General spectral flow formula for fixed maximal domain

Bernhelm Booss-Bavnbek; Chaofeng Zhu

Open Mathematics (2005)

  • Volume: 3, Issue: 3, page 558-577
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Bernhelm Booss-Bavnbek, and Chaofeng Zhu. "General spectral flow formula for fixed maximal domain." Open Mathematics 3.3 (2005): 558-577. <http://eudml.org/doc/268740>.

@article{BernhelmBooss2005,
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.},
author = {Bernhelm Booss-Bavnbek, Chaofeng Zhu},
journal = {Open Mathematics},
keywords = {58J30; 53D12},
language = {eng},
number = {3},
pages = {558-577},
title = {General spectral flow formula for fixed maximal domain},
url = {http://eudml.org/doc/268740},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Bernhelm Booss-Bavnbek
AU - Chaofeng Zhu
TI - General spectral flow formula for fixed maximal domain
JO - Open Mathematics
PY - 2005
VL - 3
IS - 3
SP - 558
EP - 577
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.
LA - eng
KW - 58J30; 53D12
UR - http://eudml.org/doc/268740
ER -

References

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