Electron in the Aharonov-Bohm potential and in the Coulomb field in 2+1 dimensions

We obtain exact solutions of the Dirac equation in 2+1 dimensions and the electron energy spectrum in the superposition of the Aharonov-Bohm and Coulomb potentials, which are used to study the Aharonov-Bohm effect for states with continuous and discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov-Bohm and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov-Bohm potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the magnetic field is taken into account in the form of the two-dimensional Dirac delta function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 502–517, December, 2006. An erratum to this article is available at .     

Quasiexact Solution of a Relativistic Finite-Difference Analogue of the Schrödinger Equation for a Rectangular Potential Well

We consider a well-posed formulation of the spectral problem for a relativistic analogue of the one-dimensional Schrödinger equation with differential operators replaced with operators of finite purely imaginary argument shifts exp(±id/dx). We find effective solution methods that permit determining the spectrum and investigating the properties of wave functions in a wide parameter range for this problem in the case of potentials of the type of a rectangular well. We show that the properties of solutions of these equations depend essentially on the relation between and the parameters of the potential and a situation in which the solution for 1 is nevertheless fundamentally different from its Schrödinger analogue is quite possible.     

Zero-mass fermions in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions

We consider the motion of a relativistic charged zero-mass fermion in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions. With these singular external potentials, we construct one-parameter self-adjoint Dirac Hamiltonians classified by self-adjoint boundary conditions. We show that if the so-called effective charge becomes overcritical, then virtual (quasistationary) bound states occur. The wave functions corresponding to these states have large amplitudes near the Coulomb center, and their energy spectrum is quasidiscrete and consists of a number of broadened levels of a width related to the inverse lifetime of the quasistationary state. We derive equations for the quasidiscrete spectra and quasistationary state lifetimes and solve these equations in physically interesting cases. We study the so-called local densities of state, which can be assessed in physical experiments, as functions of the energy and the problem parameters, investigating these densities both analytically and graphically.     

Information entropy of the relativistic Kozlov-Nikishin model

We calculate wave functions in the momentum representation for bound states in the relativistic Kozlov-Nikishin model. We investigate the asymptotic behavior of entropy in the coordinate and momentum representations at various limiting transitions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 444–458, September, 2006.     

Center-of-mass tomography and probability representation of quantum states for tunneling

We develop a representation of quantum states in which the states are described by fair probability distribution functions instead of wave functions and density operators. We present a one-random-variable tomography map of density operators onto the probability distributions, the random variable being analogous to the center-of-mass coordinate considered in reference frames rotated and scaled in the phase space. We derive the evolution equation for the quantum state probability distribution and analyze the properties of the map. To illustrate the advantages of the new tomography representations, we describe a new method for simulating nonstationary quantum processes based on the tomography representation. The problem of the nonstationary tunneling of a wave packet of a composite particle, an exciton, is considered in detail.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 371–387, February, 2005.     

The quadratic plus the tensor potentials for particle with 1/2‐spin via L2 method under the spin symmetry

In the present letter, the relativistic equation for particle 1/2‐spin have been obtained for the quadratic scalar and vector potentials in the presence of the tensor interaction that depends on the radial component either linearly and inversely. Under the spin symmetry, the relativistic equation is calculated by using the idea of L² that supports a tridiagonal matrix representation of the wave operator. By this requirement, the relativistic energy spectrum and corresponding spinor wave functions are obtained. Also, the obtained analytically result is compared with other results that are in good agreement. Some of the numerical results are given, too. Copyright © 2015 John Wiley & Sons, Ltd.     

Four-Parameter 1/<Emphasis Type="Italic">r</Emphasis><Superscript>2</Superscript> Singular Short-Range Potential with Rich Bound States and A Resonance Spectrum

We use the tridiagonal representation approach to enlarge the class of exactly solvable quantum systems. For this, we use a square-integrable basis in which the matrix representation of the wave operator is tridiagonal. In this case, the wave equation becomes a three-term recurrence relation for the expansion coefficients of the wave function with a solution in terms of orthogonal polynomials that is equivalent to a solution of the original problem. We obtain S-wave bound states for a new four-parameter potential with a 1/r² singularity but short-range, which has an elaborate configuration structure and rich spectral properties. A particle scattered by this potential must overcome a barrier and can then be trapped in the potential valley in a resonance or bound state. Using complex rotation, we demonstrate the rich spectral properties of the potential in the case of a nonzero angular momentum and show how this structure varies with the parameters of the potential.     

Peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus

We consider the peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus and discuss the long history of an incorrect interpretation of this problem in the case of a pointlike nucleus and its current correct solution. We consider the spectral problem in the case of a regularized Coulomb potential. For some special regularizations, we derive an exact equation for the point spectrum in the energy interval (-m,m) and find some of its solutions numerically. We also derive an exact equation for charges yielding bound states with the energy E = -m; some call them supercritical charges. We show the existence of an infinite number of such charges. Their existence does not mean that the oneparticle relativistic quantum mechanics based on the Dirac Hamiltonian with the Coulomb field of such charges is mathematically inconsistent, although it is physically unacceptable because the spectrum of the Hamiltonian is unbounded from below. The question of constructing a consistent nonperturbative second-quantized theory remains open, and the consequences of the existence of supercritical charges from the standpoint of the possibility of constructing such a theory also remain unclear.     

The Spectrum of Two-Particle Bound States for the Transfer Matrices of Gibbs Fields (an Isolated Bound State)

This paper initiates a general study of the spectrum of two-particle bound states of transfer matrices for a fairly wide class of Gibbs fields at high temperature T. In the present first part of this study, a detailed statement of the problem is given and the existence of a so-called "isolated level" lying at a distance 1/T ² from the boundary of the continuous spectrum is established for all values of the total quasimomentum of the system. In the concluding part of the paper, we prove that there are no other bound states provided that is far from certain singular values. In the second part, we will consider bound states for close to the singular values. The distance from these states (adjacent levels, in the authors' terminology) to the continuous spectrum is at most of the order of 1/T ⁴.     

Bound States of a Particle in the Field of Receding Point Potentials

We study bound states in a nonstationary system of receding δ-centers. We consider the states characterized by different depths of point spectrum levels in one-dimensional and three-dimensional problems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 385–392, December, 2005.     

Schr?dinger and Dirac particles in quasi-one-dimensional systems with a Coulomb interaction

We consider specific features and principal distinctions in the behavior of the energy spectra of Schr?dinger and Dirac particles in the regularized ??Coulomb?? potential V_??(z) = ?q/(|z|+??) as functions of the cutoff parameter ?? in 1+1 dimensions. We show that the discrete spectrum becomes a quasiperiodic function of ?? for ?? ? 1 in such a one-dimensional ??hydrogen atom?? in the relativistic case. This effect is nonanalytically dependent on the coupling constant and has no nonrelativistic analogue in this case. This property of the Dirac spectral problem explicitly demonstrates the presence of a physically informative energy spectrum for an arbitrarily small ?? > 0, but also the absence of a regular limit transition ?? ?? 0 for all nonzero q. We also show that the three-dimensional Coulomb problem has a similar property of quasiperiodicity with respect to the cutoff parameter for q = Z?? > 1, i.e., in the case where the domain of the Dirac Hamiltonian with the nonregularized potential must be especially refined by specifying boundary conditions as r ?? 0 or by using other methods.     

Wigner function of a relativistic particle in a time-dependent linear potential

We construct phase-space representations for a relativistic particle in both a constant and a time-dependent linear potential. We obtain explicit expressions for the Wigner distribution functions for these systems and find the correct nonrelativistic limit and free-particle limit for these functions. We derive the relativistic dynamical equation governing the time development of the Wigner distribution function and relativistic equation for the Wigner distribution function of stationary states and also calculate the amplitudes of transitions between energy states.     

Persistence of Anderson localization in Schrödinger operators with decaying random potentials

We show persistence of both Anderson and dynamical localization in Schrödinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schrödinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than |x|^-2 at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as |x|^-α at infinity, we determine the number of bound states below a given energy E<0, asymptotically as α↓0. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent α; (b) dynamical localization holds uniformly in α.     

两不同型部件并联可修系统主算子的性质

研究了两不同型部件并联可修系统.通过选取空间及定义算子A和B,将模型方程转化成了Banach空间中的抽象Cauchy问题.通过分析系统主算子A的谱分布,求出A的谱上界.利用预解正算子及共尾理论,证明了A的谱上界和增长界相等.     

Fermion bound states in the Aharonov-Bohm field in 2+1 dimensions

We find exact solutions of the Dirac equation that describe fermion bound states in the Aharonov-Bohm potential in 2+1 dimensions with the particle spin taken into account. For this, we construct self-adjoint extensions of the Hamiltonian of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions. The self-adjoint extensions depend on a single parameter. We select the range of this parameter in which quantum fermion states are bound. We demonstrate that the energy levels of particles and antiparticles intersect. Because solutions of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions describe the behavior of relativistic fermions in the field of the cosmic string in 3+1 dimensions, our results can presumably be used to describe fermions in the cosmic string field.     

Lieb–Robinson Bounds,Arveson Spectrum and Haag–Ruelle Scattering Theory for Gapped Quantum Spin Systems

We consider translation invariant gapped quantum spin systems satisfying the Lieb–Robinson bound and containing single-particle states in a ground state representation. Following the Haag–Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding S-matrix. We also obtain some general restrictions on the shape of the energy–momentum spectrum. For the purpose of our analysis, we adapt the concepts of almost local observables and energy–momentum transfer (or Arveson spectrum) from relativistic QFT to the lattice setting. The Lieb–Robinson bound, which is the crucial substitute of strict locality from relativistic QFT, underlies all our constructions. Our results hold, in particular, in the Ising model in strong transverse magnetic fields.     

WKB method for the Dirac equation with a scalar-vector coupling

We outline a recursive method for obtaining WKB expansions of solutions of the Dirac equation in an external centrally symmetric field with a scalar-vector Lorentz structure of the interaction potentials. We obtain semiclassical formulas for radial functions in the classically allowed and forbidden regions and find conditions for matching them in passing through the turning points. We generalize the Bohr-Sommerfeld quantization rule to the relativistic case where a spin-1/2 particle interacts simultaneously with a scalar and an electrostatic external field. We obtain a general expression in the semiclassical approximation for the width of quasistationary levels, which was earlier known only for barrier-type electrostatic potentials (the Gamow formula). We show that the obtained quantization rule exactly produces the energy spectrum for Coulomb- and oscillatory-type potentials. We use an example of the funnel potential to demonstrate that the proposed version of the WKB method not only extends the possibilities for studying the spectrum of energies and wave functions analytically but also ensures an appropriate accuracy of calculations even for states with n_r 1.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 83–111, April, 2005.     

Anderson localization and Lifshits tails for random surface potentials

We consider Schrödinger operators on L²(R^d) with a random potential concentrated near the surface R^d₁×{0}⊂R^d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87-97] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tails relies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of such operators is parabolic.     

Stable Blowup Dynamics for the 1‐Corotational Energy Critical Harmonic Heat Flow

We exhibit a stable finite time blowup regime for the 1‐corotational energy critical harmonic heat flow from ?² into a smooth compact revolution surface of ?³ that reduces to the semilinear parabolic problem for a suitable class of functions f. The corresponding initial data can be chosen smooth, well localized, and arbitrarily close to the ground state harmonic map in the energy‐critical topology. We give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den Berg, Hulshof, and King. Our approach lies in the continuation of the study of the 1‐equivariant energy critical wave map and Schrödinger map with ??² target by Merle, Raphaël, and Rodnianski. © 2012 Wiley Periodicals, Inc.