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Zbl 0981.58024
Carron, Gilles
The relative determinant and the function xi. (Déterminant relatif et la fonction xi.) (French)
[A] Séminaire de théorie spectrale et géométrie. Année 1999-2000. St. Martin d'Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectr. Géom. 18, 119-124 (2000).

This paper is a short summary of the author's paper ``Déterminant relatif et la fonction xi'' to appear at the American Journal of Mathematics. \par If a Riemannian manifold $(M,g)$ is complete, but not compact, it is difficult to define the determinant of the Laplacian. The author defines a relative determinant (determinant on $M$ relative to a complement of an open bounded set ${\cal O}\subset M$).\par The first result relates this relative determinant to the determinant of the Dirichlet to Neumann operator on $C^\infty(\partial{\cal O})$.\par The second result relates the relative determinant to the function of spectral shift. \par The behaviour of this spectral shift function to infinity is given.
[B.Colbois (Neuchâtel)]

MSC 2000:: *58J50 Spectral problems; spectral geometry; scattering theory
58J32 Boundary value problems on manifolds
58J20 Index theory and related fixed point theorems
47A40 Scattering theory of linear operators

Keywords: relative determinant; spectral shift function

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