Progress in Inverse Spectral Geometry by Stig I. AnderssonProgress in Inverse Spectral Geometry by Stig I. Andersson

Progress in Inverse Spectral Geometry

byStig I. Andersson, Michel L. Lapidus

Paperback | October 12, 2012

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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* ®E), locally given by 00 K(x,y; t) = L>-IAk(- 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.
Title:Progress in Inverse Spectral GeometryFormat:PaperbackDimensions:197 pagesPublished:October 12, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034898355

ISBN - 13:9783034898355

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Table of Contents

Spectral Geometry: An Introduction and Background Material for this Volume.- Geometry Detected by a Finite Part of the Spectrum.- Spectral Geometry on Nilmanifolds.- Upper Bounds for the Poincaré Metric Near a Fractal Boundary.- Construction de Variétés Isospectrales du Théorème de T. Sunada.- Inverse spectral theory for Riemannian foliations and curvature theory.- Computer Graphics and the Eigenfunctions for the Koch Snowflake Drum.- Inverse Spectral Geometry.- Inverse Spectral Geometry on Riemann Surfaces.- Quantum Ergodicity.