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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

We prove the uniform estimates for the resolvent (Δ α)1 as a map from Lq to Lq on real hyperbolic space n, where α [(n 1)24,) and 2n(n + 2) q < 2. In contrast with analogous results on Euclidean space 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 n than that on n.

hyperbolic spaces, uniform Sobolev inequalities, eigenvalues, Schrödinger operators
Mathematical Subject Classification
Primary: 35J10, 35P15, 58C40
Received: 18 October 2019
Revised: 3 March 2021
Accepted: 6 April 2021
Published: 10 February 2023
Xi Chen
Shanghai Center for Mathematical Sciences
Fudan University
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom