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A uniform controllability result for the Keller-Segel system
In this paper we study the controllability of the Keller-Segel system
approximating its parabolic-elliptic version. We show that this parabolic
system is locally uniform controllable around a constant solution of the
parabolic-elliptic system when the control is acting on the component of the
chemical
-particle sigma model: Momentum-space quantization of a particle on the sphere
We perform the momentum-space quantization of a spin-less particle moving on
the group manifold, that is, the three-dimensional sphere , by
using a non-canonical method entirely based on symmetry grounds. To achieve
this task, non-standard (contact) symmetries are required as already shown in a
previous article where the configuration-space quantization was given. The
Hilbert space in the momentum space representation turns out to be made of a
subset of (oscillatory) solutions of the Helmholtz equation in four dimensions.
The most relevant result is the fact that both the scalar product and the
generalized Fourier transform between configuration and momentum spaces deviate
notably from the naively expected expressions, the former exhibiting now a
non-trivial kernel, under a double integral, traced back to the non-trivial
topology of the phase space, even though the momentum space as such is flat. In
addition, momentum space itself appears directly as the carrier space of an
irreducible representation of the symmetry group, and the Fourier transform as
the unitary equivalence between two unitary irreducible representations.Comment: 29 pages, 3 figure
The quantum Arnold transformation
By a quantum version of the Arnold transformation of classical mechanics, all
quantum dynamical systems whose classical equations of motion are
non-homogeneous linear second-order ordinary differential equations, including
systems with friction linear in velocity, can be related to the quantum
free-particle dynamical system. This transformation provides a basic
(Heisenberg-Weyl) algebra of quantum operators, along with well-defined
Hermitian operators which can be chosen as evolution-like observables and
complete the entire Schr\"odinger algebra. It also proves to be very helpful in
performing certain computations quickly, to obtain, for example, wave functions
and closed analytic expressions for time-evolution operators.Comment: 19 pages, minor changes, references update
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