{"product_id":"progress-in-inverse-spectral-geometry-stig-i-andersson-9783764357559","title":"Progress in Inverse Spectral Geometry","description":"most polynomial growth on every half-space Re (z)::::: c. Moreover, Op(t) depends holomorphically on t for Re t \u0026gt; 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* (R)E), locally given by 00 K(x, y; t) = L\u0026gt;-IAk( k (R) '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:: \u0026gt;- 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e Stig I. Andersson,Michel L. Lapidus\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 376435755X\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9783764357559\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Birkhauser\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 10\/01\/1997\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 197\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Hardcover\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 1.05lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.21h x 6.14w x 0.50d","brand":"Stig I. Andersson","offers":[{"title":"Hardcover","offer_id":47424618692863,"sku":"9783764357559","price":54.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_e6605494-94d7-43b7-a566-1bef642d1cee.jpg?v=1761531423","url":"https:\/\/www.whiterainbookhouse.com\/products\/progress-in-inverse-spectral-geometry-stig-i-andersson-9783764357559","provider":"WR Book House","version":"1.0","type":"link"}