Conditions au bord spectrales et formules de trace
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2001-2002), Exposé no. 15, 12 p.

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 k0 c k t k/2 logt (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.

Grubb, Gerd 1

1 Copenhagen Univ. Math. Dept., Universitetsparken 5, DK-2100 Copenhague, Danemark.
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Grubb, Gerd. Conditions au bord spectrales et formules de trace. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2001-2002), Exposé no. 15, 12 p. http://archive.numdam.org/item/SEDP_2001-2002____A15_0/

[BM71] de Monvel, L. Boutet Boundary problems for pseudo-differential operators, Acta Math., Volume 126 (1971), pp. 11-51 | MR | Zbl

[DGK99] Dowker, J. S.; Gilkey, P. B.; Kirsten, K. Heat asymptotics with spectral boundary conditions, AMS Contemporary Math., Volume 242 (1999), pp. 107-124 | MR | Zbl

[FGLS96] Fedosov, B. V.; Golse, F.; Leichtnam, E.; Schrohe, E. The noncommutative residue for manifolds with boundary, J. Funct. Anal., Volume 142 (1996), pp. 1-31 | MR | Zbl

[G01] Grubb, G. A weakly polyhomogeneous calculus for pseudodifferential boundary problems, J. Functional An., Volume 184 (2001), pp. 19-76 | MR | Zbl

[G01’] Grubb, G. 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] Grubb, G. Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems (2002) (preprint) | MR | Zbl

[G02’] Grubb, G. Spectral boundary conditions for second-order elliptic operators (2002) (preprint)

[G73] Grubb, G. Weakly semibounded boundary problems and sesquilinear forms, Ann. Inst. Fourier, Volume 23 (1973), pp. 145-194 | EuDML | Numdam | MR | Zbl

[G74] Grubb, G. 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] Grubb, G. Functional Calculus of Pseudodifferential Boundary Problems, Second Edition, Progress in Mathematics, 65, Birkhäuser, Boston, 1996 | MR | Zbl

[G97] Grubb, G.; Rodino, L. Parametrized pseudodifferential operators and geometric invariants, Microlocal Analysis and Spectral Theory, Kluwer, Dordrecht, 1997, pp. 115-164 | MR | Zbl

[G99] Grubb, G. 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] Gilkey, P. B.; Grubb, G. Logarithmic terms in asymptotic expansions of heat operator traces, Comm. Part. Diff. Eq., Volume 23 (1998), pp. 777-792 | MR | Zbl

[Gi95] Gilkey, P. B. Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, CRC Press, Boca Raton, 1995 | MR | Zbl

[GK02] Gilkey, P. B.; Kirsten, K. Heat asymptotics with spectral boundary conditions II (preprint) | Zbl

[Gre71] Greiner, P. An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal., Volume 41 (1971), pp. 163-218 | MR | Zbl

[GS95] Grubb, G.; Seeley, R. Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems, Inventiones Math., Volume 121 (1995), pp. 481-529 | MR | Zbl

[GS96] Grubb, G.; Seeley, R. Zeta and eta functions for Atiyah-Patodi-Singer operators, Journal of Geometric Analysis, Volume 6 (1996), pp. 31-77 | MR | Zbl

[GSc01] Grubb, G.; Schrohe, E. 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] Minakshisundaram, S.; Pleijel, Å. Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math., Volume 1 (1949), pp. 242-256 | MR | Zbl

[MS67] McKean, H.P.; Singer, I.M. Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., Volume 1 (1967), pp. 43-69 | MR | Zbl

[S69] Seeley, R. T. The resolvent of an elliptic boundary problem, Amer. J. Math., Volume 91 (1969), pp. 889-920 | MR | Zbl

[S69’] Seeley, R. T. Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math., Volume 91 (1969), pp. 963-983 | Zbl

[S69”] Seeley, R. T. Topics in Pseudo-Differential Operators, C.I.M.E. Conf. on Pseudo-Differential Operators, Edizioni Cremonese, Roma (1969), pp. 169-305

[V01] Vassilevich, D. Spectral branes, J. High Energy Phys., Volume 0103 (2001), pp. 023 | MR

[V02] Vassilevich, D. Spectral geometry for strings and branes, Nuclear Physics B (Proc. Suppl.), Volume 104 (2002), pp. 208-211 | MR

[W84] Wodzicki, M. Spectral asymmetry and noncommutative residue, Steklov Institute of Mathematics, Moscow (1984) (Ph. D. Thesis in Russian)

[W84’] Wodzicki, M. Local invariants of spectral asymmetry, Inventiones Math., Volume 75 (1984), pp. 143-178 | Zbl