The inverse spectral problem for indefinite strings

Eckhardt, Jonathan; Kostenko, Aleksey

doi:10.1007/s00222-015-0629-1

Cited by 44 publications

(137 citation statements)

References 67 publications

(96 reference statements)

Supporting

Mentioning

135

Contrasting

Order By: Relevance

“…where z is a complex spectral parameter. Of course, this differential equation has to be understood in a distributional way in general; compare [17,23,26,33]. All of the facts stated without proofs in this section may be found in [17, Appendix A], [18, Appendices A and B], [22].…”

Section: The Basic Differential Equationmentioning

confidence: 99%

Trace formulas and continuous dependence of spectra for the periodic conservative Camassa–Holm flow

Eckhardt

Kostenko

Nicolussi

2020

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

This article is concerned with the isospectral problemfor the periodic conservative Camassa-Holm flow, where ω is a periodic real distribution in H −1 loc (R) and υ is a periodic non-negative Borel measure on R. We develop basic Floquet theory for this problem, derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak * topology.

show abstract

Section: The Basic Differential Equationmentioning

confidence: 99%

Trace formulas and continuous dependence of spectra for the periodic conservative Camassa–Holm flow

Eckhardt

Kostenko

Nicolussi

2020

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…. , q N are given recursively by (16). Upon defining the quantities l N +1 = γ , ω N +1 = α and υ N +1 = β as well as the polynomials p N +1 and q N +1 via setting…”

Section: Lemma B If M Is a Rationalmentioning

confidence: 99%

“…. , q N are defined recursively via (16). Because p N and q N must not have any common zeros, we may conclude that…”

Section: Lemma B If M Is a Rationalmentioning

confidence: 99%

Unique Solvability of a Coupling Problem for Entire Functions

Eckhardt

2017

Constr Approx

Self Cite

View full text Add to dashboard Cite

We establish the unique solvability of a coupling problem for entire functions that arises in inverse spectral theory for singular second-order ordinary differential equations/two-dimensional first-order systems and is also of relevance for the integration of certain nonlinear wave equations.Keywords Coupling problem for entire functions · Unique solvability · Inverse spectral theory Mathematics Subject Classification Primary 30D20, 34A55 · Secondary 34B05, 37K15 ResultsLet σ be a discrete set of nonzero real numbers such that the sumCommunicated by Arno Kuijlaars. For a given sequence η ∈R σ (referred to as coupling constants or data), where we denote byR = R∪{∞} the one-point compactification of R, we consider the following task. Coupling problem(G) Growth and positivity condition:Let us first assume that the pair ( − , + ) is a solution of the coupling problem with data η. The growth and positivity condition (G) means that the function Upon invoking the open mapping theorem, this first of all guarantees that all zeros of the functions − and + are real. It furthermore entails that the zeros of the function in the numerator of (3) and the zeros of the function in the denominator of (3) are interlacing (after possible cancelations); see [29, Theorem 27.2.1]. From this we may conclude that the functions − and + are actually of genus zero and satisfy the boundIndeed, this inequality follows essentially from roughly estimating the individual factors in the corresponding Hadamard representation, with the normalization condition (N) taken into account, and employing the interlacing property mentioned above.1 To be precise, this condition has to be read as + (λ) = 0 whenever η(λ) = ∞. We should emphasize here that this upper bound is always independent of the actual coupling constants η. On the other hand, the condition (G) also tells us that the residues of the function in (3) at all poles are negative. In conjunction with the coupling condition (C), this impliesfor all those λ ∈ σ for which the coupling constant η(λ) is finite. Unless it happens that λ is a zero of the function + , this constitutes a necessary restriction on the sign of the coupling constant η(λ) in order for a solution of the coupling problem to exist. Roughly speaking, the coupling constants are expected to have alternating signs beginning with nonnegative ones for those corresponding to the smallest (in modulus) positive and negative element of σ . Motivated by these considerations and the nature of our applications, we introduce the following terminology.Definition Coupling constants η ∈R σ are called admissible if the inequalityholds for all those λ ∈ σ for which η(λ) is finite.The main purpose of the present article is to prove that this simple condition is sufficient to guarantee unique solvability of the corresponding coupling problem. Theorem (Existence and Uniqueness) If the coupling constants η ∈R σ are admissible, then the coupling problem with data η has a unique solution.Apart from this result, we will also establish the fact ...

show abstract

“…By allowing q to change signs, we can study problems where the weight is related to the index of refraction of the media, and problems of population dynamics where the weight represents the intrinsic growth rate of species, and it is positive (resp., negative) in the favorable (resp., unfavorable) zone of habitat, see [3,16]. Recently, Eckhardt and Kostenko studied the spectral inverse problem for indefinite strings in terms of the Weyl-Titchmarsh function, motivated by the study of Camassa-Holm and Hunter-Saxton equations, see [7] for details. We give a different proof of Theorem 7 in [10], using only the weaker estimate (3) instead of (4), without using any information of the lengths of nodal domains, and we extend it to indefinite weights.…”

Section: Introductionmentioning

confidence: 99%