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Semiclassical measures for the Schrödinger equation on the torus
, 2011
"... Abstract. In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain ..."
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Abstract. In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the RadonNikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the L 2norm of a solution on any open subset of the torus controls the full L 2norm. hal00476829, version 2 13 Sep 2011 1.
HOMOGENIZATION ON LATTICES: SMALL PARAMETER LIMITS, HMEASURES, AND DISCRETE WIGNER MEASURES
"... Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of m ..."
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Cited by 1 (0 self)
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Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of magnetic dipoles on a Bravais lattice, (letting the lattice parameter tend to zero). In order to describe the smallparameter limit, we use discrete Wigner transforms to transform the storedenergy which is given by the double convolution of a sequence of (dipole) functions on a Bravais lattice with a kernel, homogeneous of degree with N with the cancellation property, as the lattice parameter tends to zero. By rescaling and using Fourier methods, discrete Wigner transforms in particular, to transform the problem to one on the torus, we are able to characterize the smallparameter limit of the energy depending on whether the dipoles oscillate on the scale of the lattice, oscillate on a much longer lengthscale, or converge strongly. In the case where> N, the result is simple and can be characterized by anintegral with respect to the Wigner measure limit on the torus. In the case where = N, oscillations essentially on the scale of the lattice must be separated from oscillations essentially onamuch longer lengthscale in order to characterize the energy in terms of the Wigner measure limit on the torus, an Hmeasure limit, and the limiting magnetization. We show that the classical
Strong and weak semiclassical limit for . . .
, 2011
"... We present several results concerning the semiclassical limit of the time dependent Schrödinger equation with potentials whose regularity doesn’t guarantee the uniqueness of the underlying classical flow. Different topologies for the limit are considered and the situation where two bicharacteristic ..."
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We present several results concerning the semiclassical limit of the time dependent Schrödinger equation with potentials whose regularity doesn’t guarantee the uniqueness of the underlying classical flow. Different topologies for the limit are considered and the situation where two bicharacteristics can be obtained out of the same initial point is
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, 2012
"... Abstract. We consider the classical limit of the quantum evolution, with some rough potential, of wave packets concentrated near singular trajectories of the underlying dynamics. We prove that under appropriate conditions, even in the case of BV vector fields, the correct classical limit can be sele ..."
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Abstract. We consider the classical limit of the quantum evolution, with some rough potential, of wave packets concentrated near singular trajectories of the underlying dynamics. We prove that under appropriate conditions, even in the case of BV vector fields, the correct classical limit can be selected.