Progress in Inverse Spectral Geometry:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | Englisch |
| Veröffentlicht: |
Basel
Birkhäuser Basel
1997
|
| Schriftenreihe: | Trends in Mathematics
|
| Schlagwörter: | |
| Links: | https://doi.org/10.1007/978-3-0348-8938-4 |
| 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 |
| Umfang: | 1 Online-Ressource (V, 197 p) |
| ISBN: | 9783034889384 9783034898355 |
| DOI: | 10.1007/978-3-0348-8938-4 |
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| 245 | 1 | 0 | |a Progress in Inverse Spectral Geometry |c edited by Stig I. Andersson, Michel L. Lapidus |
| 264 | 1 | |a Basel |b Birkhäuser Basel |c 1997 | |
| 300 | |a 1 Online-Ressource (V, 197 p) | ||
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| 490 | 0 | |a Trends in Mathematics | |
| 500 | |a 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 | ||
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Datensatz im Suchindex
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| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-0348-8938-4 |
| format | Electronic eBook |
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| illustrated | Not Illustrated |
| indexdate | 2024-12-20T17:10:46Z |
| institution | BVB |
| isbn | 9783034889384 9783034898355 |
| language | English |
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| physical | 1 Online-Ressource (V, 197 p) |
| psigel | ZDB-2-SMA ZDB-2-BAE ZDB-2-SMA_Archive |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Birkhäuser Basel |
| record_format | marc |
| series2 | Trends in Mathematics |
| 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 |