The Sobolev Inequalities on Real Hyperbolic Spaces and Eigenvalue Bounds for Schrödinger Operators with Complex Potentials

Abstract

In this paper, we prove the uniform estimates for the resolvent $(\Delta -\alpha {)}^{-1}$ as a map from $L^q$ to $L^{q^{\prime }}$ on real hyperbolic space ${\mathbb{H}}^n$ where $\alpha \in \mathbb{C}\setminus [(n-1{)}^2/4,\infty )$ and $2n/(n+2)\le q<2$. In contrast with analogous results on Euclidean space ${\mathbb{R}}^n$, the exponent $q$ here can be arbitrarily close to $2$. 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 Schrödinger operator with a complex potential. The improved Sobolev inequality results in a better long range eigenvalue bound on ${\mathbb{H}}^n$ than that on ${\mathbb{R}}^n$.

A revised version. In particular, a gap in the proof of Proposition 11 in the previous version is fixed

References in corpus (4)

• Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds II: spectral measure, restriction theorem, spectral multiplier
• Eigenvalue bounds for non-self-adjoint Schrödinger operators with non-trapping metrics
• Concerning $L^p$ resolvent estimates for simply connected manifolds of constant curvature
• $L_p$-spectrum and Lieb-Thirring inequalities for Schrödinger operators on the hyperbolic plane