Eigenvalue bounds for Schrödinger operators with complex potentials on compact manifolds

TL;DR

This work extends Frank's Euclidean eigenvalue bounds to Schrödinger operators with complex potentials on compact Riemannian manifolds. It develops a Birman–Schwinger framework combined with sharp resolvent estimates, leveraging Sogge’s spectral cluster bounds to obtain a spectral inclusion into a union of disks around Laplacian eigenvalues and a second region controlled by the $L^q$-norm of $V$. The bounds are shown to be sharp on the sphere and Zoll manifolds via explicit saturation constructions and a frequency-analysis Rouche argument, with scaling on tori connecting to the Euclidean theory. The results provide a robust, optimal set of eigenvalue bounds for complex potentials on manifolds, with precise dependence on $q$, the geometry, and the spectral data.

Abstract

We prove eigenvalue bounds for Schrödinger operator $-Δ_g+V$ on compact manifolds with complex potentials $V$. The bounds depend only on an $L^q$-norm of the potential, and they are shown to be optimal, in a certain sense, on the round sphere and more general Zoll manifolds. These bounds are natural analogues of Frank's \cite{MR2820160} results in the Euclidean case.

Theorems & Definitions (21)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 2.1
• Remark 2.2
• proof : Proof of \ref{['universal resolvent bound imsqrtz>1']}
• Lemma 2.3
• proof
• proof : Proof of \ref{['universal resolvent bound imsqrtz<1']}