A study of correlation functions for the delta-function Bose gas
The Gel'fand-Levitan equation for the quantum nonlinear Schrödinger field theory is used to investigate the correlation functions of the delta-function Bose gas. Operator expressions are derived for the field and for nonlocal products of fields in terms of the quantized reflection operators which create and annihilate eigenstates of the Hamiltonian. For the two-point function, an explicit series expression is obtained in which the nth term is determined by well-defined n-body combinatorics in an infinite volume. The inductive properties of this series are discussed and used to express the temperature and chemical potential dependence of the correlation functions entirely in terms of previously known thermodynamic functions. The zero separation limit of the series for the two-point function reproduces the thermodynamics derived by Yang and Yang, while the infinite coupling limit gives the Fredholm determinant result of Schultz and Lenard. The latter is related to the Painlevé V equation by the monodromy arguments of Jimbo, Miwa, Mori, and Sato. The \({\mathcal O}(1/c)\) correction to the large coupling limit is calculated from the Gel'fand-Levitan series and expressed in terms of solutions to Painlevé V. The asymptotic behavior of the relevant Painlevé function is discussed and related to the long range behavior of the correlation function.
quantum nonlinear Schrödinger field theory
scientific article; zbMATH DE number 3973923