Control of eigenfunctions on surfaces of variable curvature

Abstract

We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works [arXiv:1705.05019], [arXiv:1712.02692], which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of [arXiv:1705.05019], [arXiv:1712.02692] and rely on the fractal uncertainty principle of [arXiv:1612.09040]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used in [arXiv:1504.06589]. Instead, our argument uses Egorov's Theorem up to local Ehrenfest time and the hyperbolic parametrix of [arXiv:0706.3242], together with the $C^{1+}$ regularity of the stable/unstable foliations.

119+eps pages, 13 figures. To appear in J. Amer. Math. Soc

References in corpus (13)

• Scarred eigenstates for quantum cat maps of minimal periods
• Decay rates for the damped wave equation on the torus
• Spectral gaps without the pressure condition
• Control of eigenfunctions on surfaces of variable curvature
• Resonance projectors and asymptotics for r-normally hyperbolic trapped sets
• Entropy of semiclassical measures in dimension 2
• Entropy of semiclassical measures of the Walsh-quantized baker's map
• Spectral gaps, additive energy, and a fractal uncertainty principle
• Semiclassical measures on hyperbolic surfaces have full support
• Energy decay for the damped wave equation under a pressure condition
• An introduction to fractal uncertainty principle
• Notes on hyperbolic dynamics
• Exponential decay for products of Fourier integral operators

Cited by in corpus (20)

• Control of eigenfunctions on surfaces of variable curvature
• An introduction to fractal uncertainty principle
• Semiclassical measures for higher dimensional quantum cat maps
• Observability of the Schr{ö}dinger equation with subquadratic confining potential in the Euclidean space
• Around quantum ergodicity
• Polynomial Fourier decay for fractal measures and their pushforwards
• The full delocalization of eigenstates for the quantized cat map
• Spectral gap for obstacle scattering in dimension 2
• Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus
• Local version of Courant's nodal domain theorem
• Semiclassical measures for complex hyperbolic quotients
• Macroscopic limits of chaotic eigenfunctions
• Mass equidistribution for Saito-Kurokawa lifts
• Particle Trajectories for Quantum Maps
• Superscars for Arithmetic Point Scatterers II
• Local $L^p$ norms of Schr{ö}dinger eigenfunctions on ${\mathbb{S}}^2$
• Fractal uncertainty principle for discrete Cantor sets with random alphabets
• Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds
• Resolvent of vector fields and Lefschetz numbers
• Exact observability properties of subelliptic wave and Schr{ö}dinger equations