Algebraic aspects of Darboux transformations, quantum integrable systems and supersymmetric quantum mechanics
This volume represents the 2010 Jairo Charris Seminar in Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, which was held at the Universidad Sergio Arboleda in Santa Marta, Colombia. The papers cover the fields of Supersymmetric Quantum Me...
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| Hlavní autoři: | , , , |
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Providence, R.I
Instituto de Matemáticas y sus Aplicaciones
2012
Madrid American Mathematical Society |
| Vydání: | 1 |
| Edice: | Contemporary mathematics |
| Témata: |
Quantum theory
> General mathematical topics and methods in quantum theory
> Special quantum systems, such as solvable systems. msc
Quantum theory
> General mathematical topics and methods in quantum theory
> Supersymmetry and quantum mechanics. msc
Special functions (33-XX deals with the properties of functions as functions)
> Other special functions
> None of the above, but in this section. msc
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| ISBN: | 0821875841, 9780821875841 |
| On-line přístup: | Získat plný text |
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- The pentagram map: geometry, algebra, integrability -- 1. Introduction: The pentagram map -- 2. Complete integrability of the pentagram map on the space _{ } -- 3. Relation to the Boussinesq equation -- 4. The space _{ } and 2-frieze patterns -- Acknowledgments -- References -- Jet Bundles, symmetries, Darboux transforms -- 1. Introduction -- 2. Jet bundles and differential equations -- 3. A glimpse of Gromov's ℎ-principle -- 4. Symmetries of differential equations -- 5. Darboux transformations for the ACH equation -- References -- Explicit higher-dimensional Darboux transformations for the time-dependent Schrödinger equation -- Introduction -- 1. Preliminaries -- 2. The Darboux operator in ( +1) dimensions -- 3. Application: the free particle in (5+1) dimensions -- 4. Darboux transformations in arbitrary coordinate systems -- Concluding remarks -- References -- Elliptic beta integrals and solvable models of statistical mechanics -- 1. The simplest elliptic hypergeometric integrals -- 2. The elliptic beta integral STR solution and star-star relation -- 3. A hyperbolic beta integral STR solution -- 4. Partition functions -- 5. Conclusion -- Acknowledgments -- Appendix A. The modified -gamma function -- Appendix B. General multiple gamma functions -- References
- Intro -- Preface -- Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials -- 1. Introduction -- 2. Background -- 3. Integrability of equation (1) by Picard-Vessiot -- 4. Spectral/quadrature duality. An integration procedure -- 5. The Θ-functons -- 6. Integration as a linearly exponential Θ-extension -- 7. Integrability and differential closedness -- 8. Definition of through Liouvillian extension -- 9. Non-finite-gap integrable counterexamples -- 10. Concluding remarks -- References -- Darboux transformation, exceptional orthogonal polynomials and information theoretic measures of uncertainty -- 1. Introduction -- 2. A brief outline of supersymmetric quantum mechanics -- 3. EOP's associated with broken supersymmetry -- 4. Relation with higher order Darboux transformation -- 5. Information-theoretic lengths for the partner potential of linear harmonic oscillator via supersymmetry -- 6. Moment of the quantum mechanical states associated with the partner of harmonic oscillator via supersymmetry -- 7. Discussion: -- References -- On orthogonal polynomials spanning a non-standard flag -- 1. Introduction -- 2. Preliminaries -- 3. Codimension 1 flags -- 4. Higher codimension flags -- Acknowledgements -- References -- On the Supersymmetric Spectra of two Planar Integrable Quantum Systems -- 1. Introduction -- 2. =2 supersymmetric planar quantum systems -- 3. The planar quantum Kepler/Coulomb problem and supersymmetry -- 4. The planar quantum Euler/Coulomb problem and supersymmetry -- 5. Two center collapse in one center -- 6. Further comments -- Acknowledgements -- References -- Solvable rational extension of translationally shape invariant potentials -- Introduction -- 1. Solvable rational extensions of the harmonic oscillator -- 2. Second category potentials -- 3. Conclusion -- Acknowledgments -- References