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Inverse scattering on asymptotically hyperbolic manifolds. (English) Zbl 1142.58309

From the introduction: In this paper, we study scattering for Schrödinger operators on asymptotically hyperbolic manifolds. In particular, we show that the scattering matrix depends meromorphically on the energy \(\zeta\in\mathbb C\), and for the values of \(\zeta\) where it is defined, it is a pseudo-differential operator of order \(2\operatorname{Re}\zeta-n\) (really complex order \(2\zeta-n\)), where the dimension of the manifold is \(n+1\). We then show that the total symbol of this operator is determined locally by the metric and the potential, and that, except for a countable set of energies, the asymptotics of either the metric or the potential can be recovered from the scattering matrix at one energy. This also allows us to characterize the total symbol of the scattering matrix in the case where the manifold is of product type modulo terms vanishing to infinite order at the boundary.
We then analyze the difference of the scattering matrices when the metrics \(g_1,g_2\), and the potentials \(V_1,V_2\), agree to order \(k\geq 1\) at the boundary. We also prove:
Theorem 1.2. Let \((X,\partial X)\) be a smooth manifold with boundary, and let \(p\in\partial X\). Suppose that \(g_1\) and \(g_2\) induce asymptotically hyperbolic structures on \(X\), and that \(g_i= (dx^2+h_i(x,y,dy))/x^2\), \(i=1,2\), with respect to some product decomposition, Moreover suppose that \(h_2-h_1= x^k L(y,dy)+ O(x^{k+1})\), \(k\geq 1\), near \(p\)A, and that \(V_1,V_2\) are smooth short-range potentials such that \(V_2-V_1= x^kW(y)+ O(x^{k+1})\) near \(p\). Let \(S_i(\zeta)\) be the scattering matrix associated to \(\Delta_{g_i}+ V_i+ \zeta(\zeta-n)\). We then have that, near \(p\),
\[ S_1(\zeta)- S_2(\zeta)\in \Psi^{2\operatorname{Re}\zeta-n-k}, \]
and the principal symbol of \(S_1(\zeta)- S_2(\zeta)\) equals
\[ A_1(k,\zeta) \sum_{i,j} H_{ij}\xi_i\xi_j|\xi|^{2\zeta-n-k-2}+ A_2(k,\zeta) \bigl(W+\tfrac14 k(k+1)T\bigr)|\xi|^{2\zeta-n-k}, \]
where \(H=h_0^{-1}Lh_0^{-1}\) as matrices, \(h_0= h_1|_{x=0}= h_2|_{x=0}\), \(T= \text{trace}(h_0^{-1}L)\), \(|\xi|\) is the length of the covector \(\xi\) induced by \(h_0\), and \(A_1,A_2\) are meromorphic functions of \(\zeta\), given by
\[ \begin{aligned} A_1(k,\zeta)&= -\pi^{n/2} 2^{k+2-2\zeta+n} \frac {\Gamma(\frac12(k+2-2\zeta+n))} {\Gamma(-\frac12(k+2-2\zeta))} \cdot \frac{C(\zeta)}{M(\zeta)} T_1(k,\zeta),\\ A_2(k,\zeta)&= \pi^{n/2} 2^{k-2\zeta+n} \frac {\Gamma(\frac12(k-2\zeta+n))} {\Gamma(-\frac12(k-2\zeta))} \cdot \frac{C(\zeta)}{M(\zeta)} T_2(k,\zeta), \end{aligned} \]
where \(C(\zeta)\), \(T_j(k,\zeta)\), \(j=1,2\), the meromorphic continuation of certain functions, and \(M(\zeta)\) are given in the paper.
As an application of Theorem 1.2 we analyze the cases where the manifold is actually hyperbolic and is almost of product type.
Theorem 1.3. If \((X,\partial X)\) is such that in some product decomposition the metric is a product modulo te1ms vanishing to infinite order at the boundary, then the scattering matrix is equal to
\[ 2^{n-2\zeta} \frac {\Gamma(\frac12 n-\zeta)}{\Gamma(\zeta-\frac12n)} \Delta_{\partial X}^{\zeta-n/2}, \]
modulo smoothing. If \((X,\partial X)\) is a smooth hyperbolic manifold, the same result holds modulo pseudo-differential operators of order \(2\operatorname{Re}\zeta-n-2\). Here we have chosen a defining function \(x\) in order to trivialize the normal bundle and to induce a metric on the boundary, with respect to which we take \(\Delta_{\partial X}\).
We prove the result for almost product-type structures in §6. In the hyperbolic case, the result for principal symbols is due to P. A. Perry [J. Reine Angew. Math. 398, 67–91 (1989; Zbl 0677.58044)] (Perry’s definition of the scattering matrix was slightly different which caused an extra factor to be present). The result for hyperbolic manifolds is an immediate consequence of Theorem 1.2 and observing that a funnel is product type to second order.
As consequences of Theorem 1.2 we also obtain some inverse results. In §7, we give an application of these results, or rather of the methods used to prove them, to inverse scattering on the Schwarzschild and the De Sitter-Schwarzschild model of black holes. We show that the Taylor series at the boundary of certain perturbations of these models can be recovered from the scattering matrix at a fixed energy.
Our approach is heavily influenced by the work of Guillopc and Zworski. In particular, we compute the scattering matrix as a boundary value of the resolvent. To do this we use the calculus developed by Mazzeo and Melrose, of zero-pseudo-differential operators in order to construct the resolvent.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P25 Scattering theory for PDEs

Citations:

Zbl 0677.58044
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References:

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