turkmath.org

In this talk I shall review the main mathematical problems for the rigorous treatment of quantum systems of particles interacting through a so-called "point" (or "delta", or "contact") interactions, namely a strong interaction supported on a region of size zero. I shall focus primarily on the main issue of the rigorous definition of point interaction Hamiltonians and of their self-adjointness, for the "delta" interaction does not allow for perturbative (Kato-Rellich, KLMN, etc.) techniques unlike ordinary potentials in Schroedinger operators. To this aim various alternative strategies shall be presented, together with the most recent developments: point interactions constructed (1) as extensions of symmetric operators for free particles, by means of the Birmen-Krein-Vishik theory of self-adjoint extensions, (2) as appropriate scaling limits (in the resolvent sense) of Schroedinger operators with 'squeezing' potentials, and (3) as self-adjoint realisations of a renormalised quadratic form for the energy of the system. Time permitting, I shall also discuss the spectral properties of point interaction Hamiltonians, significantly the occurrence of the 'Thomas effect' and the 'Efimov effect' in the negative discrete spectrum. In a related talk at the Physics department I shall further focus on the interplay between this rigorous mathematical study and the physical phenomena currently observed and investigated in systems with zero-range interactions.