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The Sobolev Inequalities on Real Hyperbolic Spaces and Eigenvalue Bounds for Schr\"odinger Operators with Complex Potentials

Abstract

In this paper, we prove the uniform estimates for the resolvent (Ξ”βˆ’Ξ±)βˆ’1(\Delta - \alpha)^{-1} as a map from LqL^q to Lqβ€²L^{q'} on real hyperbolic space Hn\mathbb{H}^n where α∈Cβˆ–[(nβˆ’1)2/4,∞)\alpha \in \mathbb{C}\setminus [(n - 1)^2/4, \infty) and 2n/(n+2)≀q<22n/(n + 2) \leq q < 2. In contrast with analogous results on Euclidean space Rn\mathbb{R}^n, the exponent qq here can be arbitrarily close to 22. This striking improvement is due to two non-Euclidean features of hyperbolic space: the Kunze-Stein phenomenon and the exponential decay of the spectral measure. In addition, we apply this result to the study of eigenvalue bounds of the Schr\"{o}dinger operator with a complex potential. The improved Sobolev inequality results in a better long range eigenvalue bound on Hn\mathbb{H}^n than that on Rn\mathbb{R}^n.Comment: A revised version. In particular, a gap in the proof of Proposition 11 in the previous version is fixe

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