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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