| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 14, 2020 |
| Latest Amendment Date: | October 14, 2020 |
| Award Number: | 2053285 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Marian Bocea
mbocea@nsf.gov (703)292-2595 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 13, 2020 |
| End Date: | July 31, 2022 (Estimated) |
| Total Intended Award Amount: | $51,713.00 |
| Total Awarded Amount to Date: | $51,713.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
Baton Rouge LA US 70803-2701 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ANALYSIS PROGRAM |
| Primary Program Source: |
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| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This project concerns the spectral theory of Schrodinger operators, a central topic in quantum mechanics. Schrodinger operators describe the movement of an electron in a medium subject to a disordered system. The development of the theory of Schrodinger operators is expected to enhance the understanding of many types of physical phenomena, including conductance, quantum Hall effect, quasicrystals, and graphene. The PI intends to develop new tools to provide rigorous mathematical explanations for these phenomena. The tools that will be developed will also find applications in other branches of mathematics, including harmonic analysis, probability and number theory.
This project consists of several parts. One is to study quantum graphs in magnetic fields. The PI intends to understand the topological structure of the spectrum and spectral decompositions. The second part involves studying Laplacians on discrete periodic graphs with a goal of finding the connection between the presence of spectral gaps and the geometric structure of the underlying lattice. The third goal concerns discrete quasi-periodic Schrodinger operators, focusing on several well-known problems including measure of the spectrum, continuity of the spectrum, structure of eigenfunctions in the localization regime and quantum dynamics in the singular continuous regime for quasi-periodic operators. Another goal is to understand Schrodinger operators with potentials generated by skew-shift. These operators, although being completely deterministic, are expected to resemble random features.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The intellectual merit of this project consists of solving several open
problems originated from quantum mechanics, harmonic analysis and statistical physics. Our focus has been both on developing new methods and on applications to models that are of interest to the physics community.
Our work during the entire project period included the proof of Cantor spectrum for a tight-binding operator in graphene structure for all irrational magnetic fluxes, thus providing the theoereticl foundation for self-similar structures in graphene material, a full spectral analysis of a tight-binding model in graphene structure in external magnetic fields, the first higher dimensional version of the fractal uncertainty principle, the first sharp L^p improving estimates for arithmetic averages including prime numbers and polynomial progressions, the first higher dimensional version of polynomial Roth theorem, the first result in multi-point dynamical localization, the first multi-point correlation decay with symmetrize distances in arbitrary dimension, the first sharp spectral analysis for quasi-periodic operators in the presense of two kinds of resonances, the existence of Dirac cones for graphene for arbitrary rational magnetic fields, density of states expansion for magnetic random operators, rigorous study of the quantum Hall effect and the proof of dynamical delocalization close to the graphene conical point under disorder, the first coupling independent Holder continuity of the Lyapunov exponent in the large coupling regime, entropy and information decay of the Kac’s evolution.
Last Modified: 11/29/2022
Modified by: Rui Han
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