Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature.

*(English)*Zbl 0636.58034This 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.

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 |

##### Keywords:

degenerate elliptic boundary problems; scattering theory; blown up; parametrix; Dirichlet problem; Laplacian; Eisenstein series
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\textit{R. R. Mazzeo} and \textit{R. B. Melrose}, J. Funct. Anal. 75, 260--310 (1987; Zbl 0636.58034)

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