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 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 d’-interaction on a contour.- 11. Pozzoli, E., Quantum confinement in Grushin-type manifolds.- 12. Gallone, M. et al., Krein-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).