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Quantum nonlinear Schrödinger equation and its classical counterpart. (English) Zbl 0596.35026
The author considers the quantum nonlinear Schrödinger equation of the form \[ i\psi_ t=-\psi_{xx}+2c\psi^{†}(x)\psi (x)^ 2, \] where \(\psi^{†}(x,t)\) 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 \[ [\psi (x,t), \psi (y,t)]=0,\quad [\psi (x,t), \psi^{†}(y,t)]=\delta (x-y), \] and such that \(\psi\) (x,t) annihilate the vacuum vector \(\Omega\in {\mathcal H}_ 0\). This problem and its classical counterpart are discussed.
Reviewer: Jiang Furu
MSC:
35J10 Schrödinger operator, Schrödinger equation
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35A22 Transform methods (e.g., integral transforms) applied to PDEs
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