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Register a new userScattering and correlations
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Added by
Yves Colin de Verdiere
Publication date
2011
fields
Physics
and research's language is
English
Authors
Yves Colin De Verdi`ere
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In this paper we give an exact relation between the Greens function in a scattering problem for a wave equation and the correlation of scattered plane waves. This general relation was proved in a special case by Sanchez-Sesma and al.
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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 prove the analogue of the strong Szeg{H o} limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk cite{YP} for the next-to-diagonal correlations $langle sigma_{0,0}sigma_{N-1,N} rangle$ in the anisotropic square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. The anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations is also established. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.
We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph, and the associated unitary scattering operator. We prove that the statistics of level spacings, and moments of observations in the eigenbases coincide in the limit that all bond lengths approach a positive constant value.
Problem solutions in area of diffraction and of scattering theory are considered from one point of view. The method common for them is based on approximate orthogonality of solution constituents, which oscillate on a body long frontier. Method potentiality is discussed.
The S-matrices corresponding to PT-symmetric rho-perturbed operators are defined and calculated by means of an approach based on an operator-theoretical interpretation of the Lax-Phillips scattering theory.
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