Peter Wittwer

Localization for a class of one dimensional quasi-periodic Schrödinger operators

AbstractWe prove for small ɛ and α satisfying a certain Diophantine condition the operator

Intermittency in the presence of noise

H = - \varepsilon ^2 \Delta + \frac{1}{{2\pi }}\cos 2\pi (j\alpha + \theta ) j \in \mathbb{Z}

A non-Gaussian renormalization group fixed point for hierarchical scalar lattice field theories

has pure point spectrum for almost all θ. A similar result is established at low energy for

A renormalization group for Hamiltonians: numerical results

H = - \frac{{d^2 }}{{dx^2 }} - K^2 (\cos 2\pi x + \cos 2\pi (\alpha x + \theta ))

Fixed points of Feigenbaum's type for the equationf p (λx)≡λf(x)

providedK is sufficiently large.

A nontrivial renormalization group fixed point for the Dyson-Baker hierarchical model

The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map ℳ. This map acts on functionsΦ that are jointly analytic in a “position variable” (t) and in the parameter (μ) that controls the period doubling phenomenon. A fixed pointΦ* for this map is found. The usual renormalization group doubling operatorN acts on this functionΦ* simply by multiplication ofμ with the universal Feigenbaum ratioδ*= 4.669201..., i.e., (NΦ*(μ,t)=Φ*(δ*μ,t). Therefore, the one-parameter family of functions,Ψμ*,Ψμ*(t)=(Φ*(μ,t), is invariant underN. In particular, the functionΨ0* is the Feigenbaum fixed point ofN, whileΨμ* represents the unstable manifold ofN. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally.