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

Chen, Xi

doi:10.2140/apde.2022.15.1861

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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

DOI: 10.2140/apde.2022.15.1861

Abstract

We prove the uniform estimates for the resolvent ${(Δ - α)}^{- 1}$ as a map from $L^{q}$ to $L^{q^{'}}$ on real hyperbolic space $ℍ^{n}$ , where $α \in ℂ ∖ [{(n - 1)}^{2} ∕ 4, \infty)$ and $2 n ∕ (n + 2) \leq q < 2$ . 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 $ℝ^{n}$ .

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Keywords

hyperbolic spaces, uniform Sobolev inequalities, eigenvalues, Schrödinger operators

Mathematical Subject Classification

Primary: 35J10, 35P15, 58C40

Milestones

Received: 18 October 2019

Revised: 3 March 2021

Accepted: 6 April 2021

Published: 10 February 2023

Authors

Xi Chen
Shanghai Center for Mathematical Sciences Fudan University Shanghai China	Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom

Vol. 15, No. 8, 2022