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Introduction
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Introduction
Contents
Tools
Subsections
Introduction
Correlation functions and related quantities
Correlation functions: second quantization form
Correlation functions: first quantization form
Homogeneous system
Momentum distribution and static structure factor
Trapped system
The scattering problem
Introduction
Three-dimensional scattering problem
General approach
Scattering on a hard sphere potential
Scattering on a soft sphere potential
One-dimensional scattering problem
General approach
Scattering on a pseudopotential
Scattering on a 1D square well potential
Scattering on a hard-rod potential
Pseudopotential
The pseudopotential method
Resonance scattering
Scattering on a square-well potential
Scattering on a modified Pöschl-Teller potential
Energy of the TG and HR gas
Energy of the Tonks-Girardeau gas
Hard-rod gas
Gross Pitaevskii Equation
Variational derivation of the GPE
Coupling constant in quasi one- and two- dimensional systems
Local Density Approximation
General method
Exact solution for 1D ``perturbative'' equation of state
Exact solution for 3D ``perturbative'' equation of state
Static structure factor of a trapped Tonks-Girardeau gas
Correlation functions in a Luttinger liquid
Stationary density-density correlation function
Time-dependent density-density correlation function
Calculation with non-logarithmic accuracy
Dynamic form factor
Popov's coefficient
G.E. Astrakharchik 15-th of December 2004
