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Register a new userDiscrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices
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Alexander Sakhnovich
Publication date
2019
fields
Physics
and research's language is
English
Authors
Alexander Sakhnovich
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We consider discrete Dirac systems as an alternative (to the famous SzegH{o} recurrencies and matrix orthogonal polynomials) approach to the study of the corresponding block Toeplitz matrices. We prove an analog of the Christoffel--Darboux formula and derive the asymptotic relations for the analog of reproducing kernel (using Weyl--Titchmarsh functions of discrete Dirac systems). We study also the case of rational Weyl--Titchmarsh functions (and GBDT version of the Backlund-Darboux transformation of the trivial discrete Dirac system). We show that block diagonal plus block semi-separable Toeplitz matrices appear in this case.
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We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Greens functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.
We study matrix roots with certain commutation properties and their application to the explicit construction of Darboux matrices in the framework of the GBDT version of Backlund-Darboux transformation. The approach is demonstrated on the important case of generalised canonical systems depending rationally on the spectral parameter.
Depending on the behaviour of the complex-valued electromagnetic potential in the neighbourhood of infinity, pseudomodes of one-dimensional Dirac operators corresponding to large pseudoeigenvalues are constructed. This is a first systematic non-semi-classical approach, which results in substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including superexponential ones.
We investigate self-adjoint extensions of the minimal Kirchhoff Laplacian on an infinite metric graph. More specifically, the main focus is on the relationship between graph ends and the space of self-adjoint extensions of the corresponding minimal Kirchhoff Laplacian $mathbf{H}_0$. First, we introduce the notion of finite and infinite volume for (topological) ends of a metric graph and then establish a lower bound on the deficiency indices of $mathbf{H}_0$ in terms of the number of finite volume graph ends. This estimate is sharp and we also find a necessary and sufficient condition for the equality between the number of finite volume graph ends and the deficiency indices of $mathbf{H}_0$ to hold. Moreover, it turns out that finite volume graph ends play a crucial role in the study of Markovian extensions of $mathbf{H}_0$. In particular, we show that the minimal Kirchhoff Laplacian admits a unique Markovian extension exactly when every topological end of the underlying metric graph has infinite volume. In the case of finitely many finite volume ends (for instance, the latter includes Cayley graphs of a large class of finitely generated infinite groups) we are even able to provide a complete description of all Markovian extensions of $mathbf{H}_0$.
Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=int_0^{infty} lambda dE_{ L}({lambda})$ be the spectral resolution of ${ L}$ and $S_R({ L})f=int_0^R dE_{ L}(lambda) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a sufficient condition on $L$ such that $$ lim_{Rrightarrow infty} S_R({ L})f(x) =f(x), {rm a.e.} $$ for any $f$ such that ${rm log } (2+L) fin L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrodinger operators with inverse square potentials.
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