General spectral flow formula for fixed maximal domain

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    Resumé

    We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.
    OriginalsprogEngelsk
    TidsskriftCentral European Journal of Mathematics
    Vol/bind3
    Udgave nummer3
    Sider (fra-til)558-577
    Antal sider20
    ISSN1895-1074
    StatusUdgivet - 2005

    Citer dette

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    title = "General spectral flow formula for fixed maximal domain",
    abstract = "We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.",
    keywords = "Spectral flow, Maslov index, elliptic boundary value problems",
    author = "Bernhelm Booss-Bavnbek and Chaofeng Zhu",
    year = "2005",
    language = "English",
    volume = "3",
    pages = "558--577",
    journal = "Central European Journal of Mathematics",
    issn = "1895-1074",
    publisher = "Versita",
    number = "3",

    }

    General spectral flow formula for fixed maximal domain. / Booss-Bavnbek, Bernhelm; Zhu, Chaofeng.

    I: Central European Journal of Mathematics, Bind 3, Nr. 3, 2005, s. 558-577.

    Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

    TY - JOUR

    T1 - General spectral flow formula for fixed maximal domain

    AU - Booss-Bavnbek, Bernhelm

    AU - Zhu, Chaofeng

    PY - 2005

    Y1 - 2005

    N2 - We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.

    AB - We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.

    KW - Spectral flow, Maslov index, elliptic boundary value problems

    M3 - Journal article

    VL - 3

    SP - 558

    EP - 577

    JO - Central European Journal of Mathematics

    JF - Central European Journal of Mathematics

    SN - 1895-1074

    IS - 3

    ER -