**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:**

·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 Schr****ö****dinger operators.**** ***Commun. Math. Phys. 351, 877**–921 (2017). **Journal arXiv:
1512.02195*

· Z. Zhao: **Ballistic transport in
one-dimensional quasi-periodic continuous Schr****ö****dinger
equation. J. Diff. Eqs. 262, 4523****–****4566 (2017). ****Journal arXiv:1604.00210**

· Z. Zhao: **Ballistic motion in
one-dimensional quasi-periodic discrete Schr****ö****dinger
equation.*** **Commun. Math. Phys. 347, 511**–549 (2016).** ***Journal arXiv:1507.08909 **

· *J. Geng and Z. Zhao:** ***Reducibility of one-dimensional quasi-periodic Schrödinger equations. J.
Math. Pures Appl. 104, 436****–****453 (2015).**** Journal Article**

· J. Geng, J. You and Z. Zhao: Localization in one-dimensional
quasi-periodic nonlinear systems. Geom. And Func. Anal. 24, 116–158 (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), 3651–3689 (2013). **Journal Article**

· S. Zhang and Z. Zhao: Diffusion bound and reducibility for discrete
Schrödinger equations with tangent potential. Front. Math. China, 7(6), 1213–1235 (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, 961–997 (2011). **Journal Article **