Progress in Inverse Spectral Geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Birkhäuser Basel
1997
|
| Schriftenreihe: | Trends in Mathematics
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | 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 |
| Beschreibung: | 1 Online-Ressource (V, 197 p) |
| ISBN: | 9783034889384 9783034898355 |
| DOI: | 10.1007/978-3-0348-8938-4 |
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| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-0348-8938-4 |
| format | Electronic eBook |
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| isbn | 9783034889384 9783034898355 |
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| spelling | Andersson, Stig I. Verfasser aut Progress in Inverse Spectral Geometry edited by Stig I. Andersson, Michel L. Lapidus Basel Birkhäuser Basel 1997 1 Online-Ressource (V, 197 p) txt rdacontent c rdamedia cr rdacarrier Trends in Mathematics 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 Mathematics Mathematics, general Mathematik Inverses Problem (DE-588)4125161-1 gnd rswk-swf Spektralgeometrie (DE-588)4128531-1 gnd rswk-swf 1\p (DE-588)4143413-4 Aufsatzsammlung gnd-content Spektralgeometrie (DE-588)4128531-1 s Inverses Problem (DE-588)4125161-1 s 2\p DE-604 Lapidus, Michel L. Sonstige oth https://doi.org/10.1007/978-3-0348-8938-4 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Andersson, Stig I. Progress in Inverse Spectral Geometry Mathematics Mathematics, general Mathematik Inverses Problem (DE-588)4125161-1 gnd Spektralgeometrie (DE-588)4128531-1 gnd |
| subject_GND | (DE-588)4125161-1 (DE-588)4128531-1 (DE-588)4143413-4 |
| title | Progress in Inverse Spectral Geometry |
| title_auth | Progress in Inverse Spectral Geometry |
| title_exact_search | Progress in Inverse Spectral Geometry |
| title_full | Progress in Inverse Spectral Geometry edited by Stig I. Andersson, Michel L. Lapidus |
| title_fullStr | Progress in Inverse Spectral Geometry edited by Stig I. Andersson, Michel L. Lapidus |
| title_full_unstemmed | Progress in Inverse Spectral Geometry edited by Stig I. Andersson, Michel L. Lapidus |
| title_short | Progress in Inverse Spectral Geometry |
| title_sort | progress in inverse spectral geometry |
| topic | Mathematics Mathematics, general Mathematik Inverses Problem (DE-588)4125161-1 gnd Spektralgeometrie (DE-588)4128531-1 gnd |
| topic_facet | Mathematics Mathematics, general Mathematik Inverses Problem Spektralgeometrie Aufsatzsammlung |
| url | https://doi.org/10.1007/978-3-0348-8938-4 |
| work_keys_str_mv | AT anderssonstigi progressininversespectralgeometry AT lapidusmichell progressininversespectralgeometry |