Control of eigenfunctions on surfaces of variable curvature

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 (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. 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 by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the C 1 + C^{1+} regularity of the stable/unstable foliations.

我们证明了负曲率紧曲面（更一般地，具有阿诺索夫测地流的曲面上）拉普拉斯算子的高能本征函数质量的微局部下界。这意味着薛定谔方程可由任何非空开集控制，并表明每个半经典测度都有完全支撑。我们还证明了对于此类曲面上的阻尼波动方程的解，对于任何非平凡的阻尼系数，能量呈指数衰减。这些结果扩展了先前的工作（见谢苗·季亚特洛夫和金龙[《数学学报》220（2018），第297 - 339页]以及金龙[《数学物理通讯》373（2020），第771 - 794页]），先前的工作考虑的是常负曲率曲面的情形。证明使用了谢苗·季亚特洛夫和金龙[《数学学报》220（2018），第297 - 339页]以及金龙[《数学物理通讯》373（2020），第771 - 794页]的策略，并依赖于让·布尔甘和谢苗·季亚特洛夫[《数学年刊》（2）187（2018），第825 - 867页]的分形不确定性原理。然而，在变曲率情形下，稳定/不稳定叶状结构不光滑，所以我们不能再将谢苗·季亚特洛夫和约书亚·扎尔[《几何与泛函分析》26（2016），第1011 - 1094页]所使用的那种伪微分演算法与这些叶状结构相关联。相反，我们的论证使用了直到局部埃伦费斯特时间的叶戈罗夫定理以及斯特凡·农嫩马赫和马切伊·兹沃尔斯基[《数学学报》203（2009），第149 - 233页]的双曲拟基本解，以及稳定/不稳定叶状结构的\(C^{1 +}\)正则性。