzBaldes et al. by means o ..."
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Cited by 18 (1 self)
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The integral quantum Hall eect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to dierent de nitions of Hall conductance, which agree under appropriate hypotheses, as shown by SchulzBaldes et al. by means of Ktheory. We propose an alternative proof based on a generalization of the index of a pair of projections to more general operators. The equality of conductances is an expression of the stability of that index as a ux tube is moved from within the bulk across the boundary of a sample.
Sigal : Time dependent scattering theory for Nbody quantum systems
, 1997
"... We give a full and self contained account of the basic results in Nbody scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r−µ, µ> √ 3 − 1. Our approach is a synthesis of earlier work and of new ideas. Globa ..."
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Cited by 18 (2 self)
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We give a full and self contained account of the basic results in Nbody scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r−µ, µ> √ 3 − 1. Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to define the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like t2 as t →±∞for any orbit ψt which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre’s theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase–space analysis. As a topic of general interest we describe a method of commutator expansions. 0.
Minimal escape velocities
 Comm. Partial Differential Equations
, 1999
"... Abstract. We give a new derivation of the minimal velocity estimates [SiSo1] for unitary evolutions. Let H and A be selfadjoint operators on a Hilbert space H. The starting point is Mourre’s inequality i[H, A] ≥ θ> 0, which is supposed to hold in form sense on the spectral subspace H ∆ of H for ..."
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Cited by 15 (3 self)
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Abstract. We give a new derivation of the minimal velocity estimates [SiSo1] for unitary evolutions. Let H and A be selfadjoint operators on a Hilbert space H. The starting point is Mourre’s inequality i[H, A] ≥ θ> 0, which is supposed to hold in form sense on the spectral subspace H ∆ of H for some interval ∆ ⊂ R. The second assumption is that the multiple commutators ad (k) (H) are wellA behaved for k = 1...n (n ≥ 2). Then we show that, for a dense set of ψ’s in H ∆ and all m < n−1, ψt = exp(−iHt) is contained in the spectral subspace A ≥ θt as t → ∞, up to an error of order t−m in norm. We apply this general result to the case where H is a Schrödinger operator on Rn and A the dilation generator, proving that ψt(x) is asymptotically supported in the set x  ≥ t √ θ up to an error of order t−m in norm. 1.
Sufficient conditions for exponential stability and dichotomy of evolution equations
 Forum Math
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Localisation for Random Perturbations of Periodic Schrödinger Operators with Regular Floquet Eigenvalues
, 2000
"... We prove a localisation theorem for continuous ergodic Schrödinger operators H! := H0 + V! , where the random potential V! is a nonnegative Andersontype random perturbation of the periodic operator H0 . We consider a lower spectral band edge of (H0 ), say E = 0, at a gap which is preserved by ..."
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Cited by 11 (4 self)
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We prove a localisation theorem for continuous ergodic Schrödinger operators H! := H0 + V! , where the random potential V! is a nonnegative Andersontype random perturbation of the periodic operator H0 . We consider a lower spectral band edge of (H0 ), say E = 0, at a gap which is preserved by the perturbation V! . Assuming that all Floquet eigenvalues of H0 , which reach the spectral edge 0 as a minimum, have there a positive definite Hessian, we conclude that there exists an interval I containing 0 such that H! has only pure point spectrum in I for almost all !.
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.
A ptsymmetric periodic problem with boundary and interior singularities. arXiv:0801.0172
, 2008
"... interior singularities ∗ ..."