253 research outputs found

    On the connection between the Hilger and Radon--Nikodym derivatives

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    We show that the Hilger derivative on time scales is a special case of the Radon--Nikodym derivative with respect to the natural measure associated with every time scale. Moreover, we show that the concept of delta absolute continuity agrees with the one from measure theory in this context.Comment: 7 page

    Algebro-Geometric Constraints on Solitons with Respect to Quasi-Periodic Backgrounds

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    We investigate the algebraic conditions the scattering data of short-range perturbations of quasi-periodic finite-gap Jacobi operators have to satisfy. As our main result we provide the Poisson-Jensen-type formula for the transmission coefficient in terms of Abelian integrals on the underlying hyperelliptic Riemann surface and give an explicit condition for its single-valuedness. In addition, we establish trace formulas which relate the scattering data to the conserved quantities in this case.Comment: 9 pages. Bull. London Math. Soc. (to appear

    Relative Oscillation Theory for Sturm-Liouville Operators Extended

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    We extend relative oscillation theory to the case of Sturm--Liouville operators Hu=rβˆ’1(βˆ’(puβ€²)β€²+qu)H u = r^{-1}(-(pu')'+q u) with different pp's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.Comment: 16 page

    A coupling problem for entire functions and its application to the long-time asymptotics of integrable wave equations

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    We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case when the underlying isospectral problem has purely discrete spectrum. To this end, we introduce a natural coupling problem for entire functions, which serves as a replacement for the usual Riemann-Hilbert problem, which does not apply in these cases. As a prototypical example, we investigate the long-time asymptotics of the dispersionless Camassa-Holm equation.Comment: 11 page

    Stability of the Periodic Toda Lattice in the Soliton Region

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    We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the periodic (and slightly more generally of the quasi-periodic finite-gap) Toda lattice for decaying initial data in the soliton region. In addition, we show how to reduce the problem in the remaining region to the known case without solitons.Comment: 28 page

    Reconstruction of the Transmission Coefficient for Steplike Finite-Gap Backgrounds

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    We consider scattering theory for one-dimensional Jacobi operators with respect to steplike quasi-periodic finite-gap backgrounds and show how the transmission coefficient can be reconstructed from minimal scattering data. This generalizes the Poisson-Jensen formula for the classical constant background case.Comment: 9 page

    Singular Weyl-Titchmarsh-Kodaira Theory for Jacobi Operators

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    We develop singular Weyl-Titchmarsh-Kodaira theory for Jacobi operators. In particular, we establish existence of a spectral transformation as well as local Borg-Marchenko and Hochstadt-Liebermann type uniqueness results.Comment: 16 page
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