190 research outputs found
From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables
This article is a part of a project investigating the relationship between
the dynamics of completely integrable or close to completely integrable
billiard tables, the integral geometry on them, and the spectrum of the
corresponding Laplace-Beltrami operators. It is concerned with new isospectral
invariants and with the spectral rigidity problem for families of
Laplace-Beltrami operators with Dirichlet, Neumann or Robin boundary
conditions, associated with C^1 families of billiard tables. We introduce a
notion of weak isospectrality for such deformations. The main dynamical
assumption on the initial billiard table is that the corresponding billiard
ball map or an iterate of it has a Kronecker invariant torus with a Diophantine
frequency and that the corresponding Birkhoff Normal Form is nondegenerate in
Kolmogorov sense. Then we obtain C^1 families of Kronecker tori with
Diophantine frequencies. If the family of the Laplace-Beltrami operators
satisfies the weak isospectral condition, we prove that the average action on
the tori and the Birkhoff Normal Form of the billiard ball maps remain the same
along the perturbation. As an application we obtain infinitesimal spectral
rigidity for Liouville billiard tables in dimensions two and three.
Applications are obtained also for strictly convex billiard tables of dimension
two as well as in the case when the initial billiard table admits an elliptic
periodic billiard trajectory. Spectral rigidity of billard tables close
elliptical billiard tables is obtained. The results are based on a construction
of C^1 families of quasi-modes associated with the Kronecker tori and on
suitable KAM theorems for C^1 families of Hamiltonians.Comment: 170 pages; new results about the spectral rigidity of elliptical
billiard tables; new Modified Iterative Lemma in the proof of KAM theorem
with parameter
On the symplectic phase space of KdV
We prove that the Birkhoff map \Om for KdV constructed on H^{-1}_0(\T)
can be interpolated between H^{-1}_0(\T) and L^2_0(\T). In particular, the
symplectic phase space H^{1/2}_0(\T) can be described in terms of Birkhoff
coordinates. As an application, we characterize the regularity of a potential
q\in H^{-1}(\T) in terms of the decay of the gap lengths of the periodic
spectrum of Hill's operator on the interval
Solutions of mKdV in classes of functions unbounded at infinity
In 1974 P. Lax introduced an algebro-analytic mechanism similar to the Lax
L-A pair. Using it we prove global existence and uniqueness for solutions of
the initial value problem for mKdV in classes of smooth functions which can be
unbounded at infinity, and may even include functions which tend to infinity
with respect to the space variable. Moreover, we establish the invariance of
the spectrum and the unitary type of the Schr{\"o}dinger operator under the KdV
flow and the invariance of the spectrum and the unitary type of the impedance
operator under the mKdV flow for potentials in these classes.Comment: 35 pages, new results about spectra and eigenfunctions of
Schr\"odinger operators added, new references adde
Interpolation of nonlinear maps
Let and be complex Banach couples and assume that
with norms satisfying for
some . For any , denote by
and the complex interpolation spaces and by
, the open ball of radius in
, centered at zero. Then for any analytic map such that and
are continuous and bounded by constants and , respectively, the
restriction of to , is
shown to be a map with values in which is analytic and bounded by
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