502 research outputs found
Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross-Pitaevskii Equation
We rigorously analyze the bifurcation of stationary so called nonlinear Bloch
waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a
periodic potential, in arbitrary space dimensions. These are solutions which
can be expressed as finite sums of quasi-periodic functions, and which in a
formal asymptotic expansion are obtained from solutions of the so called
algebraic coupled mode equations. Here we justify this expansion by proving the
existence of NLBs and estimating the error of the formal asymptotics. The
analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In
addition, we illustrate some relations of NLBs to other classes of solutions of
the GP equation, in particular to so called out--of--gap solitons and truncated
NLBs, and present some numerical experiments concerning the stability of these
solutions.Comment: 32 pages, 12 figures, changes: discussion of assumptions reorganized,
a new section on stability of the studied solutions, 15 new references adde
On the singular spectrum for adiabatic quasi-periodic Schrodinger operators on the real line
In this paper, we study spectral properties of a family of quasi-periodic
Schrodinger operators on the real line in the adiabatic limit. We assume that
the adiabatic iso-energetic curves are extended along the momentum direction.
In the energy intervals where this happens, we obtain an asymptotic formula for
the Lyapunov exponent, and show that the spectrum is purely singular
Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities
The Lyapunov exponent characterizes the asymptotic behavior of long matrix
products. Recognizing scenarios where the Lyapunov exponent is strictly
positive is a fundamental challenge that is relevant in many applications. In
this work we establish a novel tool for this task by deriving a quantitative
lower bound on the Lyapunov exponent in terms of a matrix sum which is
efficiently computable in ergodic situations. Our approach combines two deep
results from matrix analysis --- the -matrix extension of the
Golden-Thompson inequality and the Avalanche-Principle. We apply these bounds
to the Lyapunov exponents of Schr\"odinger cocycles with certain ergodic
potentials of polymer type and arbitrary correlation structure. We also derive
related quantitative stability results for the Lyapunov exponent near aligned
diagonal matrices and a bound for almost-commuting matrices.Comment: 46 pages; comments welcom
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