Browsing by Author "Tiedra de Aldecoa, R."

Now showing 1 - 5 of 5
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    A formula relating sojourn times to the time of arrival in Hamiltonian dynamics
    (2012) Gournay, A.; Tiedra de Aldecoa, R.
    We consider on a manifold M equipped with a Poisson bracket {., .} a Hamiltonian H with complete flow and a family Phi = (Phi(1), ... , Phi(d)) of abstract position observables satisfying the condition {{Phi(j), H}, H} = 0 for each j. Under these assumptions, we prove a new formula relating sojourn times in dilated regions defined in terms of Phi to the time of arrival of classical orbits. The correspondence between this formula and a formula established recently in the framework of quantum mechanics is put into evidence. Among other examples, our theory applies to Stark Hamiltonians, homogeneous Hamiltonians, purely kinetic Hamiltonians, the repulsive harmonic potential, central force systems, the Poincare ball model, the wave equation, the nonlinear Schrodinger equation, the Korteweg-de Vries equation and quantum Hamiltonians defined via expectation values.
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    Decay Estimates for Unitary Representations with Applications to Continuous- and Discrete-Time Models
    (2023) Richard, S.; Tiedra de Aldecoa, R.
    We present a new technique to obtain polynomial decay estimates for the matrix coefficients of unitary operators. Our approach, based on commutator methods, applies to nets of unitary operators, unitary representations of topological groups, and unitary operators given by the evolution group of a self-adjoint operator or by powers of a unitary operator. Our results are illustrated with a wide range of examples in quantum mechanics and dynamical systems, as for instance Schrodinger operators, Dirac operators, quantum waveguides, horocycle flows, adjacency matrices, Jacobi matrices, quantum walks or skew products.
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    Discrete Laplacian in a half-space with a periodic surface potential I: Resolvent expansions, scattering matrix, and wave operators
    (2022) Nguyen, H. S.; Richard, S.; Tiedra de Aldecoa, R.
    We present a detailed study of the scattering system given by the Neumann Laplacian on the discrete half-space perturbed by a periodic potential at the boundary. We derive asymptotic resolvent expansions at thresholds and eigenvalues, we prove the continuity of the scattering matrix, and we establish new formulas for the wave operators. Along the way, our analysis puts into evidence a surprising relation between some properties of the potential, like the parity of its period, and the behaviour of the integral kernel of the wave operators.
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    Spectral and scattering theory of one-dimensional coupled photonic crystals
    (WORLD SCIENTIFIC PUBL CO PTE LTD, 2021) De Nittis, G.; Moscolari, M.; Richard, S.; Tiedra de Aldecoa, R.
    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.
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    Time delay is a common feature of quantum scattering theory
    (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2012) Richard, S.; Tiedra de Aldecoa, R.
    We prove that the existence of time delay defined in terms of sojourn times, as well as its identity with Eisenbud-Wigner time delay, is a common feature of two-Hilbert spaces quantum scattering theory. All statements are model-independent. (C) 2011 Elsevier Inc. All rights reserved.