Trace Formulas for Schrödinger Operators in Connection with Scattering Theory for Finite-gap Backgrounds

“…While several aspects in the steplike case are similar to the decaying case, there are also some distinctive differences due to the presence of spectrum of multiplicity one. Moreover, there have also been further generalizations to the case of periodic backgrounds made by Firsova [21,22,23] and to steplike finite-gap backgrounds by Boutet de Monvel and two of us [7] (see also [39]) and to steplike almost periodic backgrounds by Grunert [26,27]. We refer to these publications for more information.…”

Section: Introductionmentioning

confidence: 95%

Inverse Scattering Theory for Schrodinger Operators with Steplike Potentials

Egorova¹,

Gladka²,

Lange³

2015

Z. mat. fiz. anal. geom.

View full text Add to dashboard Cite

Abstract. We study the direct and inverse scattering problem for the onedimensional Schrödinger equation with steplike potentials. We give necessary and sufficient conditions for the scattering data to correspond to a potential with prescribed smoothness and prescribed decay to their asymptotics. These results are important for solving the Korteweg-de Vries equation via the inverse scattering transform.

show abstract

“…In this section we recall some basic facts from the inverse scattering transform for our setting. For further background and proofs we refer to [4], [14], and [12] (see also [29]).…”

Section: The Inverse Scattering Transform and The Riemann-hilbert Pro...mentioning

confidence: 99%

Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions

Mikikits-Leitner,

Teschl

2010

Preprint

Self Cite

View full text Add to dashboard Cite

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 x/t plane splits into g + 1 soliton regions which are interlaced by g + 1 oscillatory regions, where g + 1 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 a 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.

show abstract

Trace Formulas for Schrödinger Operators in Connection with Scattering Theory for Finite-gap Backgrounds

Cited by 5 publications

References 35 publications

On the Cauchy problem for the Kortewegde Vries equation with steplike finite-gap initial data II. Perturbations with finite moments

On the Cauchy problem for the Kortewegde Vries equation with steplike finite-gap initial data II. Perturbations with finite moments

Inverse Scattering Theory for Schrodinger Operators with Steplike Potentials

Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions

Contact Info

Product

Resources

About