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Lyapunov control of a quantum particle in a decaying potential. (English) Zbl 1176.35169
Summary: A Lyapunov-based approach for the trajectory generation of an $$N$$-dimensional Schrödinger equation in whole $$\mathbb R^N$$ is proposed. For the case of a quantum particle in an $$N$$-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs 35P05 General topics in linear spectral theory for PDEs 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 93C20 Control/observation systems governed by partial differential equations
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