Abstract. In the one-particle sector of Nelson's massless model, the one-particle states are constructed for an arbitrarily small infrared cutoff in the interaction term of the Hamiltonian of the system. The performed method is a constructive one which exploits only regular perturbation theory, by a suitable iteration scheme. The disappearance of one-particle states is showed in the limit of no infrared regularization. Constructive features, as regularity in some parameters, are also inquired.

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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 term.

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices under the assumption that the absolute values of the off-diagonal matrix entries have uniformly bounded fifth moment and the absolute values of 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 the results by Capitaine, and Féral [12], [13].