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Microlocal heat kernel asymptotics and inverse spectral problems. I: Reduction to the boundary and the asymptotic expansion. (English) Zbl 0702.35237
Proc. Winter Sch. Geom. Phys., Srní/Czech. 1988, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 21, 93-114 (1989).
[For the entire collection see Zbl 0672.00006.]
Given a partial differential operaor P(x,D) on a manifold M with spectrum \(\sigma\) (P). This paper is concerned with two problems: that of reconstructing the geometry and differential topology of M, given a- priori information about spectral properties of a certain class of P(x,D), and a kind of reduction process via microlocalisation. Assuming knowledge of \(\sigma\) (P), it is possible to identify certain spectral invariants, attached to P, with geometric quantities. Here the function \(\theta_ p(t):=tr(e^{-tP})\) \((t>0)\) is used, and a new expedient way to derive the form of the asymptotic expansion of \(\lim_{t\to 0_+}\theta_ p(t)\) is demonstrated for suitable classes of P(x,D) and boundary conditions.
Reviewer: K.Hardenberg
MSC:
35R30 Inverse problems for PDEs
35P99 Spectral theory and eigenvalue problems for partial differential equations
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds