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Scattering of solitons for the Schrödinger equation coupled to a particle. (English) Zbl 1118.35040

The authors consider the Schrödinger equation coupled to a nonrelativistic particle with position \(q\)
\[ i\dot{\psi}(x,t)=-\Delta\psi(x,t)+m^2\psi(x,t)+\rho(x-q(t)), \]
\[ \ddot{q}(t)=\frac{1}{2}\int\left[\overline{\psi}\nabla\rho(x-q(t))+\psi(x,t) \nabla\overline{\rho}(x-q(t))\right]\,dx, \]
where \(x\in{\mathbb R},m>0\) and \(\rho\) is the charge distribution. Their main result is the soliton asymptotics of type
\[ \psi(x,t)\sim\psi_{v_{\pm}}(x-v_{\pm}t-a_{\pm})+W_0 (t)\Psi_{\pm}, \;\;q(t)\sim v_{\pm}t+a_{\pm}\;\;\text{as}\;\;t\rightarrow\pm\infty, \]
under the assumption that the initial state is close to the solitary manifold
\[ {\mathcal{S}}:=\{(\operatorname {Re}\psi(x-a),\operatorname {Im}\psi(x-a),a,v):a,v\in{\mathbb{R}}^3,| v| <2m\}. \]
The proof uses spectral theory and the symplectic projection onto \({\mathcal{S}}\) in the Hilbert phase space. For similar results for other coupled systems of wave fields and particles, see I. Valery, A. Komech and B. Vainberg [Commun. Math. Phys. 268, 321–367 (2006; Zbl 1127.35054)].

MSC:

35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U15 Exactly and quasi-solvable systems arising in quantum theory
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems

Citations:

Zbl 1127.35054
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References:

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