Bouche, Thierry On the heat kernel associated with tensor powers of a complex line bundle. Asymptotic estimates and vanishing theorems. (Sur le noyau de la chaleur associé aux puissances tensorielles d’un fibré en droites complexes. Estimations asymptotiques et théorèmes d’annulation.) (French) Zbl 0921.58069 Séminaire de théorie spectrale et géométrie. Année 1993-1994. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Math. Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 12, 41-49 (1994). On a compact Riemannian manifold \(M\) a complex Hermitian line bundle \(L\) and a vector bundle \(E\) are given. Then on the bundle \(E(k)=E \otimes \bigotimes^k L= E \otimes L \otimes \cdots \otimes L\) the Schrödinger operator \(S_k:=k^{-1}\Delta_k+V\) is considered. Here \(\Delta_k\) is the Laplace operator on \(E(k).\) Then the spectral properties of \(k S_k\) are described. The results can be seen as a geometric version of the more analytic results obtained by the author in [Th. Bouche, Ann. Inst. Fourier 40, No. 1, 117-130 (1990; Zbl 0685.32015); Proc. Int. Conf., Trento, Italy 1994, 67-81 (1996; Zbl 0914.32010); Math. Z. 218, No. 4, 519-526 (1995; Zbl 0823.32013).For the entire collection see [Zbl 0812.00007]. Reviewer: H.-B.Rademacher (Leipzig) MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs 32L05 Holomorphic bundles and generalizations Keywords:Schrödinger operator; asymptotic estimates; tensor powers of line bundles Citations:Zbl 0823.32013; Zbl 0685.32015; Zbl 0914.32010 PDFBibTeX XMLCite \textit{T. Bouche}, in: Séminaire de théorie spectrale et géométrie. Année 1993-1994. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Math.; St. Martin d'Hères: Univ. de Grenoble I, Inst. Fourier. 41--49 (1994; Zbl 0921.58069) Full Text: EuDML