Workshop on Applied Mathematics: Quantum and Classical  Models

In this talk I will present the proof of the Fredholm alternative for the linearized Boltzmann operator. The model is describded with a distribution function with an additional continous energy variable. The collision operator is based on the Borgnakke-Larsen procedure. We present the proof in the case of a single gas and of mixtures. The cornerstone of the proof is the introduction of a kernel on the perturbation part in order to prove that the operator is Hilbert Schmidt.

The Wigner-function (WF) approach to Quantum Kinetic Theory, nowadays widely studied and used, presents open problems and difficulties of practical and theoretical nature which make the applications problematic. They arise from the fundamental assumptions under which the Wigner equation holds and which entail that the WF formalism in its standard formulation is applicable only to fully Hamiltonian, spatially infinite coherent systems, which are also confined. Among these fundamental assumptions we recall the absence of a mechanism which destroys the phase correlations of the individual states, i.e., the correlation length must be infinite. A numerical investigation aimed at studying the effect of a finite coherence length on a simple reflection-transmission problem was presented in Barletti, Bordone, Demeio, Giovannini, Phys. Rev. E. 104 , 044112 (2021), based on the decoherence model proposed by Barletti, Frosali, Giovannini, Journal of Computat. and Theor. Transport 47, 209 (2018). The results confirm the predicted broadening and flattening of the Wigner function with time; they also indicate that a reduced coherence length favors transmission of low-energy electrons through the potential barrier, inhibiting reflection.

We consider a  Nonlinear Schroedinger equation with a point interaction and power type nonlinearity and

we prove that for low power there is no scattering in dimensione two and three.

The Haldane model is a standard tight binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar, magnetic field. We consider its  interacting version and study the critical case at the transition between the trivial and the "topological" insulating phases. In previous works, we proved the quantization of the critical longitudinal conductivity for weak enough interaction strength. We now report a recent extension of the result to the critical transverse conductivity, which turns out to be quantized at half-integer values, irrespective of the interaction strength. Proofs are based on a combination of   constructive Renormalization Group methods and exact lattice Ward Identities. Joint works with S. Fabbri, V. Mastropietro, M. Porta, R. Reuvers. 

Anomalies are the breaking of classical symmetries by quantum effects, and their non-renormalization properties play a crucial role in a wide range of phenomena. I present some rigorous theorems on the (non-perturbative) anomaly non-renormalization in QFT models, based on Renormalization Group, Cluster or Tree Expansion, and Determinant Bounds, proving the exact cancellation of the terms arising from the lattice cut-offs. I discuss, in particular, a lattice fermion-vector model in D=3+1, the Sommerfield model in D=1+1, and the anomaly cancellation in a Chiral lattice D=3+1 model. Analogous results on universality in transport coefficients in graphene, Hall insulators, and Weyl semimetals in the presence of many-body interactions will also be briefly presented.