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The Sobolev inequalities on real hyperbolic spaces and eigenvalue bounds for Schrödinger operators with complex potentials

### Xi Chen

Vol. 15 (2022), No. 8, 1861–1878
##### Abstract

We prove the uniform estimates for the resolvent ${\left(\mathrm{\Delta }-\alpha \right)}^{-1}$ as a map from ${L}^{q}$ to ${L}^{{q}^{\prime }}$ on real hyperbolic space ${ℍ}^{n}$, where $\alpha \in ℂ\setminus \left[{\left(n-1\right)}^{2}∕4,\infty \right)$ and $2n∕\left(n+2\right)\le q<2$. In contrast with analogous results on Euclidean space ${ℝ}^{n}$, 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 ${ℍ}^{n}$ than that on ${ℝ}^{n}$.

##### Keywords
hyperbolic spaces, uniform Sobolev inequalities, eigenvalues, Schrödinger operators
##### Mathematical Subject Classification
Primary: 35J10, 35P15, 58C40