565 research outputs found
Huygens' principle and Dirac-Weyl equation
We investigate the validity of Huygens' principle for forward propagation in
the massless Dirac-Weyl equation. The principle holds for odd space dimension
n, while it is invalid for even n. We explicitly solve the cases n=1,2 and 3
and discuss generic . We compare with the massless Klein-Gordon equation and
comment on possible generalizations and applications.Comment: 7 pages, 1 figur
Representation of non-semibounded quadratic forms and orthogonal additivity
In this article we give a representation theorem for non-semibounded
Hermitean quadratic forms in terms of a (non-semibounded) self-adjoint
operator. The main assumptions are closability of the Hermitean quadratic form,
the direct integral structure of the underlying Hilbert space and orthogonal
additivity. We apply this result to several examples, including the position
operator in quantum mechanics and quadratic forms invariant under a unitary
representation of a separable locally compact group. The case of invariance
under a compact group is also discussed in detail
A Hodge - De Rham Dirac operator on the quantum
We describe how it is possible to describe irreducible actions of the Hodge -
de Rham Dirac operator upon the exterior algebra over the quantum spheres equipped with a three dimensional left covariant calculus.Comment: 18 page
On the theory of self-dajoint extensions of the Laplace-Beltrami operator quadratic forms and symmetry
The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to describe them. Moreover, we want to emphasise the role that quadratic forms can play in the description of quantum systems. ------------------------------------------------------------------------------------------------------------El objetivo principal de esta memoria es analizar en detalle tanto la construcciĂłn de extensiones autoadjuntas del operador de Laplace-Beltrami definido sobre una variedad Riemanniana compacta con frontera, como el papel que juegan las
formas cuadráticas a la hora de describirlas. Más aún, queremos enfatizar el papel que juegan las formas cuadráticas a la hora de describir sistemas cuánticos
On global approximate controllability of a quantum particle in a box by moving walls
We study a system composed of a free quantum particle trapped in a box whose
walls can change their position. We prove the global approximate
controllability of the system. That is, any initial state can be driven
arbitrarily close to any target state in the Hilbert space of the free particle
with a predetermined final position of the box. To this purpose we consider
weak solutions of the Schr\"odinger equation and use a stability theorem for
the time-dependent Schr\"odinger equation.Comment: 25 pages, 1 figur
On a sharper bound on the stability of non-autonomous Schr\"odinger equations and applications to quantum control
We study the stability of the Schr\"odinger equation generated by
time-dependent Hamiltonians with constant form domain. That is, we bound the
difference between solutions of the Schr\"odinger equation by the difference of
their Hamiltonians. The stability theorem obtained in this article provides a
sharper bound than those previously obtained in the literature. This makes it a
potentially useful tool for time-dependent problems in Quantum Physics, in
particular for Quantum Control. We apply this result to prove two theorems
about global approximate controllability of infinite-dimensional quantum
systems. These results improve and generalise existing results on
infinite-dimensional quantum control.Comment: arXiv admin note: text overlap with arXiv:2108.0049
On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics
This is a series of five lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics
Aspects of geodesical motion with Fisher-Rao metric: classical and quantum
The purpose of this article is to exploit the geometric structure of Quantum
Mechanics and of statistical manifolds to study the qualitative effect that the
quantum properties have in the statistical description of a system. We show
that the end points of geodesics in the classical setting coincide with the
probability distributions that minimise Shannon's Entropy, i.e. with
distributions of zero dispersion. In the quantum setting this happens only for
particular initial conditions, which in turn correspond to classical
submanifolds. This result can be interpreted as a geometric manifestation of
the uncertainty principle.Comment: 15 pages, 5 figure
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