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Bilinear quantum systems on compact graphs: well-posedness and global exact controllability.
TL;DR: In this paper, the authors studied the controllability of a particle trapped on a graph in the presence of an external electromagnetic field and determined assumptions on the graph and the potential field.
Abstract: A major application of the mathematical concept of graph in quantum mechanics is to model networks of electrical wires or electromagnetic wave-guides. In this paper, we address the dynamics of a particle trapped on such a network in presence of an external electromagnetic field. We study the controllability of the motion when the intensity of the field changes over time and plays the role of control. From a mathematical point of view, the dynamics of the particle is modeled by the so-called bilinear Schrodinger equation defined on a graph representing the network. The main purpose of this work is to extend the existing theory for bilinear quantum systems on bounded intervals to the framework of graphs. To this end, we introduce a suitable mathematical setting where to address the controllability of the equation from a theoretical point of view. More precisely, we determine assumptions on the network and on the potential field ensuring its global exact controllability in suitable spaces. Finally, we discuss two applications of our results and their practical implications to two specific problems involving a star-shaped network and a tadpole graph.
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01 Jan 2016
TL;DR: The fourier series in control theory is universally compatible with any devices to read and is available in the digital library an online access to it is set as public so you can download it instantly.
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89 citations
TL;DR: In this article, the existence/nonexistence of ground states for the L2-critical NLS equation on metric graphs with localized nonlinearity is investigated. But the authors focus on the tadpole graph, which allows to point out some specific features of the problem, whose understanding will be useful for future investigations.
Abstract: The paper aims at giving a first insight on the existence/nonexistence of ground states for the L2-critical NLS equation on metric graphs with localized nonlinearity. As a consequence, we focus on the tadpole graph, which, albeit being a toy model, allows to point out some specific features of the problem, whose understanding will be useful for future investigations. More precisely, we prove that there exists an interval of masses for which ground states do exist, and that for large masses the functional is unbounded from below, whereas for small masses ground states cannot exist although the functional is bounded.
22 citations
TL;DR: In this article, the nonlinear Dirac Equation with Kerr-type nonlinearity on non-compact metric graphs with a finite number of edges was studied and local well-posedness for the associated Cauchy problem in the operator domain was proved.
21 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
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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
Book•
18 Nov 2011
TL;DR: In this paper, the authors define the Riesz-Thorin Theorem as a necessary and sufficient condition for interpolation spaces, and apply it to approximate spaces in the context of vector spaces.
Abstract: 1. Some Classical Theorems.- 1.1. The Riesz-Thorin Theorem.- 1.2. Applications of the Riesz-Thorin Theorem.- 1.3. The Marcinkiewicz Theorem.- 1.4. An Application of the Marcinkiewicz Theorem.- 1.5. Two Classical Approximation Results.- 1.6. Exercises.- 1.7. Notes and Comment.- 2. General Properties of Interpolation Spaces.- 2.1. Categories and Functors.- 2.2. Normed Vector Spaces.- 2.3. Couples of Spaces.- 2.4. Definition of Interpolation Spaces.- 2.5. The Aronszajn-Gagliardo Theorem.- 2.6. A Necessary Condition for Interpolation.- 2.7. A Duality Theorem.- 2.8. Exercises.- 2.9. Notes and Comment.- 3. The Real Interpolation Method.- 3.1. The K-Method.- 3.2. The J-Method.- 3.3. The Equivalence Theorem.- 3.4. Simple Properties of ??, q.- 3.5. The Reiteration Theorem.- 3.6. A Formula for the K-Functional.- 3.7. The Duality Theorem.- 3.8. A Compactness Theorem.- 3.9. An Extremal Property of the Real Method.- 3.10. Quasi-Normed Abelian Groups.- 3.11. The Real Interpolation Method for Quasi-Normed Abelian Groups.- 3.12. Some Other Equivalent Real Interpolation Methods.- 3.13. Exercises.- 3.14. Notes and Comment.- 4. The Complex Interpolation Method.- 4.1. Definition of the Complex Method.- 4.2. Simple Properties of ?[?].- 4.3. The Equivalence Theorem.- 4.4. Multilinear Interpolation.- 4.5. The Duality Theorem.- 4.6. The Reiteration Theorem.- 4.7. On the Connection with the Real Method.- 4.8. Exercises.- 4.9. Notes and Comment.- 5. Interpolation of Lp-Spaces.- 5.1. Interpolation of Lp-Spaces: the Complex Method.- 5.2. Interpolation of Lp-Spaces: the Real Method.- 5.3. Interpolation of Lorentz Spaces.- 5.4. Interpolation of Lp-Spaces with Change of Measure: p0 = p1.- 5.5. Interpolation of Lp-Spaces with Change of Measure: p0 ? p1.- 5.6. Interpolation of Lp-Spaces of Vector-Valued Sequences.- 5.7. Exercises.- 5.8. Notes and Comment.- 6. Interpolation of Sobolev and Besov Spaces.- 6.1. Fourier Multipliers.- 6.2. Definition of the Sobolev and Besov Spaces.- 6.3. The Homogeneous Sobolev and Besov Spaces.- 6.4. Interpolation of Sobolev and Besov Spaces.- 6.5. An Embedding Theorem.- 6.6. A Trace Theorem.- 6.7. Interpolation of Semi-Groups of Operators.- 6.8. Exercises.- 6.9. Notes and Comment.- 7. Applications to Approximation Theory.- 7.1. Approximation Spaces.- 7.2. Approximation of Functions.- 7.3. Approximation of Operators.- 7.4. Approximation by Difference Operators.- 7.5. Exercises.- 7.6. Notes and Comment.- References.- List of Symbols.
4,025 citations
TL;DR: In this paper, the classification of π-orbitals in a cata-condensed aromatic system is like that of the orbitals of a free electron traveling in a one-dimensional loop of constant potential around the perimeter.
Abstract: The classification of π‐orbitals in a cata‐condensed aromatic system is like that of the orbitals of a free electron traveling in a one‐dimensional loop of constant potential around the perimeter. To take into account electron interaction, certain quantities corresponding to angular momenta may be added or subtracted. Introduction of the cross‐links in the molecule removes the degeneracy. The first excited configuration in such systems gives two low frequency singlet weak absorption bands and two higher singlet strong dipole absorption bands. Selection and polarization rules are given. The levels are identified from the spectra and some of their properties are determined. An explanation is given of the regularities found by Klevens and Platt. A systematic nomenclature is given. The results agree qualitatively with LCAO theory, can be applied easily to unsymmetrical molecules, and can possibly be extended to other types of ring systems.
1,206 citations