Progress in Inverse Spectral Geometry

Stig I. Andersson, Michel L. Lapidus (Autoren)

Buch | Softcover

V, 197 Seiten

2012 | 1. Softcover reprint of the original 1st ed. 1997
Springer Basel (Verlag)
978-3-0348-9835-5 (ISBN)

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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 ® '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.

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.

Erscheint lt. Verlag	12.10.2012
Reihe/Serie	Trends in Mathematics
Zusatzinfo	V, 197 p.
Verlagsort	Basel
Sprache	englisch
Maße	155 x 235 mm
Gewicht	332 g
Themenwelt	Sachbuch/Ratgeber ► Natur / Technik ► Garten
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Curvature • Fractal • inverse spectral geometry • Koch snowflake • manifold
ISBN-10	3-0348-9835-5 / 3034898355
ISBN-13	978-3-0348-9835-5 / 9783034898355
Zustand	Neuware