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BL
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| Record Number
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859713
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| Main Entry
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Kostadinova, Evdokiya Georgieva
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| Title & Author
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Spectral approach to transport problems in two-dimensional disordered lattices : : physical interpretation and applications /\ Evdokiya Georgieva Kostadinova.
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| Publication Statement
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Cham, Switzerland :: Springer,, 2018.
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| Series Statement
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Springer theses,
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| Page. NO
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1 online resource (xiii, 107 pages) :: illustrations (some color)
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| ISBN
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3030022129
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: 9783030022129
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3030022110
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9783030022112
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| Notes
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"Doctoral thesis accepted by Baylor University, Waco, Texas, USA."
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| Contents
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Intro; Supervisorś Foreword; Acknowledgments; Contents; Chapter 1: Introduction; 1.1 Formulation of the Transport Problem; 1.2 Nature of Disorder; 1.3 Relevance to Physical Systems; Bibliography; Chapter 2: Theoretical Background; 2.1 Localization Criteria; 2.2 Anderson Model; 2.3 Edwards and Thouless Model; 2.4 Scaling Theory; Bibliography; Chapter 3: Spectral Approach; 3.1 Essence of the Spectral Method; 3.1.1 Cyclic Subspaces and Equivalence Classes; 3.1.2 Spectral Decomposition of Normal Operators; 3.1.3 Extended States Conjecture and the Distance Formula
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3.2 Simplified Numerical Model (``Toy Model)́́3.2.1 Application to the Discrete Random Schrödinger Operator; 3.2.2 Preliminary Results in 2D and 3D; 3.3 Physical Interpretation; 3.3.1 Band Structure and the Spectrum of the Hamiltonian; 3.3.2 Bounded Operators and the Hilbert Space; 3.4 Scope and Limitations of the Spectral Analysis; Bibliography; Chapter 4: Delocalization in 2D Lattices of Various Geometries; 4.1 Transport in the Honeycomb, Triangular, and Square Lattices; 4.2 Orthogonality Check; 4.3 Equation Fitting; 4.4 Cluster Analysis
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4.5 Comparison Between the Honeycomb and the Triangular LatticesBibliography; Chapter 5: Transport in the Two-Dimensional Honeycomb Lattice with Substitutional Disorder; 5.1 Discrete Percolation; 5.2 Formulation of the Transport Problem; 5.2.1 Binary Alloy Model of Doping; 5.2.2 Quantum Percolation Problem; 5.2.3 Relation Between Quantum Percolation and Anderson Localization; 5.3 Distribution of Variables; 5.4 2D Honeycomb Lattice with Substitutional Disorder; Bibliography; Chapter 6: Transport in 2D Complex Plasma Crystals; 6.1 Complex Plasma Preliminaries
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6.2 Two-Dimensional Dust Crystal Analogue6.3 Transport in the Classical Regime; 6.4 Numerical Simulations of Dust Particle Dynamics; 6.4.1 Dust Crystal Formation and Defect Types; 6.4.2 Crystal Perturbation; 6.5 Spectral Analysis; Bibliography; Chapter 7: Conclusions; Bibliography; Appendix A: Basic Materials Science Terms; Comparison Between Fermi Energy and Fermi Level; Appendix B: Mathematical Preliminaries; Kindergarten Math; Measure Theory; Point-Set Topology; Group Theory; Probability Theory; Curriculum Vitae
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| Abstract
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This thesis introduces the spectral approach to transport problems in infinite disordered systems characterized by Anderson-type Hamiltonians. The spectral approach determines (with probability one) the existence of extended states for nonzero disorder in infinite lattices of any dimension and geometry. Here, the author focuses on the critical 2D case, where previous numerical and experimental results have shown disagreement with theory. Not being based on scaling theory, the proposed method avoids issues related to boundary conditions and provides an alternative approach to transport problems where interaction with various types of disorder is considered. Beginning with a general overview of Anderson-type transport problems and their relevance to physical systems, it goes on to discuss in more detail the most relevant theoretical, numerical, and experimental developments in this field of research. The mathematical formulation of the innovative spectral approach is introduced together with a physical interpretation and discussion of its applicability to physical systems, followed by a numerical study of delocalization in the 2D disordered honeycomb, triangular, and square lattices. Transport in the 2D honeycomb lattice with substitutional disorder is investigated employing a spectral analysis of the quantum percolation problem. Next, the applicability of the method is extended to the classical regime, with an examination of diffusion of lattice waves in 2D disordered complex plasma crystals, along with discussion of proposed future developments in the study of complex transport problems using spectral theory.
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| Subject
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Lattice dynamics.
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| Subject
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Nanoelectronics.
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| Subject
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Lattice dynamics.
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| Subject
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Nanoelectronics.
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| Dewey Classification
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530.4/11
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| LC Classification
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QC176.8.L3
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