Quantum scattering at low energies

Jan Derezinski, Erik Skibsted

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    Abstract

    For a class of negative slowly decaying potentials, including V(x):=−γ|x|μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki–Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.
    Original languageEnglish
    JournalJournal of Functional Analysis
    Volume257
    Issue6
    Pages (from-to)1828-1920
    Number of pages93
    ISSN0022-1236
    DOIs
    Publication statusPublished - 2009

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