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The resolvent of the Laplacian on locally symmetric spaces. (English) Zbl 0766.53044

Let \(X\) be an \(n\) dimensional Riemannian symmetric space of strictly negative curvature and let \(\Delta\) be the associated Laplace Beltrami operator. Let \(B(r,x)\) be the metric ball of radius \(r\) about the point \(x\). Let \(\zeta\) be the volume of the metric unit sphere in \(X\). Define \(h\) by the identity: \[ \text{vol}(B(r,x)) \approx \zeta e^{hr}/h \text{ as } r\rightarrow \infty. \] The authors construct a meromorphic family \(R_{\nu}(x,y)\) of smooth functions on \(X\times X\) minus the diagonal which represent the resolvent of \(\Delta-\nu^{2}+h^{2}/4\) such that:
(a) If \(\text{Re}(\nu)\geq 0\), then \(R_{\nu}\) is holomorphic and \(R_{\nu}(x,y)\approx \delta(\nu)e^{-(\nu+h/2)d(x,y)}\) as \(d(x,y)\rightarrow\infty\).
(b) \(R_{\nu}(x,y)\approx\zeta d(x,y)^{- n+2}\log(d(x,y))^{\delta_{2,n}}\) as \(d(x,y)\rightarrow 0\).
(c) If \(f\in C_{c}^{\infty}(X), \text{Re}(\nu)\geq 0\), then \(\int_{X}R_{\nu}(x,y)(\Delta-\nu^{2}+h^{2}/4) f(y)\,dy=f(x)\).
The analytic continuation involves the construction of a larger class of functions which are progressively less singular on the diagonal. The authors present several lovely applications of this theorem and use methods which perhaps extend to a more general class of spaces.

MSC:

53C35 Differential geometry of symmetric spaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J05 Elliptic equations on manifolds, general theory
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