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Alessandro Duca

Publications and source records attributed to Alessandro Duca.

21 records · Page 2Linked to original sources
arXiv2020

Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability

The aim of this work is to study the controllability of the bilinear Schrödinger equation on compact graphs. In particular, we consider the equation (BSE) $i\partial_tψ=-Δψ+u(t)Bψ$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$, with $\mathscr{G}$ being a compact graph. The Laplacian $-Δ$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We provide a new technique leading to the global exact controllability of the (BSE) in $D(|Δ|^{s/2})$ with $s\geq 3$. Afterwards, we introduce the "energetic controllability", a weaker notion of controllability useful when the global exact controllability fails. In conclusion, we develop some applications of the main results involving for instance star graphs.

math-ph↗
arXiv2019

Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall

We study the Schrödinger equation $i\partial_tψ=-Δψ+Vψ$ on $L^2((0,1),\mathbb{C})$ where $V$ is a very high and localized potential wall. We aim to perform permutations of the eigenmodes and to control the solution of the equation. We consider the process where the position and the height of the potential wall change as follows. First, the potential increases from zero to a very large value, so a narrow potential wall is formed that almost splits the interval into two parts; then the wall moves to a different position, after which the height of the wall decays to zero again. We show that even though the rate of the variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the eigenstates. Furthermore, we consider potentials with several narrow walls and we show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schrödinger equation by means of a soft, quasi-adiabatic variation of the potential.

math.OC↗