1,456 research outputs found
Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map
We show that discrete one-dimensional Schr\"odinger operators on the
half-line with ergodic potentials generated by the doubling map on the circle,
, may be realized as the half-line restrictions of
a non-deterministic family of whole-line operators. As a consequence, the
Lyapunov exponent is almost everywhere positive and the absolutely continuous
spectrum is almost surely empty.Comment: 4 page
Reflection symmetries of almost periodic functions
We study global reflection symmetries of almost periodic functions. In the
non-limit periodic case, we establish an upper bound on the Haar measure of the
set of those elements in the hull which are almost symmetric about the origin.
As an application of this result we prove that in the non-limit periodic case,
the criterion of Jitomirskaya and Simon ensuring absence of eigenvalues for
almost periodic Schr\"odinger operators is only applicable on a set of zero
Haar measure. We complement this by giving examples of limit periodic functions
where the Jitomirskaya-Simon criterion can be applied to every element of the
hull.Comment: 6 page
Autocorrelations of the characteristic polynomial of a random matrix under microscopic scaling
We calculate the autocorrelation function for the characteristic polynomial
of a random matrix in the microscopic scaling regime. While results fitting
this description have be proved before, we will cover all values of inverse
temperature . The method also differs from prior work,
relying on matrix models introduced by Killip and Nenciu
Half-line Schrodinger Operators With No Bound States
We consider Sch\"odinger operators on the half-line, both discrete and
continuous, and show that the absence of bound states implies the absence of
embedded singular spectrum. More precisely, in the discrete case we prove that
if has no spectrum outside of the interval , then it has
purely absolutely continuous spectrum. In the continuum case we show that if
both and have no spectrum outside ,
then both operators are purely absolutely continuous. These results extend to
operators with finitely many bound states.Comment: 34 page
Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory
We discuss the proof of and systematic application of Case's sum rules for
Jacobi matrices. Of special interest is a linear combination of two of his sum
rules which has strictly positive terms. Among our results are a complete
classification of the spectral measures of all Jacobi matrices J for which
J-J_0 is Hilbert--Schmidt, and a proof of Nevai's conjecture that the Szego
condition holds if J-J_0 is trace class.Comment: 69 pages, published versio
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