On the exact controllability of hyperbolic magnetic Schrödinger equations

Lu, Xiaojun; Tu, Ziheng; Lv, Xuebin

doi:10.1016/j.na.2014.06.006

Cited by 8 publications

(10 citation statements)

References 16 publications

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“…However, we only present experimental results of collectives with parameters (3,50,20) and (3,100,20) because of the page limitation. Table 3 presents qualities of collective knowledge of collectives from 3 to 50 members.…”

Section: Experimental Results and Evaluationmentioning

confidence: 99%

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Intelligent Information and Database Systems

2015

Lecture Notes in Computer Science

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Abstract. Discovering the research communities to bring techniques to the world is an interesting topic. In this paper we use the DBLP data to investigate the co-author relationship in a real bibliographic network and predict the interactions between co-authors. We analysis the research trend of authors and conferences based on extracted keywords from paper titles. We can understand research fields and change of research trend to find appropriate co-authors and conferences to submit our work. We also find potential co-authors for an existing author in DBLP data by using a variety of similarity measures and a random walk model. It can be useful for building a recommendation system.

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Section: Experimental Results and Evaluationmentioning

confidence: 99%

“…Table 4. Correlation coefficient between the number of members and the quality Parameters rho p-value (3,50,20) 0.924 < 2.2e-16 (3,100,20) 0.986 < 2.2e-16…”

Section: Fig 2 Collective Knowledge Quality Of Collectivesmentioning

confidence: 99%

Intelligent Information and Database Systems

2015

Lecture Notes in Computer Science

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“…It is known from the spectral theory in [2,24,25,32,34] that, the Dirichlet magnetic Schrödinger operator ℋ 𝐀 is a selfadjoint and positive definite operator with compact resolvent whose spectrum set Spec(ℋ 𝐀 ) = {𝜆 𝑗 } 𝑗∈ℕ + is composed of discrete points with finite multiplicity, which can be arranged in order as follows…”

Section: (𝑈)mentioning

confidence: 99%

“…It is proved that  𝑠 (𝑈) is an equivalent function space of the classical Sobolev space 𝐻 𝑠 (𝑈) if the vector potential 𝐀 ∈ (𝐶 1 (Ω)) 𝑁 [24,25]. Fractional order pseudodifferential operators are very important mathematical models describing plenty of anomalous dynamic behaviors in the sciences of physical chemistry [21,32].…”

Section: (𝑈)mentioning

confidence: 99%

“…It is worth noticing that this Hamiltonian operator phenomenologically describes a quantity of behaviors discovered in superconductors and quantum electrodynamics(QED). In these respects, Ginzburg-Landau equations, Schrödinger equations, Dirac equations and the matrix Pauli operator are famous examples concerned with the above Hamiltonian operator [2,18,25,30,31,34].…”

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On the magnetic Schrödinger hyperbolic equation with randomized coefficients

2021

Z Angew Math Mech

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Magnetic Schrödinger type equations are important physical models in the study of quantum electrodynamics (QED). This manuscript focuses on the regularity behavior of the solution of the stochastic hyperbolic equation influenced by various types of exponentially oscillating coefficients on the principal Hamiltonian operator part. Techniques from harmonic analysis and stochastic analysis are applied to explore the upper bound of loss of regularity. Moreover, appropriate counter-examples with periodic coefficients are constructed to show the lower bound of loss of regularity by the application of instability arguments. K E Y W O R D Sexponentially oscillating coefficients, Hamiltonian operator, loss of regularity, instability argument, magnetic Schrödinger type hyperbolic equations INTRODUCTIONAs is known, in the study of quantum mechanics, a magnetic field is produced by electric fields varying in time, spinning of the elementary particles, or moving electric charges, etc. Since last century, electromagnetic theory has been extensively utilized in the graphene research in magnetic fields, medical research of organs' biomagnetism, vortex study in the superconductor which carries quantized magnetic flux, and geographical forecasts for cataclysms, such as earthquakes, volcanic eruptions, geomagnetic reversal, etc. Magnetic Schrödinger type equations are important physical models in these respects [2,4,10,12]. In this background, we discuss the regularity of the solution of the magnetic Schrödinger hyperbolic equation with various types of oscillating coefficients of second-order moment stochastic processes. More importantly, we make a delicate classification of the oscillations with respect to the regularity behavior. Before introducing the main results, it is necessary to show some important relevant prerequisites. Introduction to the pseudodifferential operators associated to the Hamiltonian operatorLet 𝐀 be the vector potential of the magnetic flux density 𝐁 in the magnetic field, which does not depend on time, namely, 𝐁 = ∇ × 𝐀. Clearly, ∇ ⋅ 𝐁 = div rot𝐀 = 0.We deduce from the Maxwell's equation (𝜇 is the magnetic permeability) ∇ × 𝐄 = −𝜇 𝜕𝐁 𝜕𝑡 = 0

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