most polynomial growth on every half-space Re (z)::::: c. Moreover,
Op(t) depends holomorphically on t for Re t > O. General
references for much of the material on the derivation of spectral
functions, asymptotic expansions and analytic properties of
spectral functions are A-P-S] and Sh], especially Chapter 2. To
study the spectral functions and their relation to the geometry and
topology of X, one could, for example, take the natural associated
parabolic problem as a starting point. That is, consider the 'heat
equation' (%t + p) u(x, t) = 0 { u(x, O) = Uo(x), tP which is
solved by means of the (heat) semi group V(t) = e-; namely, u(., t)
= V(t)uoU. Assuming that V(t) is of trace class (which is
guaranteed, for instance, if P has a positive principal symbol), it
has a Schwartz kernel K E COO(X x X x Rt, E* (r)E), locally given
by 00 K(x, y; t) = L>-IAk( k (r) 'Pk)(X, y), k=O for a complete
set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we
then obtain: 00 tA Op(t) = trace(V(t)) = 2:: >- k. k=O Now,
using, e. g., the Dunford calculus formula (where C is a suitable
curve around a(P)) as a starting point and the standard for malism
of pseudodifferential operators, one easily derives asymptotic
expansions for the spectral functions, in this case for Op."
General
Imprint: |
Springer Basel
|
Country of origin: |
Switzerland |
Series: |
Trends in Mathematics |
Release date: |
October 2012 |
First published: |
October 2012 |
Authors: |
Stig I. Andersson
• Michel L Lapidus
|
Dimensions: |
235 x 155 x 11mm (L x W x T) |
Format: |
Paperback
|
Pages: |
197 |
Edition: |
Softcover reprint of the original 1st ed. 1997 |
ISBN-13: |
978-3-03-489835-5 |
Categories: |
Books >
Science & Mathematics >
Mathematics >
Geometry >
General
|
LSN: |
3-03-489835-5 |
Barcode: |
9783034898355 |
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