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Register a new userScattering theory for graphs isomorphic to a homogeneous tree at infinity
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Added by
Yves Colin de Verdiere
Publication date
2012
fields
Physics
and research's language is
English
Authors
Yves Colin De Verdi`ere
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We describe the spectral theory of the adjacency operator of a graph which is isomorphic to homogeneous trees at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operator on an homogeneous tree. We developp this scattering theory using the classical recipes for Schrodinger operators in Euclidian spaces.
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We investigate the distribution of the resonances near spectral thresholds of Laplace operators on regular tree graphs with $k$-fold branching, $k geq 1$, perturbed by nonself-adjoint exponentially decaying potentials. We establish results on the absence of resonances which in particular involve absence of discrete spectrum near some sectors of the essential spectrum of the operators.
Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associated to the coupled and decoupled operators are derived. In particular, it is shown that a CMV matrix is reflectionless iff the scattering matrix is off-diagonal which in turn provides a short proof of an important result of [Breuer-Ryckman-Simon]. These developments parallel those recently obtained for Jacobi matrices.
We describe some basic tools in the spectral theory of Schrodinger operator on metric graphs (also known as quantum graph) by studying in detail some basic examples. The exposition is kept as elementary and accessible as possible. In the later sections we apply these tools to prove some results on the count of zeros of the eigenfunctions of quantum graphs.
We study the spectral and scattering theory of light transmission in a system consisting of two asymptotically periodic waveguides, also known as one-dimensional photonic crystals, coupled by a junction. Using analyticity techniques and commutator methods in a two-Hilbert spaces setting, we determine the nature of the spectrum and prove the existence and completeness of the wave operators of the system.
In this paper we give a new and constructive approach to stationary scattering theory for pairs of self-adjoint operators $H_0$ and $H_1$ on a Hilbert space $mathcal H$ which satisfy the following conditions: (i) for any open bounded subset $Delta$ of $mathbb R,$ the operators $F E_Delta^{H_0}$ and $F E_Delta^{H_1}$ are Hilbert-Schmidt and (ii) $V = H_1- H_0$ is bounded and admits decomposition $V = F^*JF,$ where $F$ is a bounded operator with trivial kernel from $mathcal H$ to another Hilbert space $mathcal K$ and $J$ is a bounded self-adjoint operator on $mathcal K.$ An example of a pair of operators which satisfy these conditions is the Schrodinger operator $H_0 = -Delta + V_0$ acting on $L^2(mathbb R^ u),$ where $V_0$ is a potential of class $K_ u$ (see B.,Simon, {it Schrodinger semigroups,} Bull. AMS 7, 1982, 447--526) and $H_1 = H_0 + V_1,$ where $V_1 in L^infty(mathbb R^ u) cap L^1(mathbb R^ u).$ Among results of this paper is a new proof of existence and completeness of wave operators $W_pm(H_1,H_0)$ and a new constructive proof of stationary formula for the scattering matrix. This approach to scattering theory is based on explicit diagonalization of a self-adjoint operator $H$ on a sheaf of Hilbert spaces $EuScript S(H,F)$ associated with the pair $(H,F)$ and with subsequent construction and study of properties of wave matrices $w_pm(lambda; H_1,H_0)$ acting between fibers $mathfrak h_lambda(H_0,F)$ and $mathfrak h_lambda(H_1,F)$ of sheaves $EuScript S(H_0,F)$ and $EuScript S(H_1,F)$ respectively. The wave operators $W_pm(H_1,H_0)$ are then defined as direct integrals of wave matrices and are proved to coincide with classical time-dependent definition of wave operators.
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