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2020•Journal of Mathematical PhysicsOpen access

Controllability of periodic bilinear quantum systems on infinite graphs

Kaïs Ammari, Alessandro Duca

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Abstract

In this work, we study the controllability of the bilinear Schrödinger equation (BSE) on infinite graphs for periodic quantum states. We consider the BSE i∂tψ = −Δψ + u(t)Bψ in the Hilbert space Lp2 composed of functions defined on an infinite graph G verifying periodic boundary conditions on the infinite edges. The Laplacian −Δ is equipped with specific boundary conditions, B is a bounded symmetric operator, and u∈L2((0,T),R) with T > 0. We present the well-posedness of the BSE in suitable subspaces of D(|Δ|3/2). In such spaces, we study the global exact controllability and we provide examples involving tadpole graphs and star graphs with infinite spokes.

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In this work, we study the controllability of the bilinear Schrödinger equation (BSE) on infinite graphs for periodic quantum states. We consider the BSE i∂tψ = −Δψ + u(t)Bψ in the Hilbert space Lp2 composed of functions defined on an infinite graph G verifying periodic boundary conditions on the infinite edges. The Laplacian −Δ is equipped with specific boundary conditions, B is a bounded symmetric operator, and u∈L2((0,T),R) with T > 0. We present the well-posedness of the BSE in suitable subspaces of D(|Δ|3/2). In such spaces, we study the global exact controllability and we provide examples involving tadpole graphs and star graphs with infinite spokes.

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Available abstract

In this work, we study the controllability of the bilinear Schrödinger equation (BSE) on infinite graphs for periodic quantum states. We consider the BSE i∂tψ = −Δψ + u(t)Bψ in the Hilbert space Lp2 composed of functions defined on an infinite graph G verifying periodic boundary conditions on the infinite edges. The Laplacian −Δ is equipped with specific boundary conditions, B is a bounded symmetric operator, and u∈L2((0,T),R) with T > 0. We present the well-posedness of the BSE in suitable subspaces of D(|Δ|3/2). In such spaces, we study the global exact controllability and we provide examples involving tadpole graphs and star graphs with infinite spokes.

Key concepts: Controllability, Hilbert space, Mathematics, Bounded function, Linear subspace, Bilinear form, Quantum graph, Laplace operator

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