"Mathematical Challenges of Modern Zero-Range Physics:
a Survey of Old and New Results"
By
Alessandro
Michelangeli
(Ludwig Maximilian
University, Munich)
Abstract: 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.
Date: Tuesday, February 18, 2014
Time: 15:40
Place: Mathematics Seminar Room, SA-141
Tea and cookies will be served before the
seminar.