# Equations of Higher Order, II

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# Regularity of the Density of States for Random Schrodinger Operators

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In this talk, a joint work with Dhriti Dolai and Anish Mallick, I will present a proof of smoothness of the density of states for Random Schrodinger operators in any dimension. We show that the integrated density of states is almost as smooth as the single site distribution of the random potential, in the region of exponential localisation. The proof relies on the fractional moment bounds on the operator kernels in such energy region.

Our proof also gives a part of the results for the Anderson type models proved by Abel Klein and collaborators more than thirty years ago.

# Fractal properties of the Hofstadter's butterfly and singular continuous spectrum of the critical almost Mathieu operator

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Abstract: Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly. It reduces to the critical almost Mathieu family, indexed by phase. We discuss the proof of sungular continuous spectrum for this family for all phases, finishing a program with a long history. We also discuss a recent proof (with I. Krasovsky) of the Thouless' conjecture from the early 80s: that Hausdorff dimension of the spectrum of Harper's operator is bounded by 1/2 for all irrational fluxes.

# Localization and unique continuation on the integer lattice, II

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This will be a series of technical lectures on my recent work with Jian Ding. After a brief review of the mathematics of Anderson localization, I will explain our unique continuation result. To motivate our proof, I will describe the unique continuation result of Buhovski--Logunov--Malinnikova--Sodin for harmonic functions on the integer lattice. I will then explain how to modify this argument, introducing tools from probability theory, to obtain a unique continuation result for Schrodinger operators on the lattice with random potentials.

# Localization and unique continuation on the integer lattice, I

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Abstract: This will be a series of technical lectures on my recent work with Jian Ding. After a brief review of the mathematics of Anderson localization, I will explain our unique continuation result. To motivate our proof, I will describe the unique continuation result of Buhovski--Logunov--Malinnikova--Sodin for harmonic functions on the integer lattice. I will then explain how to modify this argument, introducing tools from probability theory, to obtain a unique continuation result for Schrodinger operators on the lattice with random potentials.

# Smooth infinite energy solutions to nonlinear Schrodinger equations

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We discuss smooth infinite energy solutions to nonlinear

Schrodinger equations on $R^d$. These solutions are periodic in time,

and quasi-periodic in space. This is unlike Moser's solutions, which

are space-time periodic. When d=1, we are moreover able to

construct 2-gap type solutions, i.e., space-time quasi-periodic

solutions with two frequencies each. We use the semi-algebraic geometry

technique introduced by Bourgain.