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Relative determinants of elliptic operators and scattering theory. (English) Zbl 0920.58062

The paper is a survey on relative determinants of elliptic operators on noncompact Riemannian manifolds. Relative determinants are interesting in situations in which two elliptic operators on \(C^\infty(E)\), \(E\) a Hermitian fiber-bundle over \(M\), are given in such a way that although the individual determinants have no meaning, there is a natural way to define the ratio of the two determinants. A typical example is when we consider the Schrödinger operator \(\Delta+V\), \(V\in C_0^\infty(\mathbb{R}^n)\), \(\Delta\) the Laplace operator on \(\mathbb{R}^n\), in which case one can associate a determinant with the couple \(\Delta+V,\Delta\).
In the beginning of the paper, the author reviews the zeta-function regularization of the determinant of elliptic operators in the compact case. This is then expressed in terms of the trace of the associated heat operator. To prepare for the case of pairs of operators, the author then considers an abstract setting for pairs \((H,H_0)\) of nonnegative selfadjoint operators in a separable Hilbert space \({\mathcal H}\), assuming that \(\exp(-tH)- \exp(-tH_0)\) be of trace class, and assuming asymptotic expansions for \(t\to 0\) and \(t\to\infty\) for the trace of \(\exp(-tH)- \exp(-tH_0)\). This makes it possible to define a “relative” zeta function for the pair \((H,H_0)\), and the relative determinant of the pair can then be defined in analogy with the classical case. The abstract setting is then checked to hold in a number of explicit cases (e.g., the pair \((\Delta+V,V)\) on \(\mathbb{R}^n\) mentioned above, operators on manifolds with cylindrical ends and cases of complete surfaces with finite area).

MSC:

58J52 Determinants and determinant bundles, analytic torsion
35P25 Scattering theory for PDEs
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