Exact quantization of the nonlinear Schrödinger equation

Abstract

By means of an "inverse scattering transform," we can exactly quantize the one-dimensional nonlinear Schrödinger equation iHeng hooktop sign,φ1 = -(Heng hooktop sign2/2m)φxx,- ε2(φ*φ)φ for any value of ε2 = real. When ε2 <0, the eigenvalues of the number operator, field momentum operator, and the Hamiltonian are found to be exactly the same as the linear case. In other words, by quantizing the exact theory, no effects corresponding to "renormalization" are found, and the zero point energy is independent of ε2. When ε2<0, the Hamiltonian is unbounded from below, and, in addition to the above spectra of eigenvalues, bound states can occur. Each bound state can be interpreted to be a bound state of n "excitations," moving in a coherent fashion and with a binding energy proportional to the cube of the number of excitations. This problem is also formally equivalent to the N-body problem with a delta-function interaction solved by Bethe, with which we shall contrast our results, and we shall conclude by making certain remarks concerning ordinary field quantization versus "scattering space" quantization. Copyright © 1975 American Institute of Physics.