3 Asymptotic solutions of the many-body problem

S. Takagi, S.B. Khadkikar

V. S. Buslaev, S. B. Levin, Jacob Schach Møller +1 more

The three‐body scattering problem with finite pair potentials for one‐dimensional case is investigated. The asymptotic function χ0, which satisfies the three‐body Schrödinger equation in whole configuration space outside of compact domain Ω, where the supports of all three pair potentials cross each other, has been presented in a mathematically rigorous way. For large distances |x|→∞ the function χ0 determines the asymptotics of the solution up to the circular wave with smooth coefficient in whole configuration space. The method is based on analogies between few‐body scattering problem and diffraction one of the plane wave on the system of half‐transparent infinite screens. Presented here formalism are believed to be useful also for the few‐body scattering problem of higher dimensions.

Dmitri B. Gutman, Yuval Gefen, A. D. Mirlin

Non-equilibrium bosonization technique facilitates the solution of a number of important many-body problems out of equilibrium, including the Fermiedge singularity, the tunneling spectroscopy and full counting statistics of interacting fermions forming a Luttinger liquid.We generalize the method to non-equilibrium hard-core bosons (Tonks-Girardeau gas) and establish interrelations between all these problems.The results can be expressed in terms of Fredholm determinants of the Toeplitz type.We analyze the long time asymptotics of such determinants, using Szegő and Fisher-Hartwig theorems.Our analysis yields dephasing rates as well as power-law scaling behavior, with exponents depending not only on the interaction strength but also on the non-equilibrium state of the system.

G.M. Vagradov

Fabian B. Kugler

We present an iterative algorithm to count Feynman diagrams via many-body relations. The algorithm allows us to count the number of diagrams of the exact solution for the general fermionic many-body problem at each order in the interaction. Further, we apply it to different parquet-type approximations and consider spin-resolved diagrams in the Hubbard model. Low-order results and asymptotics are explicitly discussed for various vertex functions and different two-particle channels. The algorithm can easily be implemented and generalized to many-body relations of different forms and levels of approximation.

João da Providência