Research of Zhiyan ZHAO


Research Interests:

· Perturbation theory for infinite-dimensional Hamiltonian systems, in particular:

1. Long-time behavior of Hamiltonian PDEs

2. Infinite-dimensional KAM theory

· Spectral theory of Schrödinger operator, in particular:

1. Anderson localization in quasi-crystal models

2. Reducibility and almost reducibility of Schrödinger cocycle

3. Transport property of quantum dynamics

4. Dispersive decay for the time evolution and non-linear equations

· Holomorphic dynamics and Cauchy-Riemann geometry

1. KAM theory for holomorphic dynamics

2. Local equivalence in complex spaces


Preprint:

· M. Leguil, J. You, Z. Zhao and Q. Zhou: Asymptotics of spectral gaps of quasi-periodic Schrödinger operators. arXiv:1712.04700

· Z. Liang, Z. Zhao and Q. Zhou: 1-d quantum harmonic oscillator with time quasi-periodic quadratic perturbation: reducibility and growth of Sobolev norms. arXiv:2003.13034

 

Publication:

· D. Bambusi and Z. Zhao: Dispersive estimate for quasi-periodic Schrödinger operators on 1-d lattices. Advances in Mathematics 336, 107071 (2020). Journal arXiv:1912.01528

· Y. Mi and Z. Zhao: Dispersive estimate for two-periodic discrete one-dimensional Schrödinger operator. J. Math. Anal. Appl. 485(1), 123768 (2020). Journal Article

· Z. Zhang and Z. Zhao: Ballistic transport and absolute continuity of one-frequency quasi-periodic Schrödinger operators. Commun. Math. Phys. 351, 877921 (2017). Journal arXiv: 1512.02195

· Z. Zhao: Ballistic transport in one-dimensional quasi-periodic continuous Schrödinger equation. J. Diff. Eqs. 262, 45234566 (2017). Journal arXiv:1604.00210

· Z. Zhao: Ballistic motion in one-dimensional quasi-periodic discrete Schrödinger equation. Commun. Math. Phys. 347, 511549 (2016). Journal arXiv:1507.08909

· J. Geng and Z. Zhao: Reducibility of one-dimensional quasi-periodic Schrödinger equations. J. Math. Pures Appl. 104, 436453 (2015). Journal Article

· J. Geng, J. You and Z. Zhao: Localization in one-dimensional quasi-periodic nonlinear systems. Geom. And Func. Anal. 24, 116158 (2014). Journal Article

· J. Geng and Z. Zhao: Quasi-periodic solutions for one-dimensional discrete nonlinear Schrödinger equations with tangent potential. Siam. J. Math. Anal. 45(6), 36513689 (2013). Journal Article

· S. Zhang and Z. Zhao: Diffusion bound and reducibility for discrete Schrödinger equations with tangent potential. Front. Math. China, 7(6), 12131235 (2012). Journal Article

· Z. Zhao and J. Geng: Linearly stable quasi-periodic breathers in a class of random Hamiltonian systems. J. Dyn. Diff. Eqs., 23, 961997 (2011). Journal Article