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 -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 . 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 (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 -coefficients ; special features appear when 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 vanishes, which assures that the zeta function is regular at 0.
@article{SEDP_2001-2002____A15_0, author = {Grubb, Gerd}, title = {Conditions au bord spectrales et formules de trace}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"}, note = {talk:15}, pages = {1--12}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2001-2002}, language = {fr}, url = {http://www.numdam.org/item/SEDP_2001-2002____A15_0/} }
TY - JOUR AU - Grubb, Gerd TI - Conditions au bord spectrales et formules de trace JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:15 PY - 2001-2002 SP - 1 EP - 12 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2001-2002____A15_0/ LA - fr ID - SEDP_2001-2002____A15_0 ER -
%0 Journal Article %A Grubb, Gerd %T Conditions au bord spectrales et formules de trace %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:15 %D 2001-2002 %P 1-12 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_2001-2002____A15_0/ %G fr %F SEDP_2001-2002____A15_0
Grubb, Gerd. Conditions au bord spectrales et formules de trace. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2001-2002), Talk no. 15, 12 p. http://www.numdam.org/item/SEDP_2001-2002____A15_0/
[BM71] Boundary problems for pseudo-differential operators, Acta Math., Volume 126 (1971), pp. 11-51 | MR | Zbl
[DGK99] Heat asymptotics with spectral boundary conditions, AMS Contemporary Math., Volume 242 (1999), pp. 107-124 | MR | Zbl
[FGLS96] The noncommutative residue for manifolds with boundary, J. Funct. Anal., Volume 142 (1996), pp. 1-31 | MR | Zbl
[G01] A weakly polyhomogeneous calculus for pseudodifferential boundary problems, J. Functional An., Volume 184 (2001), pp. 19-76 | MR | Zbl
[G01’] Poles of zeta and eta functions for perturbations of the Atiyah-Patodi-Singer problem, Comm. Math. Phys., Volume 215 (2001), pp. 583-589 | Zbl
[G02] Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems (2002) (preprint) | MR | Zbl
[G02’] Spectral boundary conditions for second-order elliptic operators (2002) (preprint)
[G73] Weakly semibounded boundary problems and sesquilinear forms, Ann. Inst. Fourier, Volume 23 (1973), pp. 145-194 | EuDML | Numdam | MR | Zbl
[G74] Properties of normal boundary problems for elliptic even-order systems, Ann. Sc. Norm. Sup. Pisa, Ser. IV, Volume 1 (1974), pp. 1-61 | EuDML | Numdam | MR | Zbl
[G96] Functional Calculus of Pseudodifferential Boundary Problems, Second Edition, Progress in Mathematics, 65, Birkhäuser, Boston, 1996 | MR | Zbl
[G97] Parametrized pseudodifferential operators and geometric invariants, Microlocal Analysis and Spectral Theory, Kluwer, Dordrecht, 1997, pp. 115-164 | MR | Zbl
[G99] Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems, Arkiv f. Mat., Volume 37 (1999), pp. 45-86 | MR | Zbl
[GG98] Logarithmic terms in asymptotic expansions of heat operator traces, Comm. Part. Diff. Eq., Volume 23 (1998), pp. 777-792 | MR | Zbl
[Gi95] Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, CRC Press, Boca Raton, 1995 | MR | Zbl
[GK02] Heat asymptotics with spectral boundary conditions II (preprint) | Zbl
[Gre71] An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal., Volume 41 (1971), pp. 163-218 | MR | Zbl
[GS95] Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems, Inventiones Math., Volume 121 (1995), pp. 481-529 | MR | Zbl
[GS96] Zeta and eta functions for Atiyah-Patodi-Singer operators, Journal of Geometric Analysis, Volume 6 (1996), pp. 31-77 | MR | Zbl
[GSc01] Trace expansions and the noncommutative residue for manifolds with boundary, J. Reine Angew. Math. (Crelle’s Journal), Volume 536 (2001), pp. 167-207 | Zbl
[MP49] Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math., Volume 1 (1949), pp. 242-256 | MR | Zbl
[MS67] Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., Volume 1 (1967), pp. 43-69 | MR | Zbl
[S69] The resolvent of an elliptic boundary problem, Amer. J. Math., Volume 91 (1969), pp. 889-920 | MR | Zbl
[S69’] Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math., Volume 91 (1969), pp. 963-983 | Zbl
[S69”] Topics in Pseudo-Differential Operators, C.I.M.E. Conf. on Pseudo-Differential Operators, Edizioni Cremonese, Roma (1969), pp. 169-305
[V01] Spectral branes, J. High Energy Phys., Volume 0103 (2001), pp. 023 | MR
[V02] Spectral geometry for strings and branes, Nuclear Physics B (Proc. Suppl.), Volume 104 (2002), pp. 208-211 | MR
[W84] Spectral asymmetry and noncommutative residue, Steklov Institute of Mathematics, Moscow (1984) (Ph. D. Thesis in Russian)
[W84’] Local invariants of spectral asymmetry, Inventiones Math., Volume 75 (1984), pp. 143-178 | Zbl