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Commutation relations for creation-annihilation operators associated with the quantum nonlinear Schrödinger equation. (English) Zbl 0626.46065
The quantum nonlinear Schrödinger equation (NLS) $$\sqrt{-1}\psi_ t=- \psi_{xx}+2c\psi^+\psi^ 2$$ is treated here. Let $$b^+(k),b(\ell)$$ be the creation and annihilation operators for the Bethe ansatz eigenstates (BAE) of the Hamiltonian $$H=\int^{\infty}_{-\infty}dx[- \psi^+\psi_{xx}+c\psi^{+2}\psi^ 2]$$ of NLS and let a(k), $$a^+(\ell)$$ be their companions diagonalized by the BAE. The normalized quantum reflection coefficients $$R^+(k)=b^+(k)a^{-1}(k)$$ and corresponding R($$\ell)$$ have been introduced and the commutation relation (CR) between them have been also derived by using quantum inverse scattering transform and the relation $$\int^{\infty}_{0}dxe^{ikx}=\delta (k)/2$$ etc.
In this paper the author gives a rigorous mathematical derivation of the CR based on the intertwining operators and an approach to the quantum NLS as follows:
Let $$\psi^+(x,0)=\psi^+_ 0(x)$$ and $$b_ 0^+(k)=\int^{\infty}_{-\infty}dxe^{\sqrt{-1}kx}\psi_ 0^+(x)$$. Then operators $$b_ 0^+(x)$$ create the normalized free eigenstates $$f_ 0(x_ 1,...,x_ N| k_ 1,..,k_ N)\equiv f_ 0(x| k)=(N!)^{-}\sum_{w}\exp (\sqrt{-1}<wk| x>)$$ of $$H_ 0=- \int^{\infty}_{-\infty}dx\psi^+_ 0\psi_{0xx}$$, where w is an element of the permutation group $$S_ N$$. His intertwining operators $$P^*$$, P are given as follows: $$P^{*-1}f_ 0(\cdot | k)=BAEf(\cdot | k)$$ given by $$f(\cdot | k)|_{x_ 1\geq...\geq x_ N}=(N!)^{-}\sum_{w}w[\{\prod_{i<j}\{c+\sqrt{- 1}(k_ i-k_ j)\}/\sqrt{-1}(k_ i-k_ j)\}\exp (\sqrt{-1}<k| x>]$$, and $$b^+(k)=P^{*-1}b^+_ 0(k)P^*$$, where $$w\in S_ N$$ operate to k. By the same way P and b(k) are given from $$\psi$$ (x,0). $$a_ 0(k)$$ is the operator $a_ 0(k)f_ 0(\cdot | k_ 1,...,k_ N)=[\prod^{N}_{i-1}\{c+\sqrt{-1}(k-k_ i)\}/\sqrt{-1}(k- k_ i)]f_ 0(\cdot | k_ 1,...,k_ N),$ and $$a^+_ 0(k)$$ is in a similar way. Then a(k), $$a^+(k)$$, A(k) and $$A^+(k)=Pa^+_ 0(k)[a^+_ 0(k)a_ 0(k)]^{-}P^{-1}$$ are also derived using the operators P and $$P^*$$, and the Theorems on CR between R(k) and $$R^+(\ell)$$, and on scattering states by $$R^+(\ell)$$ are proved. Finally similar arguments by using another intertwining operator $$Q=P^{*-1}(P^*P)^{1/2}$$ are also proved.
Reviewer: H.Yamagata

##### MSC:
 46N99 Miscellaneous applications of functional analysis 81T08 Constructive quantum field theory
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