Green's function estimates for lattice Schrödinger operators and applications /:
This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas...
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Princeton, N.J. :
Princeton University Press,
©2005.
©2005 |
| Schriftenreihe: | Annals of mathematics studies ;
no. 158. |
| Schlagworte: | |
| Online-Zugang: | DE-862 DE-863 |
| Zusammenfassung: | This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art." |
| Beschreibung: | 1 online resource (1 online resource :) |
| Bibliographie: | Includes bibliographical references. |
| ISBN: | 9780691120980 0691120986 9781400837144 1400837146 1322075719 9781322075716 |
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| 100 | 1 | |a Bourgain, Jean, |d 1954-2018. |1 https://id.oclc.org/worldcat/entity/E39PBJyCMdGCJ6GYJyBfC6JhpP |0 http://id.loc.gov/authorities/names/n81115866 | |
| 245 | 1 | 0 | |a Green's function estimates for lattice Schrödinger operators and applications / |c J. Bourgain. |
| 260 | |a Princeton, N.J. : |b Princeton University Press, |c ©2005. | ||
| 264 | 4 | |c ©2005 | |
| 300 | |a 1 online resource (1 online resource :) | ||
| 336 | |a text |b txt |2 rdacontent | ||
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| 490 | 1 | |a Annals of mathematics studies ; |v no. 158 | |
| 504 | |a Includes bibliographical references. | ||
| 588 | 0 | |a Print version record. | |
| 505 | 0 | 0 | |t Frontmatter -- |t Contents -- |t Acknowledgment -- |t Chapter 1. Introduction -- |t Chapter 2. Transfer Matrix and Lyapounov Exponent -- |t Chapter 3. Herman's Subharmonicity Method -- |t Chapter 4. Estimates on Subharmonic Functions -- |t Chapter 5. LDT for Shift Model -- |t Chapter 6. Avalanche Principle in SL -- |t Chapter 7. Consequences for Lyapounov Exponent, IDS, and Green's Function -- |t Chapter 8. Refinements -- |t Chapter 9. Some Facts about Semialgebraic Sets -- |t Chapter 10. Localization -- |t Chapter 11. Generalization to Certain Long-Range Models -- |t Chapter 12. Lyapounov Exponent and Spectrum -- |t Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder -- |t Chapter 14. A Matrix-Valued Cartan-Type Theorem -- |t Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts -- |t Chapter 16. Application to the Kicked Rotor Problem -- |t Chapter 17. Quasi-Periodic Localization on the Z -- |t Chapter 18. An Approach to Melnikov's Theorem on Persistency of Nonresonant Lower Dimension Tori -- |t Chapter 19. Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations -- |t Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations -- |t Appendix. |
| 520 | |a This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art." | ||
| 546 | |a In English. | ||
| 650 | 0 | |a Schrödinger operator. |0 http://id.loc.gov/authorities/subjects/sh85118496 | |
| 650 | 0 | |a Green's functions. |0 http://id.loc.gov/authorities/subjects/sh85057264 | |
| 650 | 0 | |a Hamiltonian systems. |0 http://id.loc.gov/authorities/subjects/sh85058563 | |
| 650 | 0 | |a Evolution equations. |0 http://id.loc.gov/authorities/subjects/sh85046035 | |
| 650 | 6 | |a Fonctions de Green. | |
| 650 | 6 | |a Systèmes hamiltoniens. | |
| 650 | 6 | |a Équations d'évolution. | |
| 650 | 6 | |a Opérateur de Schrödinger. | |
| 650 | 7 | |a MATHEMATICS |x Differential Equations |x General. |2 bisacsh | |
| 650 | 7 | |a Evolution equations |2 fast | |
| 650 | 7 | |a Green's functions |2 fast | |
| 650 | 7 | |a Hamiltonian systems |2 fast | |
| 650 | 7 | |a Schrödinger operator |2 fast | |
| 650 | 7 | |a Mathematische fysica. |2 gtt | |
| 650 | 7 | |a Green-functies. |2 gtt | |
| 650 | 7 | |a Hamiltonianen. |2 gtt | |
| 650 | 7 | |a Schrödingervergelijking. |2 gtt | |
| 653 | |a Almost Mathieu operator. | ||
| 653 | |a Analytic function. | ||
| 653 | |a Anderson localization. | ||
| 653 | |a Betti number. | ||
| 653 | |a Cartan's theorem. | ||
| 653 | |a Chaos theory. | ||
| 653 | |a Density of states. | ||
| 653 | |a Dimension (vector space). | ||
| 653 | |a Diophantine equation. | ||
| 653 | |a Dynamical system. | ||
| 653 | |a Equation. | ||
| 653 | |a Existential quantification. | ||
| 653 | |a Fundamental matrix (linear differential equation). | ||
| 653 | |a Green's function. | ||
| 653 | |a Hamiltonian system. | ||
| 653 | |a Hermitian adjoint. | ||
| 653 | |a Infimum and supremum. | ||
| 653 | |a Iterative method. | ||
| 653 | |a Jacobi operator. | ||
| 653 | |a Linear equation. | ||
| 653 | |a Linear map. | ||
| 653 | |a Linearization. | ||
| 653 | |a Monodromy matrix. | ||
| 653 | |a Non-perturbative. | ||
| 653 | |a Nonlinear system. | ||
| 653 | |a Normal mode. | ||
| 653 | |a Parameter space. | ||
| 653 | |a Parameter. | ||
| 653 | |a Parametrization. | ||
| 653 | |a Partial differential equation. | ||
| 653 | |a Periodic boundary conditions. | ||
| 653 | |a Phase space. | ||
| 653 | |a Phase transition. | ||
| 653 | |a Polynomial. | ||
| 653 | |a Renormalization. | ||
| 653 | |a Self-adjoint. | ||
| 653 | |a Semialgebraic set. | ||
| 653 | |a Special case. | ||
| 653 | |a Statistical significance. | ||
| 653 | |a Subharmonic function. | ||
| 653 | |a Summation. | ||
| 653 | |a Theorem. | ||
| 653 | |a Theory. | ||
| 653 | |a Transfer matrix. | ||
| 653 | |a Transversality (mathematics). | ||
| 653 | |a Trigonometric functions. | ||
| 653 | |a Trigonometric polynomial. | ||
| 653 | |a Uniformization theorem. | ||
| 758 | |i has work: |a Green's function estimates for lattice Schrödinger operators and applications (Text) |1 https://id.oclc.org/worldcat/entity/E39PCFHkWYr8THXrh9vrQygGDy |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Bourgain, Jean, 1954- |t Green's function estimates for lattice Schrödinger operators and applications. |d Princeton, N.J. : Princeton University Press, ©2005 |z 0691120978 |w (DLC) 2004104492 |w (OCoLC)56874852 |
| 830 | 0 | |a Annals of mathematics studies ; |v no. 158. |0 http://id.loc.gov/authorities/names/n42002129 | |
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| author | Bourgain, Jean, 1954-2018 |
| author_GND | http://id.loc.gov/authorities/names/n81115866 |
| author_facet | Bourgain, Jean, 1954-2018 |
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| contents | Frontmatter -- Contents -- Acknowledgment -- Chapter 1. Introduction -- Chapter 2. Transfer Matrix and Lyapounov Exponent -- Chapter 3. Herman's Subharmonicity Method -- Chapter 4. Estimates on Subharmonic Functions -- Chapter 5. LDT for Shift Model -- Chapter 6. Avalanche Principle in SL -- Chapter 7. Consequences for Lyapounov Exponent, IDS, and Green's Function -- Chapter 8. Refinements -- Chapter 9. Some Facts about Semialgebraic Sets -- Chapter 10. Localization -- Chapter 11. Generalization to Certain Long-Range Models -- Chapter 12. Lyapounov Exponent and Spectrum -- Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder -- Chapter 14. A Matrix-Valued Cartan-Type Theorem -- Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts -- Chapter 16. Application to the Kicked Rotor Problem -- Chapter 17. Quasi-Periodic Localization on the Z -- Chapter 18. An Approach to Melnikov's Theorem on Persistency of Nonresonant Lower Dimension Tori -- Chapter 19. Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations -- Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations -- Appendix. |
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Introduction --</subfield><subfield code="t">Chapter 2. Transfer Matrix and Lyapounov Exponent --</subfield><subfield code="t">Chapter 3. Herman's Subharmonicity Method --</subfield><subfield code="t">Chapter 4. Estimates on Subharmonic Functions --</subfield><subfield code="t">Chapter 5. LDT for Shift Model --</subfield><subfield code="t">Chapter 6. Avalanche Principle in SL --</subfield><subfield code="t">Chapter 7. Consequences for Lyapounov Exponent, IDS, and Green's Function --</subfield><subfield code="t">Chapter 8. Refinements --</subfield><subfield code="t">Chapter 9. Some Facts about Semialgebraic Sets --</subfield><subfield code="t">Chapter 10. Localization --</subfield><subfield code="t">Chapter 11. Generalization to Certain Long-Range Models --</subfield><subfield code="t">Chapter 12. Lyapounov Exponent and Spectrum --</subfield><subfield code="t">Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder --</subfield><subfield code="t">Chapter 14. A Matrix-Valued Cartan-Type Theorem --</subfield><subfield code="t">Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts --</subfield><subfield code="t">Chapter 16. Application to the Kicked Rotor Problem --</subfield><subfield code="t">Chapter 17. Quasi-Periodic Localization on the Z --</subfield><subfield code="t">Chapter 18. An Approach to Melnikov's Theorem on Persistency of Nonresonant Lower Dimension Tori --</subfield><subfield code="t">Chapter 19. Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations --</subfield><subfield code="t">Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations --</subfield><subfield code="t">Appendix.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. 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| id | ZDB-4-EBA-ocn795383539 |
| illustrated | Illustrated |
| indexdate | 2025-03-18T14:16:03Z |
| institution | BVB |
| isbn | 9780691120980 0691120986 9781400837144 1400837146 1322075719 9781322075716 |
| language | English |
| oclc_num | 795383539 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (1 online resource :) |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Princeton University Press, |
| record_format | marc |
| series | Annals of mathematics studies ; |
| series2 | Annals of mathematics studies ; |
| spelling | Bourgain, Jean, 1954-2018. https://id.oclc.org/worldcat/entity/E39PBJyCMdGCJ6GYJyBfC6JhpP http://id.loc.gov/authorities/names/n81115866 Green's function estimates for lattice Schrödinger operators and applications / J. Bourgain. Princeton, N.J. : Princeton University Press, ©2005. ©2005 1 online resource (1 online resource :) text txt rdacontent computer c rdamedia online resource cr rdacarrier text file Annals of mathematics studies ; no. 158 Includes bibliographical references. Print version record. Frontmatter -- Contents -- Acknowledgment -- Chapter 1. Introduction -- Chapter 2. Transfer Matrix and Lyapounov Exponent -- Chapter 3. Herman's Subharmonicity Method -- Chapter 4. Estimates on Subharmonic Functions -- Chapter 5. LDT for Shift Model -- Chapter 6. Avalanche Principle in SL -- Chapter 7. Consequences for Lyapounov Exponent, IDS, and Green's Function -- Chapter 8. Refinements -- Chapter 9. Some Facts about Semialgebraic Sets -- Chapter 10. Localization -- Chapter 11. Generalization to Certain Long-Range Models -- Chapter 12. Lyapounov Exponent and Spectrum -- Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder -- Chapter 14. A Matrix-Valued Cartan-Type Theorem -- Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts -- Chapter 16. Application to the Kicked Rotor Problem -- Chapter 17. Quasi-Periodic Localization on the Z -- Chapter 18. An Approach to Melnikov's Theorem on Persistency of Nonresonant Lower Dimension Tori -- Chapter 19. Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations -- Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations -- Appendix. This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art." In English. Schrödinger operator. http://id.loc.gov/authorities/subjects/sh85118496 Green's functions. http://id.loc.gov/authorities/subjects/sh85057264 Hamiltonian systems. http://id.loc.gov/authorities/subjects/sh85058563 Evolution equations. http://id.loc.gov/authorities/subjects/sh85046035 Fonctions de Green. Systèmes hamiltoniens. Équations d'évolution. Opérateur de Schrödinger. MATHEMATICS Differential Equations General. bisacsh Evolution equations fast Green's functions fast Hamiltonian systems fast Schrödinger operator fast Mathematische fysica. gtt Green-functies. gtt Hamiltonianen. gtt Schrödingervergelijking. gtt Almost Mathieu operator. Analytic function. Anderson localization. Betti number. Cartan's theorem. Chaos theory. Density of states. Dimension (vector space). Diophantine equation. Dynamical system. Equation. Existential quantification. Fundamental matrix (linear differential equation). Green's function. Hamiltonian system. Hermitian adjoint. Infimum and supremum. Iterative method. Jacobi operator. Linear equation. Linear map. Linearization. Monodromy matrix. Non-perturbative. Nonlinear system. Normal mode. Parameter space. Parameter. Parametrization. Partial differential equation. Periodic boundary conditions. Phase space. Phase transition. Polynomial. Renormalization. Self-adjoint. Semialgebraic set. Special case. Statistical significance. Subharmonic function. Summation. Theorem. Theory. Transfer matrix. Transversality (mathematics). Trigonometric functions. Trigonometric polynomial. Uniformization theorem. has work: Green's function estimates for lattice Schrödinger operators and applications (Text) https://id.oclc.org/worldcat/entity/E39PCFHkWYr8THXrh9vrQygGDy https://id.oclc.org/worldcat/ontology/hasWork Print version: Bourgain, Jean, 1954- Green's function estimates for lattice Schrödinger operators and applications. Princeton, N.J. : Princeton University Press, ©2005 0691120978 (DLC) 2004104492 (OCoLC)56874852 Annals of mathematics studies ; no. 158. http://id.loc.gov/authorities/names/n42002129 |
| spellingShingle | Bourgain, Jean, 1954-2018 Green's function estimates for lattice Schrödinger operators and applications / Annals of mathematics studies ; Frontmatter -- Contents -- Acknowledgment -- Chapter 1. Introduction -- Chapter 2. Transfer Matrix and Lyapounov Exponent -- Chapter 3. Herman's Subharmonicity Method -- Chapter 4. Estimates on Subharmonic Functions -- Chapter 5. LDT for Shift Model -- Chapter 6. Avalanche Principle in SL -- Chapter 7. Consequences for Lyapounov Exponent, IDS, and Green's Function -- Chapter 8. Refinements -- Chapter 9. Some Facts about Semialgebraic Sets -- Chapter 10. Localization -- Chapter 11. Generalization to Certain Long-Range Models -- Chapter 12. Lyapounov Exponent and Spectrum -- Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder -- Chapter 14. A Matrix-Valued Cartan-Type Theorem -- Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts -- Chapter 16. Application to the Kicked Rotor Problem -- Chapter 17. Quasi-Periodic Localization on the Z -- Chapter 18. An Approach to Melnikov's Theorem on Persistency of Nonresonant Lower Dimension Tori -- Chapter 19. Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations -- Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations -- Appendix. Schrödinger operator. http://id.loc.gov/authorities/subjects/sh85118496 Green's functions. http://id.loc.gov/authorities/subjects/sh85057264 Hamiltonian systems. http://id.loc.gov/authorities/subjects/sh85058563 Evolution equations. http://id.loc.gov/authorities/subjects/sh85046035 Fonctions de Green. Systèmes hamiltoniens. Équations d'évolution. Opérateur de Schrödinger. MATHEMATICS Differential Equations General. bisacsh Evolution equations fast Green's functions fast Hamiltonian systems fast Schrödinger operator fast Mathematische fysica. gtt Green-functies. gtt Hamiltonianen. gtt Schrödingervergelijking. gtt |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85118496 http://id.loc.gov/authorities/subjects/sh85057264 http://id.loc.gov/authorities/subjects/sh85058563 http://id.loc.gov/authorities/subjects/sh85046035 |
| title | Green's function estimates for lattice Schrödinger operators and applications / |
| title_alt | Frontmatter -- Contents -- Acknowledgment -- Chapter 1. Introduction -- Chapter 2. Transfer Matrix and Lyapounov Exponent -- Chapter 3. Herman's Subharmonicity Method -- Chapter 4. Estimates on Subharmonic Functions -- Chapter 5. LDT for Shift Model -- Chapter 6. Avalanche Principle in SL -- Chapter 7. Consequences for Lyapounov Exponent, IDS, and Green's Function -- Chapter 8. Refinements -- Chapter 9. Some Facts about Semialgebraic Sets -- Chapter 10. Localization -- Chapter 11. Generalization to Certain Long-Range Models -- Chapter 12. Lyapounov Exponent and Spectrum -- Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder -- Chapter 14. A Matrix-Valued Cartan-Type Theorem -- Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts -- Chapter 16. Application to the Kicked Rotor Problem -- Chapter 17. Quasi-Periodic Localization on the Z -- Chapter 18. An Approach to Melnikov's Theorem on Persistency of Nonresonant Lower Dimension Tori -- Chapter 19. Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations -- Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations -- Appendix. |
| title_auth | Green's function estimates for lattice Schrödinger operators and applications / |
| title_exact_search | Green's function estimates for lattice Schrödinger operators and applications / |
| title_full | Green's function estimates for lattice Schrödinger operators and applications / J. Bourgain. |
| title_fullStr | Green's function estimates for lattice Schrödinger operators and applications / J. Bourgain. |
| title_full_unstemmed | Green's function estimates for lattice Schrödinger operators and applications / J. Bourgain. |
| title_short | Green's function estimates for lattice Schrödinger operators and applications / |
| title_sort | green s function estimates for lattice schrodinger operators and applications |
| topic | Schrödinger operator. http://id.loc.gov/authorities/subjects/sh85118496 Green's functions. http://id.loc.gov/authorities/subjects/sh85057264 Hamiltonian systems. http://id.loc.gov/authorities/subjects/sh85058563 Evolution equations. http://id.loc.gov/authorities/subjects/sh85046035 Fonctions de Green. Systèmes hamiltoniens. Équations d'évolution. Opérateur de Schrödinger. MATHEMATICS Differential Equations General. bisacsh Evolution equations fast Green's functions fast Hamiltonian systems fast Schrödinger operator fast Mathematische fysica. gtt Green-functies. gtt Hamiltonianen. gtt Schrödingervergelijking. gtt |
| topic_facet | Schrödinger operator. Green's functions. Hamiltonian systems. Evolution equations. Fonctions de Green. Systèmes hamiltoniens. Équations d'évolution. Opérateur de Schrödinger. MATHEMATICS Differential Equations General. Evolution equations Green's functions Hamiltonian systems Schrödinger operator Mathematische fysica. Green-functies. Hamiltonianen. Schrödingervergelijking. |
| work_keys_str_mv | AT bourgainjean greensfunctionestimatesforlatticeschrodingeroperatorsandapplications |