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European Mathematical Society Publishing House
2024-03-29 14:45:44
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=3&iss=1&update_since=2024-03-29
Journal of Spectral Theory
J. Spectr. Theory
JST
1664-039X
1664-0403
Quantum theory
10.4171/JST
http://www.ems-ph.org/doi/10.4171/JST
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
3
2013
1
An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size $n$ contained in an interval of size $n^{–C}$
Ilia
Binder
University of Toronto, TORONTO, ONTARIO, CANADA
Mircea
Voda
University of Toronto, TORONTO, ONTARIO, CANADA
Eigenvalues, eigenfunctions, quasiperiodic Jacobi matrix, avalanche principle, large deviations
We consider infinite quasi-periodic Jacobi self-adjoint matrices for which the three main diagonals are given via values of real analytic functions on the trajectory of the shift $x\rightarrow x+\omega$. We assume that the Lyapunov exponent $L(E_{0})$ of the corresponding Jacobi cocycle satisfies $L(E_{0})\ge\gamma>0$. In this setting we prove that the number of eigenvalues $E_{j}^{(n)}(x)$ of a submatrix of size $n$ contained in an interval $I$ centered at $E_{0}$ with $|I|=n^{-C_{1}}$ does not exceed $\left(\log n\right)^{C_{0}}$ for any $x$. Here $n\ge n_{0}$, and $n_{0}$, $C_{0}$, $C_{1}$ are constants depending on $\gamma$ (and the other parameters of the problem).
Quantum theory
Operator theory
Statistical mechanics, structure of matter
General
1
45
10.4171/JST/36
http://www.ems-ph.org/doi/10.4171/JST/36
Scattering zippers and their spectral theory
Laurent
Marin
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Hermann
Schulz-Baldes
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Weyl theory, Sturm–Liouville oscillation theory, orthogonal polynomials on the unit circle
A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries and generalizes Blatter–Browne and Chalker–Coddington models and CMV matrices. Weyl discs are analyzed and used to prove a bijection between the set of semi-infinite scattering zipper operators and matrix valued probability measures on the unit circle. Sturm–Liouville oscillation theory is developed as a tool to calculate the spectra of finite and periodic scattering zipper operators.
Ordinary differential equations
General
47
82
10.4171/JST/37
http://www.ems-ph.org/doi/10.4171/JST/37
The Tan 2$\Theta$ Theorem for indefinite quadratic forms
Luba
Grubišić
University of Zagreb, ZAGREB, CROATIA
Vadim
Kostrykin
Johannes Gutenberg-Universität Mainz, MAINZ, GERMANY
Konstantin
Makarov
University of Missouri, COLUMBIA, UNITED STATES
Krešimir
Veselić
FernUniversität Hagen, HAGEN, GERMANY
Perturbation theory, quadratic forms, invariant subspaces
A version of the Davis–Kahan Tan 2$\Theta$ theorem [3] for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a result by Motovilov and Selin [13].
Operator theory
Ordinary differential equations
General
83
100
10.4171/JST/38
http://www.ems-ph.org/doi/10.4171/JST/38
Spectral analysis of tridiagonal Fibonacci Hamiltonians
William
Yessen
Rice University, HOUSTON, UNITED STATES
Spectral theory, quasiperiodicity, Jacobi operators, Fibonacci Hamiltonians, trace maps
We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution on two letters. We show that the spectrum is a Cantor set of zero Lebesgue measure, and discuss its fractal structure and Hausdorff dimension. We also extend some known results on the diagonal and the off-diagonal Fibonacci Hamiltonians.
Operator theory
Statistical mechanics, structure of matter
General
101
128
10.4171/JST/39
http://www.ems-ph.org/doi/10.4171/JST/39