230 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
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
Towards Deconstruction of the Type D (2,0) Theory
We propose a four-dimensional supersymmetric theory that deconstructs, in a
particular limit, the six-dimensional theory of type . This 4d
theory is defined by a necklace quiver with alternating gauge nodes
and . We test this proposal by comparing the
6d half-BPS index to the Higgs branch Hilbert series of the 4d theory. In the
process, we overcome several technical difficulties, such as Hilbert series
calculations for non-complete intersections, and the choice of
versus gauge groups. Consistently, the result matches the Coulomb
branch formula for the mirror theory upon reduction to 3d
Fractional Moment Estimates 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
parameters controlling the size of its off-diagonal elements. We adapt the
method of Aizenman-Molchanov to get exponential estimates on fractional moments
of the matrix elements of , provided the
distribution of phases is absolutely continuous and the parameters correspond
to small off-diagonal elements of . Such estimates imply almost sure
localization for
On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum
We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum
of H is semibounded and discrete, and the eigenvalues behave as E_n~n^\alpha,
with 0<\alpha<1. In particular, the gaps between successive eigenvalues decay
as n^{\alpha-1}. V(t) is supposed to be periodic, bounded, continuously
differentiable in the strong sense and such that the matrix entries with
respect to the spectral decomposition of H obey the estimate
|V(t)_{m,n}|0,
p>=1 and \gamma=(1-\alpha)/2. We show that the energy diffusion exponent can be
arbitrarily small provided p is sufficiently large and \epsilon is small
enough. More precisely, for any initial condition \Psi\in Dom(H^{1/2}), the
diffusion of energy is bounded from above as _\Psi(t)=O(t^\sigma) where
\sigma=\alpha/(2\ceil{p-1}\gamma-1/2). As an application we consider the
Hamiltonian H(t)=|p|^\alpha+\epsilon*v(\theta,t) on L^2(S^1,d\theta) which was
discussed earlier in the literature by Howland
- …