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46
Absolutely continuous spectra of quantum tree graphs with weak disorder
"... Abstract: We consider the Laplacian on a rooted metric tree graph with branching number K ≥ 2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the dis ..."
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Cited by 29 (5 self)
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Abstract: We consider the Laplacian on a rooted metric tree graph with branching number K ≥ 2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the WeylTitchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder. Contents 1. Introduction.................................
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 (4 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.
The inverse scattering problem for metric graphs and the traveling salesman problem
, 2006
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Uniform existence of the integrated density of states for . . . Z^d
"... We give an overview and extension of recent results on ergodic random Schrödinger operators for models on Zd. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that n ..."
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Cited by 14 (8 self)
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We give an overview and extension of recent results on ergodic random Schrödinger operators for models on Zd. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable, at least locally. This limit, the integrated density of states (IDS), can be expressed by a closed ShubinPastur type trace formula. The set of points of increase of the IDS supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. This applies to several examples, including various periodic operators and percolation models.
On occurrence of spectral edges for periodic operators inside the Brillouin zone”, arXiv: mathph/0702035
, 2007
"... Abstract. The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schrödinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum runn ..."
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Cited by 11 (1 self)
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Abstract. The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schrödinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of “corner ” high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community. In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiplyperiodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a “generic” case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds. AMS classification scheme numbers: 35P99, 47F05, 58J50, 81Q10Spectral edges of periodic operators 2 1.
Cantor and band spectra for periodic quantum graphs with magnetic fields
 Comm. Math. Phys
"... ABSTRACT. We provide an exhaustive spectral analysis of the twodimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lya ..."
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Cited by 11 (3 self)
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ABSTRACT. We provide an exhaustive spectral analysis of the twodimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable KronigPenney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the BetheSommerfeld conjecture fails in this case.
Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence
, 2007
"... We study equivariant families of discrete Hamiltonians on amenable geometries and their integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support. The size of a jump of the IDS is consequently given by the equivariant dimens ..."
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Cited by 11 (6 self)
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We study equivariant families of discrete Hamiltonians on amenable geometries and their integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support. The size of a jump of the IDS is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions. From this we deduce uniform convergence (w.r.t. the spectral parameter) of the finite volume approximants of the IDS. Our framework includes quasiperiodic operators on Delone sets, periodic and random operators on quasitransitive graphs, and operators on percolation graphs.
ON THE SPECTRA OF CARBON NanoStructures
, 2007
"... An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nanostructures (including graphene and all types of singlewall nanotubes) is provided. ..."
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Cited by 11 (3 self)
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An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nanostructures (including graphene and all types of singlewall nanotubes) is provided.
Harmonic coordinates on fractals with finitely ramified cell structure
 Department of Mathematics, University of Copenhagen, DK2100 Copenhagen, Denmark Email address: echris@math.ku.dk Department of Mathematics, University of Hannover
"... Abstract. We define sets with finitely ramified cell structure, which are generalizations of p.c.f. selfsimilar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local selfsimilarity, and allow countably many cells connected at each junction p ..."
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Cited by 9 (5 self)
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Abstract. We define sets with finitely ramified cell structure, which are generalizations of p.c.f. selfsimilar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local selfsimilarity, and allow countably many cells connected at each junction point. In particular, we consider postcritically infinite fractals. We prove that if Kigami’s resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates. Contents
Dirac operators and spectral triples for some fractal sets built on curvers
, 2006
"... We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and t ..."
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Cited by 9 (0 self)
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We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmiertype trace coincides with the normalized Hausdorff measure of dimension log 3 / log2.