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Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. (English) Zbl 0636.58034
This is a paper on degenerate elliptic boundary problems. Another interpretation can be given in terms of scattering theory. Let M be a $$C^{\infty}$$-manifold with boundary. By $$V_ 0=V_ 0(M)$$ is denoted the space of vector fields which vanish at the boundary. The $$(n+1)$$- dimensional hyperbolic space $$H^{n+1}$$, considered as a leading example, shows the significance of $$V_ 0$$. In this case we can think of $${\mathbb{R}}^{n+1}=\{(x,y)\in {\mathbb{R}}^+_ x\times {\mathbb{R}}^ n_ y\}$$ equipped with the metric $$ds^ 2=(dx^ 2+dy^ 2)/x^ 2$$ of curvature -1 as a model.
The Laplacian acting on functions is a polynomial in operators $$x\partial /\partial x$$ and $$x\partial /\partial y_ i$$ which span (as a $$C^{\infty}$$-module) the Lie algebra $$V_ 0(H^{n+1})$$. A natural class of manifolds M, dim M$$=n+1$$, with metric having behaviour at the boundary, similar to that on the boundary of $$H^{n+1}$$, is considered. If h is a Riemannian metric on M one can take a conformal $$C^{\infty}$$- metric g with conformal factor $$\rho^{-2}$$ $$(\rho \in C^{\infty}(M)$$, $$\rho\geq 0$$, $$\rho^{-1}(0)=\partial M$$, $$\partial \rho \neq 0$$ at $$\partial M)$$. With this metric the interior of M is a complete Riemannian manifold and along any curve approaching a point of the boundary the sectional curvature of g approaches $$-| d\rho |^ 2_ h$$. Assume that $$| d\rho |_ h$$ is constant on M.
The operators from the space of $$C^{\infty}$$-functions vanishing to all orders at the boundary, $$\dot C^{\infty}(M)$$, to the space of extendible distributions, $$C^{-\infty}(M)$$, have Schwartz kernels in the space of distribution densities extendible across all boundaries. The kernels of the operators inverting $$V_ 0$$-elliptic differential operators have specific extension properties across the corner $$\partial M\times \partial N$$. In this connection the simple product $$M\times M$$ is replaced by $$V_ 0$$ blown up product $$M\times_ 0M$$. This amounts to the introduction of singular coordinates near the corner in terms of which the structure of the kernel is particularly simple. The method used to produce a parametrix for the Laplacian is based on detailed information about the kernel of the inverse.
Now let $$\Delta_ g$$ be the Laplacian of g acting on functions. The modified resolvent $$R(\zeta)=[[ | d\rho |^ 2_ h\Delta_ g+\zeta (\zeta -n)]]^{-1}$$ is a bounded operator on $$L^ 2_ g(M)$$ provided Re $$\zeta$$ $$>n$$. Considering R($$\zeta)$$ as an operator of type $$\dot C^{\infty}(M)\to C^{- \infty}(M)$$, the authors prove that R($$\zeta)$$ extends to be meromorphic in the whole complex plane with residues of infinite rank. The analysis of R($$\zeta)$$ is, in fact, a treatment of some Dirichlet problem on M, related with the asymptotic expansion of the distributions which are essentially annihilated by the modified Laplacian $$P_{\zeta}=| d\rho |^ 2_ h\Delta +\zeta (\zeta -n).$$ The meromorphy of R($$\zeta)$$ is closely related to the solvability of the mentioned Dirichlet problem. A nice application for the Eisenstein series associated with the discrete group action of certain type on $$H^{n+1}$$ is given, as well. An attractive perspective for generalization of the given application emerges.
Reviewer: S.Dimiev

##### MSC:
 58J32 Boundary value problems on manifolds 35J40 Boundary value problems for higher-order elliptic equations 53C20 Global Riemannian geometry, including pinching 35J70 Degenerate elliptic equations 35S15 Boundary value problems for PDEs with pseudodifferential operators 58J40 Pseudodifferential and Fourier integral operators on manifolds
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