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On the asymptotics of the spectral shift function. (English. Russian original) Zbl 0541.58047
Sov. Math., Dokl. 25, 332-334 (1982); translation from Dokl. Akad. Nauk SSSR 263, 283-284 (1982).
Let X be a d-dimensional connected \(C^{\infty}\)-manifold without boundary, let \(d\geq 3\) be odd, dx be a \(C^{\infty}\)-density on X and \(\epsilon\) a Hermitian \(C^{\infty}\)-fibering on X. Suppose A: \(L^ 2(X,\epsilon)\to L^ 2(X,\epsilon)\) is a self-adjoint elliptic differential operator of order \(m=1\) or 2. If \(S(\lambda)\) is the scattering matrix the spectral shift function \(s(\lambda)\) is defined by \(s(\lambda)=(2\pi)^{-1} Arg \det S(\lambda)\). In this brief announcement the authors are concerned with the existence of complete asymptotic expansions for \(s(\lambda)\) as \(\lambda \to +\infty\). Under technical conditions which imply that the perturbations are of nontrapping type, together in case \(m=2\) the positivity of a, they show that \[ \pm s(\lambda)\sim \sum^{\infty}_{n=0}k_{2n}\lambda^{d- 2n},\quad m=1,\quad s(\lambda^ 2)\sim \sum^{\infty}_{n=0}k_{2n}\lambda^{d-2n},\quad m=2,\quad \lambda \to \pm \infty. \] These results depend on the theory of the propagation of singularities of solutions of symmetric hyperbolic systems together with known results in scattering theory.
Reviewer: M.Thompson

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
47A40 Scattering theory of linear operators
58J47 Propagation of singularities; initial value problems on manifolds
35P25 Scattering theory for PDEs
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