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Between Dynamics and Spectral Theory: June 6 – 10, 2016

Organized by: William Yessen and Zhenghe Zhang

Attendee List Download Talk ScheduleVideos

Dates: June 6 – 10, 2016

The study of 1D discrete Schroedinger equation lies across physics, spectral theory, and dynamical systems. It models the motion of a quantum particle in a disordered medium. The evolution of motion is studied via the spectral theory of the Schroedinger operator. For instance, the pure point spectrum corresponds to insulating behavior, and absolutely continuous spectrum to conductive behavior. In 1958, P. Anderson discovered the absence of diffusion for certain random lattice Hamiltonians, resulting in the Nobel prize in physics for him in 1977. Rigorous mathematical results were not available until 1980s. A key ingredient is Furstenberg’s work on positivity of Lyapunov exponents (LE) for random products of matrices, which is a result in dynamical systems. Since then, the relation between spectral theory and the dynamical object LE has been intensively studied. One of the most interesting cases other than the random case is the case of quasiperiodic (intermediate between periodic and random) potentials, which is even more subtle in the sense that metal-insulator transitions occur. The insulating behavior happens at positive LE regime; the conductive behavior occurs in the regime of zero LE. Most of the existing results concerning quasiperiodic case are for analytic potentials. There is recent progress for smooth potentials. New research directions such as inverse spectral theory, and potentials defined on other types of base dynamics are also very promising.

On the other hand, the discovery of quasicrystals by D. Shechtman in the early 1980’s, which led to his Nobel prize in chemistry in 2011, further precipitated the study of Schroedinger operators with potentials intermediate between periodic and random. One of the ways to mathematically model quasicrystals is via substitution tilings. In 1D great progress has been made by applying sophisticated dynamical systems techniques. Indeed, the spectrum can be studied as a cross section of the Julia set of the associated renormalization map, leading to a dictionary between spectral characteristics and the dynamics of the renormalization map. This in turn leads to a fine understanding of the quite subtle quantum dynamics. Similar techniques have been recently applied, with equal success, to a number of other physically relevant models, such as classical and quantum spin models and quantum walks on the line. The focus in the community has began to shift towards models of dimension 2 and higher (such as operators over the Penrose tiling), which seem to require completely new ideas. Nevertheless, somewhat simplified (but still physically relevant and mathematically nontrivial) higher dimensional models demonstrate prominence of techniques from topological dynamics and geometric measure theory — techniques that have been applied with noticeable success in recent years.

Thus, the purpose of the proposed workshop is to facilitate dissemination of recent results and discussions on current and future research directions, among the experts in spectral theory and dynamical systems.

 

Talk Schedule


Time Title Speaker Location
9:00am Registration and Welcome N/A SCGP Lobby
9:40am The Fibonacci Hamiltonian Anton Gorodetski SCGP 102
10:40am  Coffee Break N/A SCGP Cafe
11:10am The Labyrinth model and products of two Cantor sets Yuki Takahashi SCGP 102
12:10am Lunch N/A SCGP Cafe
2:15pm The Toda Lattice with Almost Periodic Initial Data Thomas VandenBoom SCGP 102
3:30pm Tea Time N/A SCGP Lobby
4:00pm Xuanji Hou SCGP 102


Time Title Speaker Location
9:00am On the almost reducibility for pseudo rotations of the disk Raphael Krikorian SCGP 102
10:00am  Coffee Break N/A SCGP Cafe
10:30am Applications of Quantitative Almost Reducibility Jiangong You SCGP 102
11:30am Lunch N/A SCGP Cafe
1:00pm SCGP Weekly Talk: Reducibility of quasi-periodic co-cycles–from Schrödinger to infinite dimension Hakan Eliasson SCGP 102
2:00pm Short Break N/A SCGP 102
2:15pm Periodic Approximations to Aperiodic Hamiltonians Jean Bellissard SCGP 102
3:30pm  Tea Time N/A SCGP Lobby


Time Title Speaker Location
9:00am Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic
Schr odinger operator
Svetlana Jitomirskaya SCGP 102
10:00am  Coffee Break N/A SCGP Cafe
10:30am Reducibility, localization, and quasiperiodic XY spin chains Ilya Kachkovskiy SCGP 102
11:30am Short Break N/A SCGP 102
11:45am Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles Silvius Klein SCGP 102
12:45am  Lunch N/A SCGP Cafe
2:15pm Quantum Transport in Quasicrystalline Environments David Damanik SCGP 102
3:30pm  Tea Time N/A SCGP Lobby
4:00pm Quantum transport for nonsingularly supported initial states Vitalii Gerbuz SCGP 102


Time Title Speaker Location
9:00am KdV equation with almost periodic initial data Milivoje Lukic SCGP 102
10:00am  Coffee Break N/A SCGP Cafe
10:30am Full measure reducibility and localization for quasi-periodic Jacobi operators: a topological criterion Rui Han SCGP 102
11:30am Short Break N/A SCGP 102
11:45am Sharp arithmetic spectral transitions and bounds on quantum dynamics for supercritical almost Mathieu operators Wencai Liu SCGP 102
12:45am  Lunch N/A SCGP Cafe
2:15pm Sub-critical behavior for quasi-periodic Jacobi operators Chris Marx SCGP 102
3:30pm  Tea Time N/A SCGP Lobby
4:00pm Ballistic Motion for Limit-Periodic Potentials Jake Fillman SCGP 102


Time Title Speaker Location
9:00am Arithmetic phase transition for meromorphic potentials Fan Yang SCGP 102
10:00am Coffee Break N/A SCGP Cafe
11:30am  Lunch N/A SCGP Cafe
3:30pm  Tea Time N/A SCGP Lobby

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