Controllability of bilinear quantum systems in explicit times via explicit control fields
TL;DR: This work considers the bilinear Schrödinger equation on a bounded one-dimensional domain and provides explicit times such that the global exact controllability is verified and shows how to construct controls for the global approximate controllable.
Abstract: We consider the bilinear Schrodinger equation on a bounded one-dimensional domain and we provide explicit times such that the global exact controllability is verified. In addition, we show how to c...
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TL;DR: In such spaces, the global exact controllability of the bilinear Schrodinger equation (BSE) is attained and examples of the main results involving star graphs and tadpole graphs are provided.
13 citations
TL;DR: In this article, the bilinear Schrodinger equation (BSE) is considered in the Hilbert space L2(G,C) with G an infinite graph. And the Laplacian −Δ is equipped with self-adjoint boundary cond...
Abstract: In this work, we consider the bilinear Schrodinger equation (BSE) i∂tψ=−Δψ+u(t)Bψ in the Hilbert space L2(G,C) with G an infinite graph. The Laplacian −Δ is equipped with self-adjoint boundary cond...
9 citations
TL;DR: In this article, the controllability of the bilinear Schrodinger equation on infinite graphs for periodic quantum states was studied and the well-posedness of the system in suitable subspaces of $D(|\Delta|^{3/2})$.
Abstract: In this work, we study the controllability of the bilinear Schr\"odinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2_p$ composed by functions defined on an infinite graph $\mathscr{G}$ verifying periodic boundary conditions on the infinite edges. The Laplacian $-\Delta$ is equipped with specific boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We present the well-posedness of the system in suitable subspaces of $D(|\Delta|^{3/2})$ . In such spaces, we study the global exact controllability and we provide examples involving for instance tadpole graphs and star graphs with infinite spokes.
5 citations
TL;DR: In this paper, the Schrodinger equation is studied in terms of a very high and localized potential wall, and it is shown that even though the rate of variation of the potentials parameters can be arbitrarily slow, this process alternates adiabatic and nonadiabatic dynamics, leading to a non-trivial permutation of the instantaneous energy eigenstates.
Abstract: We study the Schrodinger equation i∂tψ = −Δψ + Vψ on L2((0,1),C) where V is a very high and localized potential wall. We consider the process where the position and the height of the wall change as follows: First, the potential increases from zero to a very large value, and so a narrow potential wall is formed and almost splits the interval into two parts; then, the wall moves to a different position, after which the height of the wall decreases to zero again. We show that even though the rate of 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 instantaneous energy eigenstates. Furthermore, we consider potentials with several narrow walls and 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 Schrodinger equation by means of a soft, quasi-adiabatic variation of the potential.
5 citations
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TL;DR: This work studies the well-posedness of the bilinear Schrödinger equation (BSE) in suitable subspaces of preserved by the dynamics despite the dispersive behaviour of the equation in the Hilbert space with an infinite graph.
Abstract: In this work, we consider the bilinear Schrodinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathcal{G},\mathbb{C})$ with $\mathcal{G}$ an infinite graph. The Laplacian $-\Delta$ 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 study the well-posedness in suitable subspaces of $D(|\Delta|^{3/2})$ preserved by the dynamics despite the dispersive behaviour of the equation. In such spaces, we study the global exact controllability and the {\virgolette{energetic controllability}}. We provide examples involving for instance infinite tadpole graphs.
4 citations
Cites background or methods from "Controllability of bilinear quantum..."
...In addition, he exhibits the global exact controllability of the bilinear Schrödinger equation between eigenstates via explicit controls and explicit times in Duca (2019)....
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...This hypothesis is crucial for the classical arguments adopted in the previousworks as Beauchard and Laurent (2010), Duca (2018c, 2019) andMorancey (2014)....
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References
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Book•
01 Jan 1968
TL;DR: This book shows engineers how to use optimization theory to solve complex problems with a minimum of mathematics and unifies the large field of optimization with a few geometric principles.
Abstract: From the Publisher:
Engineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book.
5,667 citations
"Controllability of bilinear quantum..." refers methods in this paper
...using Ingham Theorem ([13, Theorem4.3]) for T>2π G and G := inf k,j∈N k6= j |λk−λj| = 3π2 >0. Then γ1 is surjective and the proof is achieved thanks to the Generalized Inverse Function Theorem ([14], p. 240), which ensures that the map α1 is locally surjective. Remark. We point out that one can achieve the result of Theorem 4 for any positive time T >0 by using Haraux Theorem ([13, Theorem4.5...
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...unction is equivalent to prove the local right invertibility of the map α1 for a T >0 (in other words the local surjectivity). To this end, we want to use the Generalized Inverse Function Theorem ([14], p. 240) and we study the surjectivity of the Fr¨ı¿chet derivative of α1, γ1(v) := (duα1(0)) · v, the sequence with elements γk,1(v) : = ˝ φk(T),−i ZT 0 e−iA(T−s)v(s)Be−iAsφ 1ds ˛ = −i ZT 0 v(s)ei(λ...
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Book•
01 Jan 1995TL;DR: In this paper, the authors provide global and asymptotic estimates for the eigenvalues of - + q when q is real and for -+ q when 1 is complete.
Abstract: Linear operations in Banach spaces Entropy numbers, s-numbers, and eigenvalues Unbounded linear operators Sesquilinear forms in Hilbert spaces Sobolev spaces Generalized Dirichlet and Neumann boundary-value problems Second-order differential operators on arbitrary open sets Capacity and compactness criteria Essential spectra Essential spectra of general second-order differential operators Global and asymptotic estimates for the eigen-values of - + q when q is real. Estimates for the singular values of - + q when 1 is complete Bibliography Notation index Subject index
1,792 citations
TL;DR: In this paper, the controllability of systems of the form {dw} / {dt} = \mathcal {A}w + p(t) w + √ √ {B}w$ where W is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators on a Banach space X and W is a control.
Abstract: This paper studies controllability of systems of the form ${{dw} / {dt}} = \mathcal {A}w + p(t)\mathcal {B}w$ where $\mathcal{A}$ is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators $e^{\mathcal{A}t} $ on a Banach space X, $\mathcal{B}:X \to X$ is a $C^1$ map, and $p \in L^1 ([0,T];\mathbb{R})$ is a control. The paper (i) gives conditions for elements of X to be accessible from a given initial state $w_0$ and (ii) shows that controllability to a full neighborhood in X of $w_0$ is impossible for $\dim X = \infty $. Examples of hyperbolic partial differential equations are provided.
335 citations
285 citations