Ruijsenaars–Schneider model

Template:Short description Ruijsenaars–Schneider models (short RS models) are relativistic generalizations of Calogero–Moser–Sutherland models (short CMS models), which are closely connected with relativistic field theories like the Sine-Gordon model. RS models are named after Simon N. M.  Ruijsenaars and Harald Schneider. Both of them introduced the classical models in 1986 and Ruijsenaars introduced the quantum models in 1987.

Description

For $N$ particles on the real line $ℝ$, the Hamiltonian of the RS model is given by:

$H=m{c}^2\sum _{i=1}^N\cosh \left(\frac{p_i}{mc}\right)\prod _{\stackrel{1\le j\le N}{j\ne i}}f(x_i-x_j).$

Different potentials lead to different RS models, which the four types most often considered being:

• Type I/rational:

  $f(x{)}^2=1+{\left(\frac{g}{mcx}\right)}^2.$

• Type II/hyperbolic:

  $f(x{)}^2=1+\frac{\sin {\left(\frac{ag}{mc}\right)}^2}{\sinh (ax{)}^2}.$

• Type III/trigonometric:

  $f(x{)}^2=1+\frac{\sinh {\left(\frac{ag}{mc}\right)}^2}{\sin (ax{)}^2}.$

• Type IV/elliptic:

  $f(x{)}^2=a+b\mathrm{\wp }(x;{\omega }_1,{\omega }_2).$

Literature

• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite arXiv
• Template:Cite book