Exact quantization of the nonlinear Schrödinger equation

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Abstract
By means of an ’’inverse scattering transform,’’ we can exactly quantize the one‐dimensional nonlinear Schrödinger equation ih/Ψt=−(h/2/2m) Ψxx−ε2(Ψ*Ψ) Ψ for any value of ε2=real. When ε2exactly 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 ε2n ’’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.