Constructive Semiclassical Asymptotics of Bound States of Graphene in a Constant Magnetic Field with Small Mass

Mathematical Notes - The paper deals with constructive semiclassical asymptotics of eigenfunctions of the Dirac operator describing graphene in a constant magnetic field. Two cases are considered:...     

Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R ⁿ , n ≥ 3. The magnetic potential is assumed to be continuous with L ^∞ divergence and zero boundary values. The method is based on semiclassical pseudodifferential calculus and the construction of complex geometrical optics solutions in weighted Sobolev spaces.     

Equilibrium of a Degenerate Gas of Nucleons and Electrons in a Strong Magnetic Field

A model of a degenerate ideal gas of nucleons and electrons in a superstrong magnetic field is used to describe the state of matter in the central region of a strongly magnetized neutron star. The influence of a constant uniform superstrong magnetic field on the equilibrium conditions and the equation of state for the degenerate gas of neutrons, protons, and electrons is investigated in the framework of this model. The contribution determined by the interaction of the anomalous magnetic moments of the fermions with the magnetic field is taken into account. The influence of the superstrong magnetic field on the process of gravitational collapse of a magnetized neutron star is discussed under the assumption that the central region of the star consists mostly of degenerate neutrons. We show that if the densities of electrons, protons, and neutrons are relatively low depending on the field strength, the fermion gases in a superstrong uniform magnetic field become totally polarized with respect to the spin. We discuss the possibility of spontaneous magnetization occurring in a system of degenerate neutrons where the exchange interaction effects are taken into account.     

Semiclassical Spectral Series of a Helium-like Atom in a Magnetic Field

We apply the complex WKB method (the Maslov complex germ theory) to the model of two electrons in a field with a fixed center. We construct semiclassical spectral series of the Pauli operator eigenvalues in the external magnetic field with the spin–orbital and spin–spin interactions and the quadrupole moment of the nucleus taken into account. These series correspond to a new type of closed phase trajectories, the relative equilibrium positions of the corresponding classical nonintegrable system. Explicit effective formulas are derived for the fine (Zeeman effect) and the hyperfine splitting of semiclassical energy levels of a helium-like ion with an arbitrary nucleus charge Z in the entire range of the magnetic field magnitude, including the extreme cases of weak and ultrastrong fields.     

Polarization of the Electron–Positron Vacuum by a Strong Magnetic Field in the Theory with a Fundamental Mass

In the framework of the theory with a fundamental mass in the one-loop approximation, we evaluate the exact Lagrange function of the strong constant magnetic field, replacing the Heisenberg–Euler Lagrangian in the traditional QED. We establish that the derived generalization of the Lagrange function is real for arbitrary values of the magnetic field. In the weak field, the evaluated Lagrangian coincides with the known Heisenberg–Euler formula. In extremely strong fields, the field dependence of the Lagrangian completely disappears; in this range, the Lagrangian tends to the limit value determined by the ratio of the fundamental and lepton masses.     

具有磁荷,磁矩和质量四极矩的天体的引力场

该文的工作是工作[3]的继续．用微扰论的方法求得了具有磁荷、磁矩和质量四极矩的中心天体的引力场．这对于揭示这类天体的引力性质是有用的．     

On the Drift-Diffusion Model for a Two-Band Quantum Fluid at Zero Temperature

By using a scale transformation, we obtain hydrodynamic equations in the quasiclassical approximation from the two-band Schrodinger equation. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 723–730, June, 2005.     

The Asymptotic Form of the Lower Landau Bands in a Strong Magnetic Field

The asymptotic form of the bottom part of the spectrum of the two-dimensional magnetic Schrödinger operator with a periodic potential in a strong magnetic field is studied in the semiclassical approximation. Averaging methods permit reducing the corresponding classical problem to a one-dimensional problem on the torus; we thus show the almost integrability of the original problem. Using elementary corollaries from the topological theory of Hamiltonian systems, we classify the almost invariant manifolds of the classical Hamiltonian. The manifolds corresponding to the bottom part of the spectrum are closed or nonclosed curves and points. Their geometric and topological characteristics determine the asymptotic form of parts of the spectrum (spectral series). We construct this asymptotic form using the methods of the semiclassical approximation with complex phases. We discuss the relation of the asymptotic form obtained to the magneto-Bloch conditions and asymptotics of the band spectrum.     

下部有常热通量加热作用时非均匀温度梯度和磁场对Marangoni对流作用的影响

在一个水平流体层中,下部加热和上部致冷,热通量为常数时,研究磁场和非均匀温度梯度对Marangoni对流作用的影响.对线性稳定分析进行了详细的研究.分析了各种参数对对流作用的影响.考虑了6种基本的温度分布曲线,给出了造成失稳影响的一些普遍结论.     

Equation for the Complex Energy of a Bound Particle in a Laser Radiation Field in the Presence of Strong Constant Electromagnetic Fields

We construct exact solutions of the Schrodinger and Pauli equations for charged particles in the external field of the Redmond generalized configuration. We calculate the Green's functions of scalar and spinning particles in this field. Using them, we calculate the equations for the complex quasienergy of a bound particle (bound by a short-range potential) two different ways. In the example of an external constant electric field, we discuss the applicability domain of the obtained equations and the differences between their solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 198–211, November, 2005.     

On the Sharp Constant in the Small Ball Asymptotics of Some Gaussian Processes under L 2-Norm

We find the sharp constant in the small L ₂-deviation asymptotics for a wide class of Gaussian processes including the m-times integrated Wiener process and the m-times integrated Ornstein–Uhlenbeck process. Extremal properties of usual and Euler integration are proved. Bibliography: 19 titles.     

Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

We consider an electrically charged particle on the Euclidean plane subjected to a perpendicular magnetic field which depends only on one of the two Cartesian co-ordinates. For such a “unidirectionally constant” magnetic field (UMF), which otherwise may be random or not, we prove certain spectral and transport properties associated with the corresponding one-particle Schr?dinger operator (without scalar potential) by analysing its “energy-band structure”. In particular, for an ergodic random UMF we provide conditions which ensure that the operator’s entire spectrum is almost surely absolutely continuous. This implies that, along the direction in which the random UMF is constant, the quantum-mechanical motion is almost surely ballistic, while in the perpendicular direction in the plane one has dynamical localization. The conditions are verified, for example, for Gaussian and Poissonian random UMF’s with non-zero mean-values. These results may be viewed as “random analogues” of results first obtained by A. Iwatsuka [Publ. RIMS, Kyoto Univ. 21 (1985) 385] and (non-rigorously) by J.E. Müller [Phys. Rev. Lett. 68 (1992) 385].

Heinz BAUER (31 January 1928 - 15 August 2002)

of Erlangen-Nürnberg

Communicated by Frank den Hollander submitted 30/12/04, accepted 13/06/05     

含埋藏椭圆形裂纹的金属构件脉冲放电瞬间温度场分析

对带有椭圆埋藏裂纹金属构件在脉冲放电瞬间的温度场进行了理论分析．在带有裂纹问题温度场的理论求解中,根据相似性原理,将电流通过带有椭圆面裂纹导体类比于流体流过障碍物的情形,从而引入了裂纹的边界条件,得到域内电流密度的分布,进而得到了放电瞬间埋藏椭圆裂纹尖端的温度场表达式,为空间裂纹电磁热止裂技术的实际应用提供了理论基础．     

On the Algorithmic Undecidability of the Embeddability Problem for Algebraic Varieties over a Field of Characteristic Zero

Mathematical Notes -     

On Distribution Semigroups with a Singularity at Zero and Bounded Solutions of Differential Inclusions

We study distribution semigroups with a singularity at zero and their generators, and establish a relationship between this semigroup and a degenerate semigroup of linear operators on the open right half-line. The study makes an intensive use of spectral theory of linear relations. Applications to the existence problem for bounded solutions of linear differential inclusions are obtained.     

On the Temporal and the Spatial Kelvin-Helmholtz Instability of a Hypersonic Jet Confined by a Longitudinal Magnetic Field

Both the temporal and the spatial Kelvin-Helmholtz instability of a hypersonic jet confined by a longitudinal magnetic field are investigated. The reflection modes are calculated for a set of typical jet parameters. Analytical treatments for both temporal and spatial resonant modes, i.e., the most important reflection modes with the local maximum growth rates, are given. Various properties of the resonant modes are plotted against the jet temperature. The growth rates of the temporal and the spatial resonant modes are connected by the group velocity. The astrophysical implications of the present investigation are discussed.     

On Some Spectral Properties of the Energy Operator for an Infinite System in a Magnetic Field

For systems in a magnetic field, we investigate the form sum of an infinite-dimensional energy operator perturbed by a potential. We also investigate changes in the spectrum of the energy operator in the case of its perturbation by a potential.     

Asymptotics of the Wave Field in the Vicinity of the Symmetry Axis of a Transversely Isotropic Homogeneous Medium

Asymptotic formulas for the wave field, which describe the kiss singularity in a homogeneous transversely isotropic medium, are obtained. A physical interpretation of the results is presented. Bibliography: 10 titles.     

The Effect of an Axial Magnetic Field at the Interface of a Crystal Grown by the Czochralski Method

A mathematical model has been developed which aims to give insightinto the transport phenomena in the vicinity of the interfaceof a crystal grown by the Czochralski method in the presenceof an axial magnetic field. The fluid flow, temperature andconcentration fields in this region have a strong effect onthe distribution of impurities and the occurrence of cracks,dislocations and other physical defects in the crystal and soknowledge and ultimately control of these factors is of greatpractical importance. The model incorporates rotation of both the crystal and crucibleby considering the crystal to be an infinite disc rotating ina semi-infinite fluid which may be rotating at infinity. Axialsymmetry is assumed and the magnetic Prandtl number is consideredto be very much less than unity. This means that induced currentscan be neglected and allows a similarity solution to be developed.The system of partial differential equations can then be replacedby an ordinary differential boundary-value problem which issolved numerically.