Oberseminar: Mathematical Physics (SoSe 2019) [16222]

Organized by: Prof. Dr. Douglas Lundholm

Schedule:
ordinarily Fridays 14-16 (in B 045).

Program:
 Date Speaker Topic 14.6 Luca Oddis (Rome) Title: Point Interaction Hamiltonians for two Anyons Abstract: We review the main issues concerning the well-posedness (as suitable self-adjoint operators) of the Hamiltonians of two non interacting spinless anyons. We show that such operators can be identiﬁed with a one-parameter family of self-adjoint extensions of a suitable symmetric operator with Aharonov-Bohm-like magnetic potential. We also derive the explicit expressions of the corresponding quadratic forms and prove their closedness and boundedness from below. We present a decomposition of the domains of these self-adjoint operators different from the one coming from the Von-Neumann theorem in terms of the behaviour of the wave functions near the coincidence set. The proof can be extended to the case with interaction under suitable assumptions on the potential. Then we explore some perspectives on how to generalise the construction of the quadratic forms for the N-anyon case. Joint work with M. Correggi (”Sapienza” University of Rome). 28.6 Luca Fresta (Zurich) Title: A hierarchical supersymmetric model for weakly disordered three-dimensional semimetals Abstract: An important open problem in quantum mechanics is to prove that three-dimensional lattice Schr\"odinger operators with extensive disorder exhibit a localization/delocalization transition as a function of the disorder strengh. We studied a hierarchical supersymmetric lattice model for Weyl semimetals with weak Anderson-type disorder. In the talk I will present a theorem about the algebraic decay of the disorder-averaged two-point correlation function, compatible with delocalization. Our method is based on a rigorous implementation of the renormalization group, reminiscent of the Gaw\k edzki-Kupiainen block spin transformations. The main technical novelty is the multi-scale analysis of massless Gaussian convolutions with purely imaginary covariances via stationary phase expansions. Joint work with G. Antinucci and M. Porta. 5.7 12-14 in A348 Belen Paredes (Munich) Cancelled 12.7 Adrien Hardy (Lille) Title: A CLT for the circular beta-ensemble at high temperature Abstract: The circular beta-ensemble is a well-studied particle system arising from random matrix theory, which represents when beta=2 the eigenvalues a Haar random unitary matrix. It can also be seen as a 2D Coulomb interacting system of N particles constrained to stay on the unit circle, where beta represents now the inverse temperature of the system. One can also add an external field, to model interaction with the exterior. After describing the large N limit of this particle system at the macroscopic scale, a natural question is to investigate how the system fluctuates around its macroscopic limit. We have an answer to this question in the high temperature regime where beta scales like 1/N, which interpolates what is known at fixed temperature (the usual random matrix theory regime) and at infinite temperature (independent random variables) in an explicit way. This is a joint work with Gaultier Lambert. 11.7& 15-17.7 Søren Fournais (Aarhus) Mini-course on this topic: The Dilute Bose -------- Soeren Fournais (Aarhus University) is visiting until July 27th (he is sitting in B 406). While here, he will give a number of talks on "The energy of dilute Bose gases" based on https://arxiv.org/abs/1904.06164 (see also https://arxiv.org/abs/1901.00539) More precisely: (a) In Oberseminar 'PDG und Spektraltheorie' on *** Thursday 11. July *** (14.15 Uhr, B 134) Title: "The Dilute Bose Gas" (b) In four (4) talks ('mini-course'), he will then give _more_ details on the proofs, for those (including students, PhD students, post-docs) interested in knowing more. (1) Monday 15. July at 10-12 Uhr in ** B 046 ** (2) Monday 15. July (!) at 14-16 Uhr in ** B 046 ** (3) Tuesday 16. July at 10-12 Uhr in ** B 252 ** (4) Wednesday 17. July at 16-18 Uhr in ** B 132 ** (Oberseminar `Calculus of Variations and Applications', Lundholm/Nam) Abstract: The energy of the dilute Bose gas has attracted much attention in recent years. The expected formula is given by the expression $$e(\rho) = 4\pi\rho^2 a (1 +\frac{128}{15\sqrt{\pi}} \sqrt{\rho a^3} + o((\rho a^3)^{1/2}),$$ and was already known to physicists more than 50 years ago based on non-rigorous approximations. To establish this formula rigorously has since been a fundamental problem in mathematical physics. The proof of the leading order term was only rigorously established by Lieb and Yngvason in 1998. Recently, the correction term (the Lee-Huang-Yang term) has been rigorously proved by Søren Fournais and Jan Philip Solovej (Copenhagen). The talk will give an overview of the main ideas of the proof of this result.The focus will be on the lower bound - the upper bound being a consequence of previous work by Yau and Yin.