Absence of Principal Eigenvalues for Higher Rank Locally Symmetric Spaces

Abstract Given a geometrically finite hyperbolic surface of infinite volume it is classical result Patterson that the positive Laplace–Beltrami operator has no $$L^2$$ L 2 -eigenvalues $$\ge 1/4$$ ≥ 1 / 4 . In this article we prove generalization for joint algebra commuting differential operators on Riemannian locally symmetric spaces $$\Gamma \backslash G/K$$ Γ \ G K higher rank. We derive dynamical assumptions $$ -action geodesic and Satake compactifications which imply absence corresponding principal eigenvalues. A large class examples fulfilling these are non-compact quotients by Anosov subgroups.