Conditions au bord spectrales et formules de trace

Gerd Grubb[1]

  • [1] Copenhagen Univ. Math. Dept., Universitetsparken 5, DK-2100 Copenhague, Danemark.

Séminaire Équations aux dérivées partielles (2001-2002)

  • page 1-12

Abstract

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The lecture presents current results on heat trace expansions, and the related resolvent trace and zeta function expansions, for elliptic operators with boundary conditions on n -dimensional compact manifolds. As a background, we recall the set-up of elliptic differential operators with differential boundary conditions having heat trace expansions in powers ∑ k ≥ - n c k t k / m . Then we consider the spectral boundary conditions of Atiyah, Patodi and Singer for Dirac-type first-order operators, leading to expansions with additional logarithmic terms ∑ k ≥ 0 c k ′ t k / 2 log t (joint work with Seeley 1995) ; an extension to “well-posed” problems is included in a general study of pseudo-normal boundary conditions (1999). New results are presented on the vanishing or stability of the c k ′ -coefficients ; special features appear when n is odd. Finally, we study the pseudodifferential projection boundary conditions proposed by Vassilevich (2001) in string- and brane-theory, showing that they too have heat expansions with log-terms, under suitable hypotheses. In all cases, the lowest log-coefficient c 0 ′ vanishes, which assures that the zeta function is regular at 0.

How to cite

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Grubb, Gerd. "Conditions au bord spectrales et formules de trace." Séminaire Équations aux dérivées partielles (2001-2002): 1-12. <http://eudml.org/doc/11034>.

@article{Grubb2001-2002,
abstract = {The lecture presents current results on heat trace expansions, and the related resolvent trace and zeta function expansions, for elliptic operators with boundary conditions on $n$-dimensional compact manifolds. As a background, we recall the set-up of elliptic differential operators with differential boundary conditions having heat trace expansions in powers $\sum _\{k\ge -n\}c_kt^\{k/m\}$. Then we consider the spectral boundary conditions of Atiyah, Patodi and Singer for Dirac-type first-order operators, leading to expansions with additional logarithmic terms $\sum _\{k\ge 0\}c^\{\prime\}_kt^\{k/2\}\log t$ (joint work with Seeley 1995) ; an extension to “well-posed” problems is included in a general study of pseudo-normal boundary conditions (1999). New results are presented on the vanishing or stability of the $c^\{\prime\}_k$-coefficients ; special features appear when $n$ is odd. Finally, we study the pseudodifferential projection boundary conditions proposed by Vassilevich (2001) in string- and brane-theory, showing that they too have heat expansions with log-terms, under suitable hypotheses. In all cases, the lowest log-coefficient $c^\{\prime\}_0$ vanishes, which assures that the zeta function is regular at 0.},
affiliation = {Copenhagen Univ. Math. Dept., Universitetsparken 5, DK-2100 Copenhague, Danemark.},
author = {Grubb, Gerd},
journal = {Séminaire Équations aux dérivées partielles},
language = {fre},
pages = {1-12},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Conditions au bord spectrales et formules de trace},
url = {http://eudml.org/doc/11034},
year = {2001-2002},
}

TY - JOUR
AU - Grubb, Gerd
TI - Conditions au bord spectrales et formules de trace
JO - Séminaire Équations aux dérivées partielles
PY - 2001-2002
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 12
AB - The lecture presents current results on heat trace expansions, and the related resolvent trace and zeta function expansions, for elliptic operators with boundary conditions on $n$-dimensional compact manifolds. As a background, we recall the set-up of elliptic differential operators with differential boundary conditions having heat trace expansions in powers $\sum _{k\ge -n}c_kt^{k/m}$. Then we consider the spectral boundary conditions of Atiyah, Patodi and Singer for Dirac-type first-order operators, leading to expansions with additional logarithmic terms $\sum _{k\ge 0}c^{\prime}_kt^{k/2}\log t$ (joint work with Seeley 1995) ; an extension to “well-posed” problems is included in a general study of pseudo-normal boundary conditions (1999). New results are presented on the vanishing or stability of the $c^{\prime}_k$-coefficients ; special features appear when $n$ is odd. Finally, we study the pseudodifferential projection boundary conditions proposed by Vassilevich (2001) in string- and brane-theory, showing that they too have heat expansions with log-terms, under suitable hypotheses. In all cases, the lowest log-coefficient $c^{\prime}_0$ vanishes, which assures that the zeta function is regular at 0.
LA - fre
UR - http://eudml.org/doc/11034
ER -

References

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