Coulomb scattering in the massless Nelson model III. Ground state wave functions and non-commutative recurrence relations

Let $H_{P,\sigma}$ be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum $P$ and infrared cut-off $\sigma>0$. We establish detailed regularity properties of the corresponding $n$-particle ground state wave functions $f^n_{P,\sigma}$ as functions of $P$ and $\sigma$. In particular, we show that $$|\partial_{P^j}f^{n}_{P,\sigma}(k_1,\ldots, k_n)|, \ \ |\partial_{P^j} \partial_{P^{j'}} f^{n}_{P,\sigma}(k_1,\ldots, k_n)| \leq \frac{1}{\sqrt{n!}} \frac{(c\lambda_0)^n}{\sigma^{\delta_{\lambda_0}}} \prod_{i=1}^n\frac{ \chi_{[\sigma,\kappa)}(k_i)}{|k_i|^{3/2}},$$ where $c$ is a numerical constant, $\lambda_0\mapsto \delta_{\lambda_0}$ is a positive function of the maximal admissible coupling constant which satisfies $\lim_{\lambda_0\to 0}\delta_{\lambda_0}=0$ and $\chi_{[\sigma,\kappa)}$ is the (approximate) characteristic function of the energy region between the infrared cut-off $\sigma$ and the ultraviolet cut-off $\kappa$. While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy. By solving a non-commutative recurrence relation we derive a novel formula for $f^n_{P,\sigma}$ with improved infrared properties. In this representation $\partial_{P^{j'}}\partial_{P^{j}}f^n_{P,\sigma}$ is amenable to sharp estimates obtained by iterative analytic perturbation theory in part II of this series of papers. The bounds stated above are instrumental for scattering theory of two electrons in the Nelson model, as explained in part I of this series.Comment: 45 pages, minor revision

Coulomb scattering in the massless Nelson model I. Foundations of two-electron scattering

We construct two-electron scattering states and verify their tensor product structure in the infrared-regular massless Nelson model. The proof follows the lines of Haag-Ruelle scattering theory: Scattering state approximants are defined with the help of two time-dependent renormalized creation operators of the electrons acting on the vacuum. They depend on ground state wave functions of the (single-electron) fiber Hamiltonians with infrared cut-off. Convergence of these approximants as $t\to \infty$ is shown with the help of Cook's method combined with a non-stationary phase argument. Removal of the infrared cut-off in the limit $t\to \infty$ requires sharp estimates on the derivatives of these ground state wave functions w.r.t. electron and photon momenta, with mild dependence on the infrared cut-off. These key estimates, which carry information about the localization of electrons in space, are obtained in a companion paper with the help of iterative analytic perturbation theory. Our results hold in the weak coupling regime.Comment: 39 page

Lie-Schwinger block-diagonalization and gapped quantum chains

We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain

Scattering of an Infraparticle: The One Particle Sector in Nelson's Massless Model

Abstract.: In the one-particle sector of Nelson's massless model, we construct scattering states in the time-dependent approach. On the so-defined scattering subspaces, the convergence of the asymptotic Weyl operators related to the boson field as well as the asymptotic limit of the mean velocity of the infraparticle are established. The construction relies on some spectral results concerning the one-particle (improper) states of the system. Moreover, in the region of physical interest, we assume a positive bound from below for the second derivative of the ground state energy as a function of the total momentum, uniform in the limit of no infrared cut-off in the interaction ter

On Finite Rank Deformations of Wigner Matrices

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrice under the assumption that the off-diagonal matrix entries have uniformly bounded fifth moment and the diagonal entries have uniformly bounded third moment. Using our recent results on the fluctuation of resolvent entries [31],[28], and ideas from [9], we extend results by M.Capitaine, C.Donati-Martin, and D.F\'eral [12], [13].Comment: accepted for publication in Annales de l'Institut Henri Poincar\'e (B) Probabilit\'es et Statistique