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Benoit, William L.; Fulmer, Hal W.; Dunn, M. G.; Siltanen, Susan A.

doi:10.1080/10417948809372749

Cited by 1 publication

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“…A density argument shows that we may take ψ j and ϕ j in S(C) without loss. Thus, it suffices to show that ϕ, e iτ S ψ = 0 (32) and that, for each fixed T > 0,…”

Section: Continuous Dependence Of Reconstructed Q On Tmentioning

confidence: 99%

“…In section 4, we establish continuity in q for the direct scattering map T , and continuity in s(k) of the inverse scattering map Q. In section 5, we exploit ideas of Grinevich and Manakov [13] to prove necessary and sufficient conditions for the scattering data t(k) to be the scattering data of a real potential q(x) repairing a gap [32] in the proof of [22, theorem 4.1]. We apply the ideas of Grinevich-Manakov [13] directly to the zero-energy case instead of treating it as a limit of negative-energy inverse scattering (see also Grinevich [12]).…”

Section: Introductionmentioning

confidence: 99%

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Global Solutions for the zero-energy Novikov–Veselov equation by inverse scattering

Music

Perry

2018

Nonlinearity

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Using the inverse scattering method, we construct global solutions to the Novikov-Veselov equation for real-valued decaying initial data q 0 with the property that the associated Schrödinger operator −∂ x ∂ x + q 0 is nonnegative. Such initial data are either critical (an arbitrarily small perturbation of the potential makes the operator nonpositive) or subcritical (sufficiently small perturbations of the potential preserve non-negativity of the operator). Previously, Lassas, Mueller, Siltanen and Stahel proved global existence for critical potentials, also called potentials of conductivity type. We extend their results to include the much larger class of subcritical potentials. We show that the subcritical potentials form an open set and that the critical potentials form the nowhere dense boundary of this open set. Our analysis draws on previous work of the first author and on ideas of Grinevich and Manakov.

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“…A density argument shows that we may take ψ j and ϕ j in S(C) without loss. Thus, it suffices to show that ϕ, e iτ S ψ = 0 (32) and that, for each fixed T > 0,…”

Section: Continuous Dependence Of Reconstructed Q On Tmentioning

confidence: 99%