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A random necklace model
Vadim Kostrykin,Robert Schrader +1 more
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In this paper, a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length is considered and the arc lengths of the loops are considered to be independent, identically distributed random variables.Abstract:
We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.read more
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Quantum graphs: Applications to quantum chaos and universal spectral statistics
Sven Gnutzmann,Uzy Smilansky +1 more
TL;DR: In this paper, the spectral theory of quantum graphs is discussed and exact trace formulae for the spectrum and the quantum-to-classical correspondence are discussed, as well as its application to quantum chaos.
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Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
TL;DR: In this paper, 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 is considered.
Journal ArticleDOI
Spectral analysis of percolation Hamiltonians
TL;DR: In this paper, the convergence of the integrated densities of states of finite box Hamiltonians to the one on the whole space holds even at the points of discontinuity, where the integrated density of states has discontinuities precisely at this set of energies.
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Anderson localization for radial tree-like random quantum graphs
Peter D. Hislop,Olaf Post +1 more
TL;DR: In this paper, it was shown that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies, including the random Kirchhoff model.
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Anderson Localization for radial tree-like random quantum graphs
Peter D. Hislop,Olaf Post +1 more
TL;DR: In this paper, it was shown that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies, including the random Kirchhoff model.
References
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Journal ArticleDOI
Kirchhoff's rule for quantum wires
Vadim Kostrykin,Robert Schrader +1 more
TL;DR: In this paper, one-particle quantum scattering theory on an arbitrary finite graph with n open ends is formulated and discussed, where the Hamiltonian is defined to be (minus) the Laplace operator with general boundary conditions at the vertices.
Journal ArticleDOI
Electronic states on a Penrose lattice.
Mahito Kohmoto,Bill Sutherland +1 more
TL;DR: Etude numerique pour differentes conditions aux limites, mettant en evidence l'existence d'un pic central de Largemmille a l'energie zero, constitue par environ 10% des etats, strictement localises.
Journal ArticleDOI
Log hölder continuity of the integrated density of states for stochastic Jacobi matrices
Walter Craig,Barry Simon +1 more
TL;DR: In this paper, the integrated density of states,k(E), of a general operator on l 2 (ℤv) of the formh=h0+v, where h_0 u(n) = \sum\limits_{\left| i \right| = 1} {u(n + i)} \) and (vu)(n)=vn)u (n), wherev is a general bounded ergodic stationary process on ℤ v.
Journal ArticleDOI
On the density of states of Schrodinger operators with a random potential
Werner Kirsch,Fabio Martinelli +1 more
TL;DR: In this paper, the existence of the density of states for a wide class of random Schrodinger operators was proved for superadditive processes on roads. But the density was not shown to be constant.
Journal ArticleDOI
Periodic Schrödinger operators with large gaps and Wannier-Stark ladders.
Joseph E. Avron,Pavel Exner +1 more
TL;DR: In this article, the Wannier-Stark ladder of the Schr\"odinger operator has no absolutely continuous spectrum and the widths of forbidden gaps increase at large energies and the gap to band ratio is not small.