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Determinants, finite-difference operators and boundary value problems. (English) Zbl 0767.58043
The author studies the zeta-function determinant of a positive elliptic differential operator acting on sections of a vector bundle over a compact manifold. Following D. B. Ray and I. M. Singer [ Adv. Math. 7, 145-210 (1971; Zbl 0239.58014)], \(\log\det L\) is defined by analytically continuing the function \(\sum_{\lambda_ i\in Spec(L)} \lambda_ i^{-s} \log \lambda_ i\) from \(Re(s)\) large to \(s=0\),
In a previous paper [Invent. Math. 88, 447-493 (1987; Zbl 0602.58044)], the author related this determinant, the determinant of an operator acting on an infinite dimensional space of functions, to the determinant of a finite dimensional matrix whose entries are constructed from boundary values of the solutions of the operator. Previous proofs of this result follow indirect approaches. This paper presents a new and simpler proof using a result concerning matrices of \(SU(n,\mathbb{C})\).
Some applications are given: Laplacians on graphs, with relation to electrical network theory, and boundary value problems for finite difference operators defined on an interval.

MSC:
58J52 Determinants and determinant bundles, analytic torsion
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
58J32 Boundary value problems on manifolds
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
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