an:03960048
Zbl 0596.35026
Gutkin, Eugene
Quantum nonlinear Schr??dinger equation and its classical counterpart
EN
Physica D 18, 378-379 (1986).
00151624
1986
j
35J10 81Q05 35A22
inverse scattering transform; quantum nonlinear Schr??dinger equation
The author considers the quantum nonlinear Schr??dinger equation of the form
\[
i\psi_ t=-\psi_{xx}+2c\psi^{\dag}(x)\psi (x)^ 2,
\]
where \(\psi^{\dag}(x,t)\) and \(\psi\) (y,t) are the time-dependent creation and annihilation operators respectively in the Fock space \({\mathcal H}=\oplus^{\infty}_{N=0}{\mathcal H}_ N\) satisfying the canonical commutation relations
\[
[\psi (x,t), \psi (y,t)]=0,\quad [\psi (x,t), \psi^{\dag}(y,t)]=\delta (x-y),
\]
and such that \(\psi\) (x,t) annihilate the vacuum vector \(\Omega\in {\mathcal H}_ 0\). This problem and its classical counterpart are discussed.
Jiang Furu