Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

摘要: We prove that if (M, g) is a compact Riemannian manifold with ergodic geodesic flow, and \({H \subset M}\) smooth hypersurface satisfying generic microlocal asymmetry condition, then restrictions \({\varphi_j |_H}\) of an orthonormal basis \({\{\varphi_j\}}\) Δ-eigenfunctions to H are quantum on H. The condition satisfied by circles, closed horocycles geodesics hyperbolic surface. A key step in the proof matrix elements \({\langle{F}\varphi_j, \varphi_{j}\rangle}\) Fourier integral operators F whose canonical relation almost nowhere commutes flow must tend zero.