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Estimates on Functional Integrals of Non-Relativistic Quantum Field Theory, with Applications to the Nelson and Polaron Models

Gonzalo Bley
(University of Virginia)
Onsdag, 18 maj, 2016, at 14:15-15:00, in Aud. D1 (1531-113)
In this talk I will present a novel method to bound from above  certain functional integrals appearing in non-relativistic quantum  mechanics and field theory. The Feynman-Kac formula for the ground state  energy of a quantum-mechanical Hamiltonian then yields in particular a lower bound for the Hamiltonian in question. 

I will first present the method and give concrete applications to specific functional integrals. I will then apply the bounds found to the class of Schrodinger operators with potential of the form $-\alpha/|x|^{\theta}$, with $\alpha$ a positive coupling constant and $0 \leq \theta \leq 2$.   The lower bounds found turn out to be sharp for both $\theta = 1$ and $\theta = 2$. 

The  method is also applied to the polaron model of H. Frohlich, yielding an improvement of about $25\%$ over a previous result of E. Lieb and K. Yamazaki, for large coupling. I will then mention how the method can be applied to the renormalized Nelson model, giving an explicit, rigorous numerical lower bound. Finally, I will present results concerning the no-binding of bipolarons and two particles interacting via the effective, Coulomb-like attractive massless Nelson interaction and also a mutual Coulomb repulsion. For bipolarons, a global improvement of more than $50\%$ on the threshold repulsion parameter estimate is found, with respect to previous work by R. Frank, E. Lieb, R. Seiringer, and L. Thomas. For the massless Nelson model, an analog, new result is found, showing that no-binding also occurs for strong enough Coulomb repulsion. The estimate in this last case is also explicit, as in the bipolaron model. 

In this talk mathematical points of rigor will be simply touched on, and the focus will be on physical applications. In a second talk the focus will be reversed, this time concentrating on the mathematical aspects behind the method. Part of the content of this talk is from joint work with L. Thomas from the University of Virginia.
Kontaktperson: Jacob Schach Møller