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Register a new userThe Classical Moment Problem and Generalized Indefinite Strings
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We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. Namely, we introduce the class of Krein-Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of M. G. Krein on the connection between the Stieltjes moment problem and Krein-Stieltjes strings and I. S. Kac on the connection between the Hamburger moment problem and 2x2 canonical systems with Hamburger Hamiltonians.
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We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set $(-infty,-1/2]cup [1/2,infty)$.
We establish the unique solvability of a coupling problem for entire functions which 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.
Let $ast_P$ be a product on $l_{rm{fin}}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{infty}$ of real polynomials on $mathbb{R}$. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of $ast_P$-positive functionals on $l_{rm{fin}}$. If $(P_n)_{n=0}^{infty}$ is a family of the Newton polynomials $P_n(x)=prod_{i=0}^{n-1}(x-i)$ then the corresponding product $star=ast_P$ is an analog of the so-called Kondratiev--Kuna convolution on a Fock space. We get an explicit expression for the product $star$ and establish a connection between $star$-positive functionals on $l_{rm{fin}}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).
We employ some results about continued fraction expansions of Herglotz-Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. In particular, this solves the corresponding inverse spectral problem through explicit formulas.
In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced both in Riemann-Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.
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