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Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator. (English) Zbl 1125.35092

Summary: The Schrödinger equation with the nonlinearity concentrated at a single point \[ i\dot\psi(x,t)= -\psi''(x,t)- \delta(x) F(\psi(0,t)), \quad x\in\mathbb R, \] proves to be an interesting and important model for the analysis of long-time behavior of solutions, including asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space \(H^1\).

MSC:

35Q40 PDEs in connection with quantum mechanics
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:

[1] V. S. Buslaev, Al. Komech, E. A. Kopylova, and D. Stuart, ”On Asymptotic Stability of Solitary Waves in a Nonlinear Schrödinger Equation,” arXiv math-ph/0702013 (MPI Preprint No. 37/2007, available at http://www.mis.mpg.de/preprints/2005/prepr2007_37.html ).
[2] Al. Komech and An. Komech, ”On Global Attraction to Solitary Waves for the Klein-Gordon Equation Coupled to Nonlinear Oscillator,” C. R. Acad. Sci. Paris Ser. I Math. 343, 111–114 (2006). · Zbl 1096.35020
[3] Al. Komech and An. Komech, ”On Global Attraction to Quantum Stationary States I. Nonlinear Oscillator Coupled to Massive Scalar Field,” Max-Planck Institute for Mathematics in the Sciences, Preprint No. 121/2005, http://www.mis.mpg.de/preprints/2005/prepr2005_121.html .
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