arXiv subjects

Eugenio Pozzoli

Publications and source records attributed to Eugenio Pozzoli.

26 records · Page 2Linked to original sources

arXiv2021

Heat equation with inverse-square potential of bridging type across two half-lines

The heat equation with inverse square potential on both half-lines of $\mathbb{R}$ is discussed in the presence of \emph{bridging} boundary conditions at the origin. The problem is the lowest energy (zero-momentum) mode of the transmission of the heat flow across a Grushin-type cylinder, a generalisation of an almost Riemannian structure with compact singularity set. This and related models are reviewed, and the issue is posed of the analysis of the dispersive properties for the heat kernel generated by the underlying positive self-adjoint operator. Numerical integration is shown that provides a first insight and relevant qualitative features of the solution at later times.

math.AP↗

arXiv2021

Lie algebra for rotational subsystems of a driven asymmetric top

We present an analytical approach to construct the Lie algebra of finite-dimensional subsystems of the driven asymmetric top rotor. Each rotational level is degenerate due to the isotropy of space, and the degeneracy increases with rotational excitation. For a given rotational excitation, we determine the nested commutators between drift and drive Hamiltonians using a graph representation. We then generate the Lie algebra for subsystems with arbitrary rotational excitation using an inductive argument.

quant-ph↗

arXiv2021

Classical and quantum controllability of a rotating asymmetric molecule

We study both the classical and quantum rotational dynamics of an asymmetric top molecule, controlled through three orthogonal electric fields that interact with its dipole moment. The main difficulties in studying the controllability of these infinite-dimensional quantum systems are the presence of severe spectral degeneracies in the drift Hamiltonian and the nonsolvability of the stationary free Schrödinger equation, which lead us to apply a perturbative Lie algebraic approach. In this paper we show that, while the classical equations given by the Hamiltonian system on ${\rm SO}(3)\times \mathbb{R}^3$ are controllable for all values of the rotational constants and all dipole configurations, the Schödinger equation for the quantum evolution on $L^2({\rm SO}(3))$ is approximately controllable for almost all values of the rotational constants if and only if the dipole is not parallel to any of the principal axes of inertia of the asymmetric rigid body.

math-ph↗

arXiv2021

Classical and quantum controllability of a rotating symmetric molecule

In this paper we study the controllability problem for a symmetric-top molecule, both for its classical and quantum rotational dynamics. The molecule is controlled through three orthogonal electric fields interacting with its electric dipole. We characterize the controllability in terms of the dipole position: when it lies along the symmetry axis of the molecule neither the classical nor the quantum dynamics are controllable, due to the presence of a conserved quantity, the third component of the total angular momentum; when it lies in the orthogonal plane to the symmetry axis, a quantum symmetry arises, due to the superposition of symmetric states, which has no classical counterpart. If the dipole is neither along the symmetry axis nor orthogonal to it, controllability for the classical dynamics and approximate controllability for the quantum dynamics are proved to hold. The approximate controllability of the symmetric-top Schroedinger equation is established by using a Lie-Galerkin method, based on block-wise approximations of the infinite-dimensional systems.

math-ph↗

arXiv2021

Quantum confinement for the curvature Laplacian $-Δ+cK$ on 2D-almost-Riemannian manifolds

Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set $Z$, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator $Δ$) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement. In this paper we study the self-adjointness of the curvature Laplacian, namely $-Δ+cK$, for $c\in(0,1/2)$ (here $K$ is the Gaussian curvature), which originates in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.

math.FA↗

arXiv2021

Quantum confinement in $α$-Grushin planes

We consider here a family of singular Laplace-Beltrami operators, focussing our attention on the problem of so-called quantum confinement on the half-plane equipped with Riemannian metrics of Grushin type degenerate at the boundary. By introducing a costant-fiber direct integral scheme we are able to rigorously characterize the presence or absence of self-adjointness of these operators. We also compare our technique and results with the already studied problem of quantum confinement on the half-cylinder.

math-ph↗

arXiv2020

Complete Controllability Despite Degeneracy: Quantum Control of Enantiomer-Specific State Transfer in Chiral Molecules

We prove complete controllability for rotational states of an asymmetric top molecule belonging to degenerate values of the orientational quantum number M. Based on this insight, we construct a pulse sequence that energetically separates population initially distributed over degenerate M-states, as a precursor for orientational purification. Introducing the concept of enantio-selective controllability, we determine the conditions for complete enantiomer-specific population transfer in chiral molecules and construct pulse sequences realizing this transfer for population initially distributed over degenerate M-states. This degeneracy presently limits enantiomer-selectivity for any initial state except the rotational ground state. Our work thus shows how to overcome an important obstacle towards separating, with electric fields only, left-handed from right-handed molecules in a racemic mixture.

quant-ph↗

arXiv2019

On Geometric Quantum Confinement in Grushin-type Manifolds

We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl's analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace-Beltrami operator.

math.DG↗