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Marilena Ligabò

Publications and source records attributed to Marilena Ligabò.

27 records · Page 2Linked to original sources
arXiv2017

Self-adjoint extensions and unitary operators on the boundary

We establish a bijection between the self-adjoint extensions of the Laplace operator on a bounded regular domain and the unitary operators on the boundary. Each unitary encodes a specific relation between the boundary value of the function and its normal derivative. This bijection sets up a characterization of all physically admissible dynamics of a nonrelativistic quantum particle confined in a cavity. More- over, this correspondence is discussed also at the level of quadratic forms. Finally, the connection between this parametrization of the extensions and the classical one, in terms of boundary self-adjoint operators on closed subspaces, is shown.

math-ph↗
arXiv2017

Large-time limit of the quantum Zeno effect

If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, $t\propto N^α$. The limit survival probability exhibits a sudden jump from 1 to 0 at $α=1/2$, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for $α\geq 1$ the limit probability becomes sensitive to the spectral properties of the initial state and to arithmetic properties of the measurement periods.

quant-ph↗
arXiv2016

Random walks in a one-dimensional Lévy random environment

We consider a generalization of a one-dimensional stochastic process known in the physical literature as Lévy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. Then we derive the quenched and annealed CLTs for the continuous-time process.

math.PR↗
arXiv2015

Tomography: mathematical aspects and applications

In this article we present a review of the Radon transform and the instability of the tomographic reconstruction process. We show some new mathematical results in tomography obtained by a variational formulation of the reconstruction problem based on the minimization of a Mumford-Shah type functional. Finally, we exhibit a physical interpretation of this new technique and discuss some possible generalizations.

math-ph↗
arXiv2014

Torus as phase space: Weyl quantization, dequantization and Wigner formalism

The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.

math-ph↗
arXiv2010

Classical and quantum aspects of tomography

We present here a set of lecture notes on tomography. The Radon transform and some of its generalizations are considered and their inversion formulae are proved. We will also look from a group-theoretc point of view at the more general problem of expressing a function on a manifold in terms of its integrals over certain submanifolds. Finally, the extension of the tomographic maps to the quantum case is considered, as a Weyl-Wigner quantization of the classical case.

math-ph↗
arXiv2009

Classical limit of the quantum Zeno effect

The evolution of a quantum system subjected to infinitely many measurements in a finite time interval is confined in a proper subspace of the Hilbert space. This phenomenon is called "quantum Zeno effect": a particle under intensive observation does not evolve. This effect is at variance with the classical evolution, which obviously is not affected by any observations. By a semiclassical analysis we will show that the quantum Zeno effect vanishes at all orders, when the Planck constant tends to zero, and thus it is a purely quantum phenomenon without classical analog, at the same level of tunneling.

quant-ph↗
arXiv2009

Quantum Zeno effect and dynamics

If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula we will give a characterization of the quantum Zeno effect for finite-rank projections, in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its (not necessarily lower-bounded) generator as a generalized mean value Hamiltonian.

math-ph↗