an:04132731
Zbl 0692.35089
Gutkin, Eugene
Quantum nonlinear Schr??dinger equation. II. Explicit solution
EN
J. Funct. Anal. 77, No. 2, 326-345 (1988).
00167934
1988
j
35Q99 35K55 35C05 81T99
nonlinear Schr??dinger equation; Fock spaces; collision expansions; creation and annihilation operators
[For Part I see Ann. Inst. Henri Poincar??, Anal. Nonlin??aire 3, 285- 314 (1986; Zbl 0614.35086).]
This is the second in a series of papers on the nonlinear Schr??dinger equation (NLS)
\[
\sqrt{-1}\psi_ t=-\psi_{xx}+2c\psi^+\phi^ 2,
\]
an evolution equation on the time-dependent ``annihilation operators'' \(\psi\) (x,t) on the Fock spaces \(\hat {\mathcal R}=\oplus^{\infty}_{N=0}{\mathcal H}_ N.\) Explicit formulas for the solution of the NLS in terms of ``collision expansions'' are obtained. The approach is based on the method of intertwining operators for the Hamiltonian \(\hat H\) on \(\hat {\mathcal H}\) which is formally given in terms of the creation and annihilation operators \(\psi^+\), \(\psi\) by
\[
\hat H=\int^{\infty}_{-\infty}dx[-\psi^+\psi_{xx}+c\psi^+\psi^ 2].
\]
J.Weidmann
Zbl 0614.35086