Mathematical Challenges of Zero-Range Physics

1. Front Matter

   Pages i-viii

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2. Thermodynamic Properties of Ultracold Fermi Gases Across the BCS-BEC Crossover and the Bertsch Problem

   • Martina Iori, Tommaso Macrì, Andrea Trombettoni

   Pages 1-33

3. Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

   • Claudio Cacciapuoti, Davide Fermi, Andrea Posilicano

   Pages 35-55

4. The Massless Nelson Hamiltonian and Its Domain

   • Julian Schmidt

   Pages 57-80

5. A Note on the Dirac Operator with Kirchoff-Type Vertex Conditions on Metric Graphs

   • William Borrelli, Raffaele Carlone, Lorenzo Tentarelli

   Pages 81-104

6. Dirac Operators and Shell Interactions: A Survey

   • Thomas Ourmières-Bonafos, Fabio Pizzichillo

   Pages 105-131

7. Ultraviolet Properties of a Polaron Model with Point Interactions and a Number Cutoff

   • Jonas Lampart

   Pages 133-147

8. Zero Modes and Low-Energy Resolvent Expansion for Three Dimensional Schrödinger Operators with Point Interactions

   • Raffaele Scandone

   Pages 149-162

9. Spectral Properties of Point Interactions with Fermionic Symmetries

   • Andrea Ottolini

   Pages 163-176

10. Born-Oppenheimer Type Approximation for a Simple Renormalizable System

    • Haci Akbas, O. Teoman Turgut

    Pages 177-214

11. Spectral Isoperimetric Inequality for the δ′-Interaction on a Contour

    • Vladimir Lotoreichik

    Pages 215-227

12. Quantum Confinement in α-Grushin Planes

    • Eugenio Pozzoli

    Pages 229-237

13. Kreı̆n-Višik-Birman Self-Adjoint Extension Theory Revisited

    • Matteo Gallone, Alessandro Michelangeli, Andrea Ottolini

    Pages 239-304

14. Translation and Adaptation from Russian of M. Sh. Birman, “On the Theory of Self-Adjoint Extensions of Positive Definite Operators”, Math. Sb. 28 (1956), 431–450

    • Mikhail Khotyakov, Alessandro Michelangeli

    Pages 305-329

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.

• Zero-range (contact) interactions
• Singular perturbation of elliptic operators
• Self-adjoint extensions theories
• Schroedinger and Dirac operators
• Ultra-cold gases

• Institute for Applied Mathematics, and Hausdorff Center for Mathematics, University of Bonn, Bonn, Germany

  Alessandro Michelangeli

Alessandro Michelangeli is at present a von Humboldt Experienced Researcher at the Institute of Applied Mathematics and at the Hausdorff Center for Mathematics at the university of Bonn. His research is at the interface of mathematical physics, functional analysis, partial differential equations, and operator theory, with a special focus on problems and methods for quantum mechanics. He graduated in theoretical physics in Pisa and in mathematical physics at SISSA Trieste, held faculty positions at the LMU Munich and SISSA Trieste, and visiting positions at the University of Cambridge, SISSA Trieste, and Bilkent. 

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