Marilena Ligabò

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 bounded from below) generator as a generalized mean value Hamiltonian.

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.

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.

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.

On the inversion of the Radon transform: standard versus M 2 approach

We compare the Radon transform in its standard and symplectic formulations and argue that the analytical inversion of the latter is easier to perform.

Quantum fluctuation relations

We present here a set of lecture notes on exact fluctuation relations. We prove the Jarzynski equality and the Crooks fluctuation theorem, two paradigmatic examples of classical fluctuation relations. Finally, we consider their quantum versions, and analyze analogies and differences with the classical case.

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.

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. Moreover, 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.

Quantum cavities with alternating boundary conditions

We consider the quantum dynamics of a free nonrelativistic particle moving in a cavity and we analyze the effect of a rapid switching between two different boundary conditions. We show that this procedure induces, in the limit of infinitely frequent switchings, a new effective dynamics in the cavity related to a novel boundary condition. We obtain a dynamical composition law for boundary conditions which gives the emerging boundary condition in terms of the two initial ones.

Nonlinear Waves in Adhesive Strings

We study a 1D semilinear wave equation modeling the dynamic of an elastic string interacting with a rigid substrate through an adhesive layer. The constitutive law of the adhesive material is assumed elastic up to a finite critical state, beyond such a value the stress discontinuously drops to zero. Therefore the semilinear equation is characterized by a source term presenting jump discontinuity. Well-posedness of the initial boundary value problem of Neumann type, as well as qualitative properties of the solutions are studied and the evolution of different initial conditions are numerically investigated.