The local and global parts of the basic zeta coefficient for operators on manifolds with boundary |
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Authors: | Gerd Grubb |
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Institution: | (1) Mathematics Department, Copenhagen University, Universitetsparken 5, 2100 Copenhagen, Denmark |
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Abstract: | For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C
0(B, P
1,T
) is the regular value at s = 0 of the zeta function , where B = P
+ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical
elliptic problem—and P
1,T
is a realization of an elliptic differential operator P
1, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices
of P
1,T
) have been known for some time and are local. We here determine C
0(B, P
1,T
) itself (with even-order P
1), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin,
Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich
and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds
without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P
1,T
. Since the complex powers of P
1,T
lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here;
instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators
can be used. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35S15 58J42 |
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