Spectral properties of Hamiltonians with a magnetic field at a fixed pseudomoment. II

The discrete spectrum of multiparticle Hamiltonians H₀ of neutral systems in a homogeneous magnetic field is studied at a fixed pseudomoment. A general theorem is proved, which describes the discrete spectrum of H₀ under certain conditions in terms of constructed effective one-dimensional differential operators with a known spectrum structure. Based on this theorem, the conditions for a finite or infinite spectrum and the spectral asymptotic forms of the operator H₀ are obtained. The results can be applied to Hamiltonians of any atoms. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 15–39, January, 1999.     

Polarization of vacuum by an external magnetic field in the (2+1)-dimensional quantum electrodynamics with a nonzero fermion density

The Green's function of the Dirac equation with an external stationary homogeneous magnetic field in the (2+1)-dimensional quantum electrodynamics (QED ₂₊₁) with a nonzero fermion density is constructed. An expression for the polarization operator in an external stationary homogenous magnetic field with a nonzero chemical potential is derived in the one-loopQED ₂₊₁ approximation. The contribution of the induced Chern—Simons term to the polarization operator and the effective Lagrangian for the fermion density corresponding to the occupation of n relativistic Landau levels in an external magnetic field are calculated. An expression of the induced Chern—Simons term in a magnetic field for the case of a finite temperature and a nonzero chemical potential is obtained. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 132–151, October, 2000.     

QED2+1 radiation effects in a strong magnetic field

We develop the eigenfunction method for the Dirac operator in a background magnetic field in the (2+1)-dimensional quantum electrodynamics (QED₂₊₁). In the eigenfunction repressentation, we find the exact solutions and the Green's functions of the Dirac equation in a strong constant homogeneous magnetic field in 2+1 dimensions. In the one-loop QED₂₊₁ approximation, we derive the effective Lagrangian, the density of vacuum fermions induced by the field, and the electron mass operator in a homogeneous background magnetic field. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 412–423, December, 1999.     

On the discrete spectrum of a given SO(2) symmetry of multiparticle systems in potential and homogeneous magnetic fields

For a system of n identical particles in a homogeneous magnetic field, the discrete spectrum of the Hamiltonian H^{α, m} on the subspaces of functions with permutational symmetry α and rotational (SO(2)) symmetry m is studied as m→∞. It is proved that the discrete spectrum of the operator H^α,m contains only one eigenvalue if certain conditions are satisfied. The asymptotic behavior of this eigenvalue as m→∞ is found. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 28–41, 1992. Translated by A. V. Lyakhovskaya     

Spectral properties of Hamiltonians with a magnetic field at a fixed pseudomomentum. III

The general results (see previous Part II) on the structure of the discrete spectra of energy operators of neutral systems in a homogeneous magnetic field at a fixed pseudomomentum are proved to be applicable to Hamiltonians of arbitrary atoms. Asymptotic expressions for the discrete spectra of Hamiltonians in the presence of a homogeneous magnetic field are found for arbitrary atoms. This paper completes the investigation of the spectral properties of Hamiltonians of neutral systems in a homogeneous magnetic field at a fixed pseudomomentum. The essential and discrete parts of the spectrum for such systems were found previously; however, whether the theorems in Part II were valid for actual n-particle systems remained an open question for the case n>3. Translated from Teoreticheskaya i Mathematicheskaya Fizika, Vol. 120, No. 2, pp. 291–308, August, 1999.     

Discrete Spectrum of a Three-Particle Schrodinger Operator with a Homogeneous Magnetic Field

The discrete spectrum of the Schrödinger operator is studiedfor a system of three identical particles with short-range interactionsin a homogeneous magnetic field. All the two-particle subsystemsare supposed to be unstable. Finiteness of the discrete spectrumis established under some assumptions about the solutions ofthe corresponding two-particle Schrödinger equation.     

A quantum model of a real scalar field

A quantum model of a real scalar field with local operator gauge symmetry is discussed. In the localized theory, in order to keep the local operator gauge symmetry, an operator gauge potential BB _μ, is needed. By combining the constraint of operator gauge potentialB _μ, and the microscopic causality theorem, the usual canonical quantization condition of a real scalar field is obtained. Therefore, a quantum model of a real scalar field without the usual procedure of quantizing a related classical model can be directly constructed. Project supported in part by T.D. Lee’s NNSF Grant, National Natural Science Foundation of China, Foundation of Ph. D. Directing Programme of Chinese Universities and the Chinese Academy of Sciences.     

Spectrum of the Magnetic Schrödinger Operator in a Waveguide with Combined Boundary Conditions

We consider the magnetic Schrödinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann {For the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.submitted 11/05/04, accepted 21/09/04     

A (2+1)-dimensional fermion in the Coulomb and magnetic field backgrounds

In the problem of a two-dimensional hydrogen-like atom in a magnetic field background, we construct quasi-classical solutions and the energy spectrum of the Dirac equation in a strong Coulomb field and in a weak constant homogeneous magnetic field in 2+1 dimensions. We find some “exact” solutions of the Dirac and Pauli equations describing the “spinless” fermions in strong Coulomb fields and in homogeneous magnetic fields in 2+1 dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 105–118, April, 1999.     

Lifschitz tail in a magnetic field: the nonclassical regime

We obtain the Lifschitz tail, i.e. the exact low energy asymptotics of the integrated density of states (IDS) of the two-dimensional magnetic Schr?dinger operator with a uniform magnetic field and random Poissonian impurities. The single site potential is repulsive and it has a finite but nonzero range. We show that the IDS is a continuous function of the energy at the bottom of the spectrum. This result complements the earlier (nonrigorous) calculations by Brézin, Gross and Itzykson which predict that the IDS is discontinuous at the bottom of the spectrum for zero range (Dirac delta) impurities at low density. We also elucidate the reason behind this apparent controversy. Our methods involve magnetic localization techniques (both in space and energy) in addition to a modified version of the “enlargement of obstacles” method developed by A.-S. Sznitman. Received: 20 July 1997 / Revised version: 20 April 1998     

Splitting of lower energy levels in a quantum double well in a magnetic field and tunneling of wave packets in Nanowires

We consider the problem of the splitting of lower eigenvalues of the two-dimensional Schrödinger operator with a double-well-type potential in the presence of a homogeneous magnetic field. The main result is the observation that the partial Fourier transformation takes the operator under study to a Schrödingertype operator with a (new) double-well-type potential but already without any magnetic field. We use this observation to investigate the influence of the magnetic field on the tunneling effects. We discuss two methods for calculating the splitting of lower eigenvalues: based on the instanton and based on the so-called libration. We use the obtained result to study the tunneling of wave packets in parallel quantum nanowires in a constant magnetic field.     

Index theory on locally homogeneous spaces

We describe how the equivariant K homology class of an invariant elliptic operator on a homogeneous space of a linear semisimple Lie group determines the L ²-index of the associated operator on a finite volume locally homogeneous space. The machinery of equivariant K homology and of KK theory can be used to prove theorems about L ²-indices. We give an application motivated by the problem of calculating multiplicities of subrepresentations of quasi-regular representations.Supported by the National Science Foundation under Grant No. DMS-8903472.Supported by the National Science Foundation under Grant No. DMS-8901436.     

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     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.     

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T₁(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x₁ ..., x_p} (2p=q+1), and Γ, a subgroup of finite index in G. T₁(Γ, 1) is defined on the subspace L²(Γ/G, 1) ⊖ E_ex, where E_ex is the subspace of eigenfunctions of T₁(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 92–109. Translated by A. M. Nikitin.     

Spectrum of a homogeneous graph

We describe the spectrum of the Laplacian for a homogeneous graph acted on by a discrete group. This follows from a more general result which describes the spectrum of a convolution operator on a homogeneous space of a locally compact group. We also prove a version of Harnack inequality for a Schrödinger operator on an invariant homogeneous graph.     

Spectrum of three-dimensional landau operator perturbed by a periodic point potential

A study is made of a three-dimensional Schrödinger operator with magnetic field and perturbed by a periodic sum of zero-range potentials. In the case of a rational flux, the explicit form of the decomposition of the resolvent of this operator with respect to the spectrum of irreducible representations of the group of magnetic translations is found. In the case of integer flux, the explicit form of the dispersion laws is found, the spectrum is described, and a qualitative investigation of it is made (in particular, it is established that not more than one gap exists).Mordovian State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 2, pp. 283–294, May, 1995.     

On the Spectrum of Non-Selfadjoint Schrödinger Operators with Compact Resolvent

We determine the Schatten class for the compact resolvent of Dirichlet realizations, in unbounded domains, of a class of non-selfadjoint differential operators. This class consists of operators that can be obtained via analytic dilation from a Schrödinger operator with magnetic field and a complex electric potential. As an application, we prove, in a variety of examples motivated by physics, that the system of generalized eigenfunctions associated with the operator is complete, or at least the existence of an infinite discrete spectrum.     

On the spectrum of magnetic Dirac operators with Coulomb-type perturbations

We carry out the spectral analysis of singular matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the perturbations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Constant, periodic as well as diverging magnetic fields are covered, and Coulomb potentials up to the physical nuclear charge Z<137 are allowed. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.