| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 1, 2017 |
| Latest Amendment Date: | July 1, 2018 |
| Award Number: | 1758326 |
| Award Instrument: | Continuing 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 16, 2017 |
| End Date: | June 30, 2019 (Estimated) |
| Total Intended Award Amount: | $71,837.00 |
| Total Awarded Amount to Date: | $71,837.00 |
| Funds Obligated to Date: |
FY 2018 = $33,158.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
426 AUDITORIUM RD RM 2 EAST LANSING MI US 48824-2600 (517)355-5040 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
MI US 48824-2600 |
| 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: |
01001819DB NSF RESEARCH & RELATED ACTIVIT |
| 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
The main goals of this research project are to develop and study mathematical models of quantum particles in periodic and quasiperiodic media, such as crystals or quasicrystals, based on spectral theory of Schrodinger operators. Periodic media correspond to crystalline structures, such as metals or semiconductors, which can conduct electrons freely at certain energies. It is proposed to study mathematically rigorous models of electron transport near the edges of the "forbidden zones" and develop new approaches to the effective mass approximation. Quasiperiodic operators are examples of disordered systems which, depending on the regime, can look like pure crystals, or crystals with random impurities, while being completely deterministic. One of the models under study demonstrates random-like behavior at arbitrarily small disorder and can potentially be a suitable replacement for a random environment without having to employ a large parameter space. Special emphasis will be given to multi-dimensional and multi-particle models, with possible applications to quantum spin systems and quantum information theory. The project provides research opportunities for undergraduate and graduate students.
The activities of this research project fall into several groups distinguished by the classes of the operators under study and the types of their spectra. In the area of Anderson localization for quasiperiodic operators ("random-like behavior"), the project studies multi-particle models with analytic potentials at perturbatively large disorder and low regularity models, with the latter results expected to be non-perturbative. The methods here include operator theory, harmonic analysis, real algebraic geometry, and large deviation theorems for subharmonic or piecewise-monotonic functions. In the area of absolutely continuous spectrum ("crystalline behavior"), the project investigates the relation between low regularity reducibility of Schrodinger cocycles and strong ballistic transport for the corresponding Schrodinger operators, which, in turn, is related to transport properties of quantum spin systems. In the area of periodic operators, it is intended to study possible singularities of the Bloch varieties at the edges of spectral bands, both in 2D and 3D cases. Finally, on the more abstract side, the project aims to develop a quantitative classification of almost commuting matrices in topologically non-trivial cases, which demonstrates connections both with Cantor spectra for quasiperiodic operators and with some quantum spin systems.
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.
Intellectual merit:
It is shown that a system of interacting particles on a 1D lattice under a quasiperiodic potential demonstrates two-particle Anderson localization at large disorder. However, if the quasiperiodic potential has symmetries, the proof fails in some energy regions, associated with the values of the interaction potential. Moreover, in the regime of strong interaction, it is shown that the system may demonstrade delocalization. This conclusions are in line with the numerical predictions from the works of physicists who studied similar models.
It was also shown that a large class of 1D quasiperiodic operators with monotone potentials demonstrates Anderson localization even at small coupling. It can be considered as an example of a non-random disordered quantum system which demonstrates ``random-like'' behavior. Several methods were developed to further study such systems.
In spectral theory of periodic operators, spectral band edges correspond to minima and maxima of band functions (Bloch eigenvalues). Analysis of the effective mass approximation in solid state physics leads naturally to the following question: under which conditions can one transform those extrema into generic Morse-type, by a small perturbation of the potential? A partial answer to this question was obtained in the 2D case: it was shown that the level sets of band functions at the extremal points are finite.
Broader impacts:
During the fellowship period, the PI has organized and co-organized several conferences, including a session at AMS sectional meeting. The project contributed to training of a postdoctoral fellow. During the fellowship period, the PI, together with a postdoctoral scholar, advised a group of six undergraduate students on a research project in spectral theory.
Last Modified: 10/27/2019
Modified by: Ilya Kachkovskiy
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