5 research outputs found
Splitting Methods for SU(N) Loop Approximation
The problem of finding the correct asymptotic rate of approximation by
polynomial loops in dependence of the smoothness of the elements of a loop
group seems not well-understood in general. For matrix Lie groups such as
SU(N), it can be viewed as a problem of nonlinearly constrained trigonometric
approximation. Motivated by applications to optical FIR filter design and
control, we present some initial results for the case of SU(N)-loops, N>1. In
particular, using representations via the exponential map and ideas from
splitting methods, we prove that the best approximation of an SU(N)-loop
belonging to a Hoelder-Zygmund class Lip_alpha with alpha>1/2 by a polynomial
SU(N)-loop of degree n is of the order O(n^{-\alpha/(1+\alpha)}) as n tends to
infinity. Although this approximation rate is not considered final (and can be
improved in special cases), to our knowledge it is the first general,
nontrivial result of this type.Comment: 14 pages, still submitted to J. Approx. Th. typos corrected, part of
the proof of Lemma 4 concerning the auxiliary statement on page 8 rewritten
in a clearer wa
Spectral Relationships Between Kicked Harper and On-Resonance Double Kicked Rotor Operators
Kicked Harper operators and on-resonance double kicked rotor operators model
quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent
computational studies indicate a striking resemblance between the spectrums of
these operators. In this paper we apply C*-algebra methods to explain this
resemblance. We show that each pair of corresponding operators belong to a
common rotation C*-algebra B_\alpha, prove that their spectrums are equal if
\alpha is irrational, and prove that the Hausdorff distance between their
spectrums converges to zero as q increases if \alpha = p/q with p and q coprime
integers. Moreover, we show that corresponding operators in B_\alpha are
homomorphic images of mother operators in the universal rotation C*-algebra
A_\alpha that are unitarily equivalent and hence have identical spectrums.
These results extend analogous results for almost Mathieu operators. We also
utilize the C*-algebraic framework to develop efficient algorithms to compute
the spectrums of these mother operators for rational \alpha and present
preliminary numerical results that support the conjecture that their spectrums
are Cantor sets if \alpha is irrational. This conjecture for almost Mathieu
operators, called the Ten Martini Problem, was recently proved after intensive
efforts over several decades. This proof for the almost Mathieu operators
utilized transfer matrix methods, which do not exist for the kicked operators.
We outline a strategy, based on a special property of loop groups of semisimple
Lie groups, to prove this conjecture for the kicked operators.Comment: 26 pages, 8 figure