# University of Virginia Mathematical Physics Seminar 2010-11

 August 25 Ira Herbst Seminar Organization Meeting September 1 Juliane Rama Title: "Time evolution of quantum resonance states I+II" Abstract: Let $E_0$ be an embedded (possibly degenerate) eigenvalue of the unperturbed Hamiltonian $H_0$: $H_0\psi_0=E_0\psi_0$, $\|\psi_0\|=1$. For "small" perturbations $W$ and $H=H_0+W$ we shall investigate the asymptotics of the (naive) resonance state $e^{-itH}\psi_0$ in the limit $W\to 0$ I) in an abstract Hilbert space setting, where $H$ fulfills a $C^k$-property, $k>3$; II) in an abstract dilation analytic setting. September 8 Juliane Rama Title: "Time evolution of quantum resonance states I+II" Abstract: Let $E_0$ be an embedded (possibly degenerate) eigenvalue of the unperturbed Hamiltonian $H_0$: $H_0\psi_0=E_0\psi_0$, $\|\psi_0\|=1$. For "small" perturbations $W$ and $H=H_0+W$ we shall investigate the asymptotics of the (naive) resonance state $e^{-itH}\psi_0$ in the limit $W\to 0$ I) in an abstract Hilbert space setting, where $H$ fulfills a $C^k$-property, $k>3$; II) in an abstract dilation analytic setting. September 15 I. Herbst Uniqueness in the BFS renormalization analysis I I will give a proof in the framework of BFS renormalization that ground state eigenvalues are non-degenerate. This is joint work with David Hasler. September 22 I. Herbst Uniqueness in the BFS renormalization analysis II I will give a proof in the framework of BFS renormalization that ground state eigenvalues are non-degenerate. This is joint work with David Hasler. September 29 A. Chandra "The energy of immersed fractional charges in a Hierarchical Coulomb Gas in 2 dimensions" This presentation will follow material from D. Brydges' lectures at Park City in 2006. This will serve as an introduction to how the RG formalism developed by Brydges, Slade, and others can be used to gain control of a two-point function. October 6 A. Chandra "The energy of immersed fractional charges in a Hierarchical Coulomb Gas in 2 dimensions", part II This presentation will follow material from D. Brydges' lectures at Park City in 2006. This will serve as an introduction to how the RG formalism developed by Brydges, Slade, and others can be used to gain control of a two-point function. October 13 A. Chandra "The energy of immersed fractional charges in a Hierarchical Coulomb Gas in 2 dimensions", part III This presentation will follow material from D. Brydges' lectures at Park City in 2006. This will serve as an introduction to how the RG formalism developed by Brydges, Slade, and others can be used to gain control of a two-point function. October 15 3:30 PM Friday, 326 Kerchof Rafael Benguria "Universal Lower Bound on the Indirect Coulomb Energy in Atomic Physics" In this talk I will present new lower bounds on the indirect Coulomb Energy. The new universal lower bound is an alternative to the classical Lieb-Oxford bound, but involving the gradient of the single particle density as well. This is joint work with Gonzalo Bley (PUC) October 20 L. Thomas "Stability of multi-polaron systems, Part I" (with Frank, Lieb and Seiringer.) We show that by including a sufficiently strong Coulomb repulsion, N-polaron systems have an energy bounded below by constant x N. For still stronger Coulomb repulsion, no polaron complexes form. October 27 L. Thomas "Stability of multi-polaron systems, Part II" (with Frank, Lieb and Seiringer.) We show that by including a sufficiently strong Coulomb repulsion, N-polaron systems have an energy bounded below by constant x N. For still stronger Coulomb repulsion, no polaron complexes form. November 3 A. Abdesselam A rigorous renormalization group study of a p-adic quantum field theory This talk will be a progress report on an ongoing research project which is joint work with Ajay Chandra and Gianluca Guadagni and which concerns a p-adic analog of the Brydges-Mitter-Scoppola phi-4 model with fractional Laplacian in 3 dimensions. The goal is to develop a mathematical theory of the renormalization group which allows the construction of field theory models in the continuum and the analysis of the short and long distance behavior of the correlation functions. The p-adic model is used as a case study for the development of such a theory, since it is considerably simpler than the model over the reals. November 10 J. Rama Almost exponential decay of quantum resonance states and Paley-Wiener type estimates in Gevrey spaces Abstract February 2 A. Abdesselam A convergent expansion for the spin-Boson model (I) In this series of lectures I will explain in detail a new convergent combinatorial expansion method for the ground state energy of the spin-Boson model. If time permits, I will also treat Euclidean correlation functions. The spin-Boson model is a very simple, yet very rich, model in quantum physics which describes the interaction of a two-state system with a radiation field. The situation I will address is that when the strength of this interaction or coupling is small. February 9 Wojciech De Roeck, Heidelberg Equilibration in quantum systems: an approach with polymer expansions Consider a quantum spin interacting with a free boson field (spin-boson model). Physical wisdom dictates that the spin will be thermalized by the field, i.e., it will assume the temperature of the field, regardless of its original state. In the last decade, this problem has been intensively studied in the mathematical physics community. With the help of tools like spectral perturbation theory, Mourre theory, and spectral renormalization group, one has been able to prove thermalization rigorously. We attack this problem with a different technique,namely resummation of Feynman diagrams. Our approach is inspired by the fact that, for weak coupling, the evolution of the spin is nearly markovian and we set up the expansion around the markovian limit. This allows us to obtain stronger results than those previously known, and we hope that this technique will yield asymptotic completeness for the massless spin-boson model. February 16 A. Abdesselam A convergent expansion for the spin-Boson model (II) In this series of lectures I will explain in detail a new convergent combinatorial expansion method for the ground state energy of the spin-Boson model. If time permits, I will also treat Euclidean correlation functions. The spin-Boson model is a very simple, yet very rich, model in quantum physics which describes the interaction of a two-state system with a radiation field. The situation I will address is that when the strength of this interaction or coupling is small. February 18 (Friday) David Nualart, U. Kansas Clark-Ocone formula and central limit theorems for the Brownian local time increments The purpose of this talk is to discuss some applications of the Clark-Ocone representation formula. This formula provides an explicit expression for the stochastic integral representation of functionals of the Brownian motion in terms of the derivative in the sense of Malliavin calculus. We will compare this formula with the classical Ito formula and we will discuss its application to derive a central limit theorem for the modulus of continuity in the space variable of the Brownian local time increments. February 23 A. Abdesselam A convergent expansion for the spin-Boson model (III) In this series of lectures I will explain in detail a new convergent combinatorial expansion method for the ground state energy of the spin-Boson model. If time permits, I will also treat Euclidean correlation functions. The spin-Boson model is a very simple, yet very rich, model in quantum physics which describes the interaction of a two-state system with a radiation field. The situation I will address is that when the strength of this interaction or coupling is small. March 2 A. Abdesselam A convergent expansion for the spin-Boson model (IV) In this series of lectures I will explain in detail a new convergent combinatorial expansion method for the ground state energy of the spin-Boson model. If time permits, I will also treat Euclidean correlation functions. The spin-Boson model is a very simple, yet very rich, model in quantum physics which describes the interaction of a two-state system with a radiation field. The situation I will address is that when the strength of this interaction or coupling is small. March 23 A. Abdesselam A convergent expansion for the spin-Boson model (V) In this series of lectures I will explain in detail a new convergent combinatorial expansion method for the ground state energy of the spin-Boson model. If time permits, I will also treat Euclidean correlation functions. The spin-Boson model is a very simple, yet very rich, model in quantum physics which describes the interaction of a two-state system with a radiation field. The situation I will address is that when the strength of this interaction or coupling is small. March 30 I. Herbst Persistence of embedded eigenvalues Generically an embedded eigenvalue of a self-adjoint operator disappears under small perturbations. In this talk I will give some results for Schrödinger-like operators which prove the existence of infinite dimensional "manifolds" of small perturbations which keep an embedded eigenvalue embedded nearby. April 6 Gianluca Guadagni Renormalization, as it is done in Utah I will try to explain in detail the hierarchical and euclidean RG as it is done in David Brydges' Park City lectures. We have seen at the RG meeting in Oberwolfach that it may be simpler than Brydges-Mitter-Scoppola(2003), and it may suggest some idea to our RG on p-adics. If time permits, I will present a finite range decomposition for the (massless) Gaussian covariance on a 2-dim lattice, and similar decomposition for a covariance with a generated mass. April 13 Gianluca Guadagni Renormalization, as it is done in Utah (II) I will try to explain in detail the hierarchical and euclidean RG as it is done in David Brydges' Park City lectures. We have seen at the RG meeting in Oberwolfach that it may be simpler than Brydges-Mitter-Scoppola(2003), and it may suggest some idea to our RG on p-adics. If time permits, I will present a finite range decomposition for the (massless) Gaussian covariance on a 2-dim lattice, and similar decomposition for a covariance with a generated mass. *2PM April 15 Rafael Benguria, Catholic University of Chile Exact asymptotic behavior of the Pekar--Tomasevich functional An explicit asymptotic expression for the ground--state energy of the Pekar--Tomasevich functional for the N--polaron is found, when the repulsion parameter $U$ of the electrons satisfies the inequality $0 \leq U \leq 2\alpha$, where $\alpha$ is the coupling constant of the polaron. If $\mathcal{E}_U^N$ denotes this ground--state energy for the case of $N$ electrons and repulsion parameter $U$, we prove that $\mathcal{E}_U^N/N^3 \to -c_p(U - 2\alpha)^2/4$ as $N \to \infty$, where, $c_p = 0.10851\dots$. Moreover, we show that $\mathcal{E}_0^N = -c_p\alpha^2N^3$, for all $N$. April 20 Gianluca Guadagni Renormalization, as it is done in Utah (III) I will try to explain in detail the hierarchical and euclidean RG as it is done in David Brydges' Park City lectures. We have seen at the RG meeting in Oberwolfach that it may be simpler than Brydges-Mitter-Scoppola(2003), and it may suggest some idea to our RG on p-adics. If time permits, I will present a finite range decomposition for the (massless) Gaussian covariance on a 2-dim lattice, and similar decomposition for a covariance with a generated mass. April 27 Gianluca Guadagni Renormalization, as it is done in Utah (IV) I will try to explain in detail the hierarchical and euclidean RG as it is done in David Brydges' Park City lectures. We have seen at the RG meeting in Oberwolfach that it may be simpler than Brydges-Mitter-Scoppola(2003), and it may suggest some idea to our RG on p-adics. If time permits, I will present a finite range decomposition for the (massless) Gaussian covariance on a 2-dim lattice, and similar decomposition for a covariance with a generated mass.

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