Progress in Inverse Spectral Geometry (Trends in Mathematics) - Hardcover

Lapidus, Michel; Andersson, Stig I.

 
9783764357559: Progress in Inverse Spectral Geometry (Trends in Mathematics)
Hardcover
ISBN 10: 376435755X ISBN 13: 9783764357559
Verlag: Birkhäuser, 1997

Inhaltsangabe

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.

Die Inhaltsangabe kann sich auf eine andere Ausgabe dieses Titels beziehen.

Reseña del editor

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.

Reseña del editor

This volume aims to present a comprehensive overview of the research frontline in inverse spectral geometry in the late-1990s. It contains 19 papers originating from a conference dealing with this field of study.

„Über diesen Titel“ kann sich auf eine andere Ausgabe dieses Titels beziehen.

Verlag
Birkhäuser
Erscheinungsdatum
1997
Sprache
Englisch
ISBN 10 
376435755X
ISBN 13 
9783764357559
Einband
Tapa dura
Anzahl der Seiten
212
Kontakt zum Hersteller
Nicht verfügbar
Verantwortliche Person
Nicht verfügbar

Weitere beliebte Ausgaben desselben Titels

9780817657550: Progress in Inverse Spectral Geometry (Trends in Mathematics)

Vorgestellte Ausgabe

ISBN 10:  081765755X ISBN 13:  9780817657550
Verlag: Birkhauser
Hardcover