YMSC Assistant Professor Jin Long worked with foreign mathematicians professor Semyon Dyatlov and published the paper Semiclassical measures on hyperbolic surfaces have full support
in Acta Mathematica, Volume 220 (2018) , one of the world’s top four mathematic journals.
Eigenfunctions for Laplacian operators on surfaces or domains are simple models for quantum states. Their diestribution in the semiclassical limit is described by the semiclassical measures and is connected to the properties of the corresponding classical dynamical system. The geodesic flow on negatively curved surfaces is a classical example of chaotic dynamical system. Understanding semiclassical measures on such surfaces is one of the most important question in the field of quantum chaos. The Quantum Unique Ergodicity Conjecture of Rudnick and Sarnack suggests that the semiclassical measures must be the Liouville measure, that is, the eigenfunctions are equidistributed in the semiclassical limit. Previous studies of semiclassical measures focus on obtaining lower bounds for the entropy of semiclassical measures. The paper of Dyatlov and Jin breaks this restriction by using a newly developed tool, called Fractal Uncertainty Principle. In particular, they proved that for hyperbolic surfaces with constant negative curvature, the semiclassical measures have full support. In other words, the eigenfunctions cannot be completely localized on a subset in the semiclassical limit. This also implies that the linear Schrodinger equation on hyperbolic surfaces can be controlled by any nonempty open set.
Web link: https://intlpress.com/site/pub/pages/journals/items/acta/content/vols/0220/0002/a003/index.html
Note: The world’s top four mathematic journals are Acta Mathematica, Annals of Mathematics, Inventiones Mathematicae, Journal of the American Mathematical Society (in alphabetical order).
