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
Is the information for this product incomplete, wrong or inappropriate?
Let us know about it.
Does this product have an incorrect or missing image?
Send us a new image.
Is this product missing categories?
Add more categories.
Review This Product
No reviews yet - be the first to create one!
Welcome to Loot.co.za!
Sign in / Register |Wishlists & Gift Vouchers |Help | Advanced search
