712 research outputs found
Commutation Relations for Unitary Operators
Let be a unitary operator defined on some infinite-dimensional complex
Hilbert space . Under some suitable regularity assumptions, it is
known that a local positive commutation relation between and an auxiliary
self-adjoint operator defined on allows to prove that the
spectrum of has no singular continuous spectrum and a finite point
spectrum, at least locally. We show that these conclusions still hold under
weak regularity hypotheses and without any gap condition. As an application, we
study the spectral properties of the Floquet operator associated to some
perturbations of the quantum harmonic oscillator under resonant AC-Stark
potential
Commutation Relations for Unitary Operators III
Let be a unitary operator defined on some infinite-dimensional complex
Hilbert space . Under some suitable regularity assumptions, it is
known that a local positive commutation relation between and an auxiliary
self-adjoint operator defined on allows to prove that the
spectrum of has no singular continuous spectrum and a finite point
spectrum, at least locally. We prove that under stronger regularity hypotheses,
the local regularity properties of the spectral measure of are improved,
leading to a better control of the decay of the correlation functions. As shown
in the applications, these results may be applied to the study of periodic
time-dependent quantum systems, classical dynamical systems and spectral
problems related to the theory of orthogonal polynomials on the unit circle
A generalization of the Heine--Stieltjes theorem
We extend the Heine-Stieltjes Theorem to concern all (non-degenerate)
differential operators preserving the property of having only real zeros. This
solves a conjecture of B. Shapiro. The new methods developed are used to
describe intricate interlacing relations between the zeros of different pairs
of solutions. This extends recent results of Bourget, McMillen and Vargas for
the Heun equation and answers their question on how to generalize their results
to higher degrees. Many of the results are new even for the classical case.Comment: 12 pages, typos corrected and refined the interlacing theorem
Localization Properties of the Chalker-Coddington Model
The Chalker Coddington quantum network percolation model is numerically
pertinent to the understanding of the delocalization transition of the quantum
Hall effect. We study the model restricted to a cylinder of perimeter 2M. We
prove firstly that the Lyapunov exponents are simple and in particular that the
localization length is finite; secondly that this implies spectral
localization. Thirdly we prove a Thouless formula and compute the mean Lyapunov
exponent which is independent of M.Comment: 29 pages, 1 figure. New section added in which simplicity of the
Lyapunov spectrum and finiteness of the localization length are proven. To
appear in Annales Henri Poincar
Localization for Random Unitary Operators
We consider unitary analogs of dimensional Anderson models on
defined by the product where is a deterministic
unitary and is a diagonal matrix of i.i.d. random phases. The
operator is an absolutely continuous band matrix which depends on a
parameter controlling the size of its off-diagonal elements. We prove that the
spectrum of is pure point almost surely for all values of the
parameter of . We provide similar results for unitary operators defined on
together with an application to orthogonal polynomials on the unit
circle. We get almost sure localization for polynomials characterized by
Verblunski coefficients of constant modulus and correlated random phases
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