Research Interests

My research interests center around the dynamical object, the Lyapunov exponent, and lie across dynamical systems, spectral theory, and mathematical physics. Concretely, I study spectral analysis of ergodic Schrodinger operators and dynamics of the associated Schrodinger cocycles.

Currently, I am focusing on positivity and large deviation estimates for the Lyapunov exponent for different types of ergodic base dynamics. In mathematical physics, these two properties are strong indications of Anderson Localization. 

I am particular interested in quasiperiodic potentials and potentials generated by hyperbolic dynamics.

Papers 

The following papers are or will be available on my arXiv.org page.

  Publications:

• Johnson-Schwartzman Gap Labelling for Ergodic Jacobi Matrices
  (with D. Damanik and J. Fillman) 2022 preprint (18 pages), Journal of Spectral Theory, to appear.

• Schrödinger Operators with Potentials Generated by Hyperbolic Transformations: I. Positivity of the Lyapunov Exponent
  (with A. Avila and D. Damanik) 2020 preprint (54 pages), Inventiones Mathematicae, to appear.

• On the Correspondence Between Domination and the Spectrum of Jacobi Operators
  (with K. Alkorn) 2021 preprint (44 pages), Trans. Amer. Math. Soc. , to appear.

• Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger Operators
  2013 preprint, extended in 2018 and in 2019, Journal of Spectral Theory (2020), no. 4, 1471-1517

• Positive Lyapunov Exponents and a Large Deviation Theorem for Continuum Anderson Models, Briefly
  (with V. Bucaj, D. Damanik, J. Fillman, V. Gerbuz, T. VandenBoom, and F. Wang),  J. Funct. Anal. 277 (2019), no. 9, (8 pages)

• Localization for the one-dimensional Anderson model via positivity and large deviations of the Lyapunov exponent  
  (with V. Bucaj, D. Damanik, J. Fillman, V. Gerbuz, T. VandenBoom, and F. Wang),  Trans. Amer. Math. Soc. 372 (2019), no. 5, (49 pages)

• Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator
  (with J. Fillman and D. C. Ong), Comm. Math. Phys. 351 (2017), (37 pages).

• Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators
  (with Y. Wang), Int. Math. Res. Notices. 2017, no. 8, (37 pages).

• Uniform positivity and continuity of Lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles 
  (with Y. Wang), J. Funct. Anal. 268 (2015), (61 pages).

• Singular density of states measure for subshift and quasi-periodic Schrödinger operators
  (with A. Avila and D. Damanik), Comm. Math. Phys. 330 (2014), (30 pages).

• Positive Lyapunov exponents for quasiperiodic Szegő cocycles,
  Nonlinearity 25 (2012), (27 pages).

  Preprints:

• Schrödinger Operators with Potentials Generated by Hyperbolic Transformations: II. Large Deviations and Anderson Localization
  (with A. Avila and D. Damanik), 2024 preprint (36 pages, submitted).

• Uniform positivity of the Lyapunov exponent for monotone potentials generated by the doubling map
  2016 preprint (13 pages, submitted).

  Papers in Preparation:

• Schrödinger Operators with Potentials Generated by Hyperbolic Transformations: III. Large Deviations without Fiber Bunching
  (with A. Avila and D. Damanik).