Mathematical challenges of zero-range physics models, methods, rigorous results, open problems /
Since long over the decades there has been a large transversal community of mathematicians grappling with the sophisticated challenges of the rigorous modelling and the spectral and scattering analysis of quantum systems of particles subject to an interaction so much localised to be considered with...
| Other Authors: | Michelangeli, Luigi Alessandro,, SpringerLink (Online service) |
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| Format: | eBook |
| Language: | English |
| Published: |
Cham, Switzerland :
Springer,
[2021]
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| Physical Description: |
1 online resource. |
| Series: |
Springer INdAM series ;
42. |
| Subjects: |
| Summary: |
Since long over the decades there has been a large transversal community of mathematicians grappling with the sophisticated challenges of the rigorous modelling and the spectral and scattering analysis of quantum systems of particles subject to an interaction so much localised to be considered with zero range. Such a community is experiencing fruitful and inspiring exchanges with experimental and theoretical physicists. This volume reflects such spirit, with a diverse range of original contributions by experts, presenting an up-to-date collection of most relevant results and challenging open problems. It has been conceived with the deliberate two-fold purpose of serving as an updated reference for recent results, mathematical tools, and the vast related literature on the one hand, and as a bridge towards several key open problems that will surely form the forthcoming research agenda in this field. |
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| Item Description: |
Includes bibliographical references. Since long over the decades there has been a large transversal community of mathematicians grappling with the sophisticated challenges of the rigorous modelling and the spectral and scattering analysis of quantum systems of particles subject to an interaction so much localised to be considered with zero range. Such a community is experiencing fruitful and inspiring exchanges with experimental and theoretical physicists. This volume reflects such spirit, with a diverse range of original contributions by experts, presenting an up-to-date collection of most relevant results and challenging open problems. It has been conceived with the deliberate two-fold purpose of serving as an updated reference for recent results, mathematical tools, and the vast related literature on the one hand, and as a bridge towards several key open problems that will surely form the forthcoming research agenda in this field. 1. Iori, M. et al., Thermodynamic properties of ultracold Fermi gases across the BCS-BEC crossover and the Bertsch problem -- 2. Cacciapuoti, C., et al., Scattering theory for delta-potentials supported by locally deformed planes -- 3. Schmidt, J., The Massless Nelson Hamiltonian and its Domain -- 4. Borrelli, W. et al., A note on the Dirac operator with Kirchoff-type vertex conditions on metric graphs -- 5. Ourmières-Bonafos, T., Dirac operators and shell interactions: a survey -- 6. Lampart, J., Ultraviolet properties of a polaron model with point interactions and a number cutoff -- 7. Scandone, R., Zero modes and low-energy resolvent expansion for three dimensional Schrödinger operators with point interactions -- 8. Ottolini, A., Spectral properties of point interactions with fermionic symmetries -- 9. Akbas, H. and Teoman Turgut, O., Oppenheimer Type Approximation for a Simple Renormalizable System -- 10. Lotoreichik, V., Spectral isoperimetric inequality for the '-interaction on a contour -- 11. Pozzoli, E., Quantum confinement in Grushin-type manifolds -- 12. Gallone, M. et al., Kreın-Višik-Birman self-adjoint extension theory revisited -- 13. Khotyakov, M. and Michelangeli, A., Translation and adaptation from Russian of M. Sh. Birman On the theory of selfadjoint extensions of positive definite operators Math. Sb. 28 (1956), 431-450 (1956). Intro -- Preface -- Contents -- Thermodynamic Properties of Ultracold Fermi Gases Across the BCS-BEC Crossover and the Bertsch Problem -- 1 Introduction -- 2 BCS-BEC Crossover in the Mean Field Approximation -- 2.1 Mean Field Results at T = 0 -- 3 BCS-BEC Crossover at Finite Temperature -- 3.1 Effective Action -- 3.2 Saddle Point Approximation -- 4 Gaussian Fluctuations -- 4.1 Equations for the Energy Gap and for the Number of Particles -- 4.2 Results for the Critical Temperature and the Bertsch Parameter -- 5 Conclusions -- Appendix -- References. 4.3 Self-adjointness: Proof of Theorem 3.3 -- 5 Concluding Remarks -- References -- A Note on the Dirac Operator with Kirchoff-Type Vertex Conditions on Metric Graphs -- 1 Introduction -- 2 Functional Setting -- 3 Self-adjointness and Spectrum -- 3.1 Self-adjointness -- 3.2 Essential Spectrum -- 3.3 Absence of Eigenvalues in the Spectral Gap -- 3.4 Graphs with Eigenvalues at the Thresholds -- 3.5 Graphs with Embedded Eigenvalues -- 4 A Model Case: The Triple Junction -- 5 The Form-Domain -- References -- Dirac Operators and Shell Interactions: A Survey -- 1 Introduction. 3.3 Remarks and Open Problems -- 4 Structure of the Spectrum -- 4.1 A Resolvent Formula and Its Consequences -- 4.2 Spectral Asymptotics -- References -- Ultraviolet Properties of a Polaron Model with Point Interactions and a Number Cutoff -- 1 Introduction -- 2 Extension of the Free Operator and Boundary Conditions -- 2.1 Boundary Values -- 2.2 The Hamiltonian for Nmax=1 -- 2.3 The Hamiltonians for Nmax>1 -- References -- Zero Modes and Low-Energy Resolvent Expansion for Three Dimensional Schrödinger Operators with Point Interactions -- 1 Introduction and Main Results. |
| Physical Description: |
1 online resource. |
| Bibliography: |
Includes bibliographical references. |
| ISBN: |
9783030604530 3030604535 |