The Quantum Nonlinear Schrödinger Model; Gelfand-Levitan Equation and Classical Soliton

Abstract

Through the quantum inverse scattering method the nonlinear Schrödinger model is studied for the attractive case. Making use of the bound state operators and the quantum Gelfand-Levitan equation, the matrix elements of the field operator φ(x) and the density operator φ⁺(x)φ(x) are explicitly calculated. Each matrix element is found to be expressed in a product form. Based on the results, the connection between classical soliton and the quantum field theory is investigated. It is shown that the wave form of the classical soliton is related to the matrix element of the field operator in the limit n→∞, where n is the particle number making the bound state. The powerfulness of the quantum Gelfand-Levitan equation is stressed.