7 research outputs found
Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function
We develop an analog of classical oscillation theory for Sturm-Liouville
operators which, rather than measuring the spectrum of one single operator,
measures the difference between the spectra of two different operators.
This is done by replacing zeros of solutions of one operator by weighted
zeros of Wronskians of solutions of two different operators. In particular, we
show that a Sturm-type comparison theorem still holds in this situation and
demonstrate how this can be used to investigate the finiteness of eigenvalues
in essential spectral gaps. Furthermore, the connection with Krein's spectral
shift function is established.Comment: 26 page
Lieb-Robinson Bounds for the Toda Lattice
We establish locality estimates, known as Lieb-Robinson bounds, for the Toda
lattice. In contrast to harmonic models, the Lieb-Robinson velocity for these
systems do depend on the initial condition. Our results also apply to the
entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable
assumptions, our methods also yield a finite velocity for certain perturbations
of these systems
Stability of the periodic Toda lattice under short range perturbations
We consider the stability of the periodic Toda lattice (and slightly more
generally of the algebro-geometric finite-gap lattice) under a short range
perturbation. We prove that the perturbed lattice asymptotically approaches a
modulated lattice.
More precisely, let be the genus of the hyperelliptic curve associated
with the unperturbed solution. We show that, apart from the phenomenon of the
solitons travelling on the quasi-periodic background, the -pane contains
areas where the perturbed solution is close to a finite-gap solution in
the same isospectral torus. In between there are regions where the
perturbed solution is asymptotically close to a modulated lattice which
undergoes a continuous phase transition (in the Jacobian variety) and which
interpolates between these isospectral solutions. In the special case of the
free lattice () the isospectral torus consists of just one point and we
recover the known result.
Both the solutions in the isospectral torus and the phase transition are
explicitly characterized in terms of Abelian integrals on the underlying
hyperelliptic curve.
Our method relies on the equivalence of the inverse spectral problem to a
matrix Riemann--Hilbert problem defined on the hyperelliptic curve and
generalizes the so-called nonlinear stationary phase/steepest descent method
for Riemann--Hilbert problem deformations to Riemann surfaces.Comment: 38 pages, 1 figure. This version combines both the original version
and arXiv:0805.384
Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data
We develop direct and inverse scattering theory for Jacobi operators (doubly
infinite second order difference operators) with steplike coefficients which
are asymptotically close to different finite-gap quasi-periodic coefficients on
different sides. We give necessary and sufficient conditions for the scattering
data in the case of perturbations with finite second (or higher) moment.Comment: 23 page
On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data I. Schwartz-Type Perturbations
We solve the Cauchy problem for the Korteweg-de Vries equation with initial
conditions which are steplike Schwartz-type perturbations of finite-gap
potentials under the assumption that the respective spectral bands either
coincide or are disjoint.Comment: 29 page
Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions
We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of solutions of the Korteweg--de Vries equation which are decaying
perturbations of a quasi-periodic finite-gap background solution. We compute a
nonlinear dispersion relation and show that the plane splits into
soliton regions which are interlaced by oscillatory regions, where
is the number of spectral gaps.
In the soliton regions the solution is asymptotically given by a number of
solitons travelling on top of finite-gap solutions which are in the same
isospectral class as the background solution. In the oscillatory region the
solution can be described by a modulated finite-gap solution plus a decaying
dispersive tail. The modulation is given by phase transition on the isospectral
torus and is, together with the dispersive tail, explicitly characterized in
terms of Abelian integrals on the underlying hyperelliptic curve.Comment: 45 pages. arXiv admin note: substantial text overlap with
arXiv:0705.034