Rellich’s theorem, limiting absorption principle and radiation condition on manifold with ends

Rellich's theorem, limiting absorption principle and radiation condition on manifold with ends Kenichi ITO $*$ and Erik SKIBSTED\dagger 1 Setting and results In this article we present without proofs a part of our results in an ongoing joint project [IS2].In particular, Rellich's theorem, the limiting absorption principle and the radiation condition are discussed for the Schr\"odinger operator $H=H_{0}+V$ ; $H_{0}=-\frac{1}{2}\triangle=\frac{1}{2}p_{i}^{*}g^{ij}p_{j},$ $p_{i}=-i\partial_{i},$ on a connected Riemannian manifold $(M_{9)}$ with asymptotically Euclidean and/or hyperbolic ends.Here $\triangle$ is the Laplace-Beltrami operator, and $V$ is a potential-type perturbation.In this section we introduce a set of assumptions on the geometry of $(M, g)$ and on the perturbation $V$ , and we state some of our main results.Our conditions are formulated abstractly in terms of a certain "escape function"' They apply to a wide class of manifolds and potentials.See Section 2 for concrete examples satisfying the conditions.Condition 1.1.Let $(M, g)$ be a connected Riemannian manifold of dimension $d\geq 1.$There exist a function $r\in C^{\infty}(M)$ with image $r(M)=[1, \infty$ ) and constants $c>0$ and $r_{0}\geq 2$ such that:1.The gradient vector field $\nabla r\in X(M)$ is forward complete, i.e., the integral curve of $\nabla r$ exists for any initial point $x\in M$ and any non-negative time $t\geq 0.$ The bound $|\nabla r|\geq c$ holds on $\{x\in M;r(x)>r_{0}/2\}.$We call each component of the open subset $E=\{x\in M;r(x)>r_{0}\}$ an end of $M.$The function $r$ may model a distance function there.In fact, only with Condition 1.1

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