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Refined Algebraic Quantization: Systems with a single constraint

Donald Marolf (1997)

Banach Center Publications

This paper explores in some detail a recent proposal (the Rieffel induction/refined algebraic quantization scheme) for the quantization of constrained gauge systems. Below, the focus is on systems with a single constraint and, in this context, on the uniqueness of the construction. While in general the results depend heavily on the choices made for certain auxiliary structures, an additional physical argument leads to a unique result for typical cases. We also discuss the 'superselection laws' that...

Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics

Othmar Koch, Christian Lubich (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We discuss the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of the time-dependent Schrödinger equation in quantum molecular dynamics. This method approximates the high-dimensional nuclear wave function by a linear combination of products of functions depending only on a single degree of freedom. The equations of motion, obtained via the Dirac-Frenkel time-dependent variational principle, consist of a coupled system of low-dimensional nonlinear partial differential...

Relativistic stability of matter (I).

Charles L. Fefferman, Rafael de la Llave (1986)

Revista Matemática Iberoamericana

In this article, we study the quantum mechanics of N electrons and M nuclei interacting by Coulomb forces. Motivated by an important idea of Chandrasekhar and following Herbst [H], we modify the usual kinetic energy -∆ to take into account an effect from special relativity. As a result, the system can implode for unfavorable values of the nuclear charge Z and the fine structure constant α. This is analogous to the gravitational collapse of a heavy star. Our goal here is to find those values of α...

Resonance of minimizers for n-level quantum systems with an arbitrary cost

Ugo Boscain, Grégoire Charlot (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem describing a laser-induced population transfer on a n -level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n = 2 and n = 3 ): instead of looking...

Resonance of minimizers for n-level quantum systems with an arbitrary cost

Ugo Boscain, Grégoire Charlot (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem describing a laser-induced population transfer on a n-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n=2 and n=3): instead...

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