New Approach to Quantum Scattering Near the Lowest Landau Threshold for a Schrödinger Operator with a Constant Magnetic Field

 For a fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schr?dinger operator with a constant magnetic field and an axisymmetrical electric potential V. Asymptotic expansions for the resolvent of the Hamiltonian H _m  = H _om  + V are deduced as the spectral parameter tends to the lowest Landau threshold E ₀. In particular it is shown that E ₀ can be an eigenvalue of H _m . Furthermore, asymptotic expansions of the scattering matrix associated with the pair (H _m , H _om ) are derived as the energy parameter tends to E ₀. Received December 11, 2000; accepted in final form June 16, 2001 Published online June 10, 2002     

The Scattering Amplitude for the Schrödinger Equation with a Long-Range Potential

We consider the Schr?dinger operator with a long-range potential V(x) in the space . Our goal is to study spectral properties of the corresponding scattering matrix and a diagonal singularity of its kernel (the scattering amplitude). It turns out that in contrast to the short-range case the Dirac-function singularity of at the diagonal disappears and the spectrum of the scattering matrix covers the whole unit circle. For an asymptotically homogeneous function V(x) of order we show that typically , where the module w and the phase ψ are asymptotically homogeneous functions, as , of orders and , respectively. Leading terms of asymptotics of w and ψ at are calculated. In the case ρ=1 our results generalize (in the limit ) the well-known formula of Gordon and Mott. As a by-product of our considerations we show that the long-range scattering fits into the theory of smooth perturbations. This gives an elementary proof of existence and completeness of wave operators in the theory of long-range scattering. In this paper we concentrate on the case ρ>1/2 when the theory of pseudo-differential operators can be extensively used. Received: 29 January 1997 / Accepted: 6 May 1997     

A dynamic scattering approach for a gated interacting wire

A scattering approach for correlated one-dimensional systems is developed. The perfect contact to charge reservoirs is encoded in time-dependent boundary conditions. The conductance matrix for an arbitrary gated wire, respecting charge conservation, is expressed through a dynamic scattering matrix. Two applications are developed. First, it is shown that the dc conductance is equal to e ²/h for any model with conserved total left- and right-moving charges. Second, the ac conductance matrix is explicitly computated for the Tomonaga-Luttinger model (TLL). Received 31 August 1998     

Zeeman effects on d-wave superconductor and tunneling spectrum in normal-metal/d-wave superconductor tunnel junction

We study the Zeeman effect on the d-wave superconductor and tunneling spectrum in normal-metal(N)/d-wave superconductor(S) junction by applying a Zeeman magnetic field to the S. It is shown that: (1) the Zeeman magnetic field can lead to the S gap decreasing, and with the increase in Zeeman energy, the superconducting state is changed to the normal state, exhibiting a first-order phase transition; (2) the Zeeman magnetic field may make the zero-bias conductance peak split into two peaks, and the energy difference between the two splitting peaks in the conductance spectrum is equal to 2h ₀ (h ₀ is the Zeeman energy); (3) both the barrier strength of interface scattering and the temperature can lower the magnitudes of splitting peaks, of which the barrier strength can lead to the splitting peaks becoming sharp and the temperature can smear out the peaks, however, neither of them can influence the Zeeman effect.     

Positivity of energy in Abelian principal bundles over spacetime

Five-dimensional classical unified field theories may be described in terms of the geometry of a five-dimensional Lorentz space V ₅.This space may be considered as a principal bundle over spacetime V ₄with metric g _ij and gauge group U(1). It has been shown that when V ₄is asymptotically flat, the total energy momentum vector of V ₅is nonspacelike, and that the gravitational mass of V ₄has a lower bound. These results obtain when V ₅is replaced by a Lorentzian space V _N ,a principal bundle over an asymptotically flat V ₄with an Abelian group G as its gauge group.     

Dispersion of quasi-particles in high Tc cuprates

The interplay between superconductivity (SC) and antiferromagnetism (AFM) is studied in strongly correlated systems of high T _c Cuprate superconductors. It is assumed that superconductivity arises due to BCS pairing mechanism in presence of AFM in Cu lattices of Cu-O planes. The total Hamiltonian of the system is mean field one and has been solved exactly by writing the equations of motion for the single particle Green’s functions. Equations for the appropriate single particle co-relation functions are derived and the order parameters corresponding to SC and AFM are determined. It is assumed that the Fermi energy ∈ _F = 0 and the renormalized localized f energy level coincide with the Fermi level. All the quantities in the final equation for h and Δ are made dimensionless by dividing by 2t, where t is the hopping integral. The temperature dependent values of staggered magnetic field (h) and SC gap (Δ) were determined by solving self-consistent equations for h and Δ. The quasiparticle energy bands are function of AFM gap (h), SC gap (Δ) and hybridization (V). Then the dispersion of quasi-particles are studied at different temperatures by considering temperature dependent values of h and Δ and varying other different model parameters.      

Scattering Matrices in Many-Body Scattering

In this paper we show that generalized eigenfunctions of many-body Hamiltonians H with short-range two-body interactions have distributional asymptotics at non-threshold channels. The leading terms of the asymptotics can be used to define a scattering matrix, and we show that this is the same (up to normalization) as that arising from the standard wave-operator approach. We also prove the existence of local distributional asymptotics for locally approximate generalized eigenfunctions in the more general setting of short range perturbations of a scattering metric, defined by Melrose in [13]. Received: 29 October 1997 / Accepted: 19 June 1998     

Klein-Gordon thermal equation for a planck gas

In this paper the quantum hyperbolic equation formulated in our earlier paper [Found. Phys. Lett. 10, 599 (1997)] is applied to the study of the propagation of the initial thermal state of the universe. It is shown that the propagation depends on the barrier height. The Planck wall potential is introduced,V _P = ħ/8t_P = 1.125 10¹⁸ GeV, wheret _P is a Planck time. For the barrier heightV <V _P , the master thermal equation isthe modified telegrapher’sequation, and for barrier heightV >V _P the master equation is theKlein- Gordon equation. The solutions of both type equations for Cauchy boundary conditions are discussed.     

Neutron scattering by anharmonic crystals and the effect of sublattice displacements

A theory has been given for the scattering of neutrons by anharmonic crystals, for which terms of the typeV ⁽³⁾ (k _1j1; —k _1j1;o _j) which contribute to the sublattice displacements are not neglected. Using the standard perturbation theory in the interaction picture or Green’s function method, an expression has been derived for the differential scattering cross-section which brings in the shift and the width of the phonons in one-phonon energy exchange processes. It is shown that the sublattice displacements will modify the phase factor arising from the scattering by any atom in the unit cell, and the Debye-Waller factor also gets altered both by the sublattice displacements as well as by higher order terms arising from anharmonicity. It is shown that the differential scattering cross-section contains a term linearly depending on the third order anharmonicity coefficientV ⁽³⁾ (k _1j1;k _2j2;k _3j3) and neutron scattering by crystals should provide a useful method for evaluating the third order anharmonicity coefficients.     

The proton optical-model potential near the Coulomb barrier

Proton elastic scattering data from ¹⁹⁷Au, ²⁰⁸Pb and ²⁰⁹Bi at energies near the Coulomb barrier are analyzed. The energy dependences of the real volume and imaginary surface-derivative potential depths V_R and W_SF of a local optical-model potential with fixed geometric parameters are found to be much more rapid than at higher energies. The strong energy dependence of V_Rnear the Coulomb barrier is explained in terms of the non-locality of the nucleon-nucleus interaction.     

Measurement of the differential and integral distribution of diffuse x-ray scattering intensity from defects in thin strained layers

Double-and triple-crystal diffractometry have been used to study structural perfection of a ∼1 μm-thick Ga_1−x In_xSb_1−y As_y epitaxial film (x=0.9, y=0.8) on GaSb. It is shown that scattering from samples of this system can be divided into coherent and diffuse. The arrangement of reciprocal-lattice points of the film and substrate in the two-dimensional intensity distribution for asymmetrical reflections argues for the absence of elastic-strain relaxation. No dislocation networks are formed, and the diffuse scattering is produced by Coulomb-type defects. Localization of diffuse scattering in reciprocal space suggests that these defects reside in the epitaxial film. The diffuse-scattering distribution in asymmetrical reflections is shown to be anomalous; namely, it extends in a direction parallel to the surface and is split into two maxima. Schemes have been proposed and realized for measuring integral distributions of diffracted intensity along the surface and perpendicular to it, and their potential for studying diffuse scattering from defects is explored. Fiz. Tverd. Tela (St. Petersburg) 39, 1188–1193 (July 1997)     

The measurement of frequency separations in high-resolution nuclear magnetic resonance spectra using a double-lock spectrometer

The rotation-ring-puckering problem of four-membered ring cyclobutane-type molecules with D _4h -D _2d conformations is formulated as a dynamical system in the eulerian angles and the polar coordinate ?. Classical and quantum-mechanical hamiltonians and their symmetry group O ⁺⁺(3) × D _∞h are discussed. The kinetic energy matrix coefficients g_mm are found to be irrational functions of the trigonometric polynomial 3 + cos 2? in the internal coordinate ?. It is shown that the energy matrix coefficients may all be expressed as Laurent coefficients by means of the hypergeometric function ₂ F ₁, the convergence being determined by the roots of the kinetic determinant. Selection rules and matrix elements for rotation-ring-puckering Raman transitions are given.     

Levinson's theorem for the Klein-Gordon equation in one dimension

In terms of the modified Sturm-Liouville theorem, the Levinson theorem for the one-dimensional Klein-Gordon equation with a symmetric potential V(x) is established. It is shown that the number N₊ (N_-) of bound states with even (odd) parity is related to the phase shift of the scattering states with the same parity at zero momentum as and The solution of the one-dimensional Klein-Gordon equation with the energy M or -M is called as a half bound state if it is finite but does not decay fast enough at infinity to be square integrable. Received 22 December 1999     

Damping of the localized surface plasmon polariton resonance of gold nanoparticles

We report on a strong damping of the localized surface plasmon polariton resonance of gold nanoparticles. The ultra-fast dephasing time of localized surface plasmon polariton resonances in gold nanoparticles was systematically studied as a function of the particle size at a fixed photon energy of h ν=1.85 eV. Dephasing times ranging from T₂^expT_{2}^{\mathrm{exp}} = 5.5 fs to 15.0 fs were extracted and an influence of the reduced dimensions was detected. We have identified two dominant damping mechanisms: the well-known surface scattering and, for the first time, band structure changes. We have quantified the influence of these band structure changes on the optical properties by determining the essential damping parameter A to be A ^exp=0.32 nm/fs.     

Covering group and graph of discretized volumes

We consider a discretized volume V consisting of finite, congruent and attached copies of a tile t. We find a group L _V the orbit of which, when applied to t, is just V. We show the connection between the structural matrixQ in the formal solution of a boundary value problem formulated for volume V and the so called auxiliary matrix of the graph Γ _v associated with V. We show boundary value problems to be isomorphic if the graphs associated with the volumes are isomorphic, or, if the covering groups are Sunada pairs.     

On the reconstruction of an optical potential on the basis of experimental data in the WKB approximation

The inverse scattering problem at a fixed energy for a complex-valued potential is solved in the WKB approximation. The method is used to reconstruct the optical potential for elastic ¹⁶O + ¹⁶O scattering at E _lab = 350 MeV. The stability of the solution against small changes in the scattering matrix is studied.     

Fermion Fields in Theories of Gravitation

After an introduction to the formalism used throughout the paper there follows a concise presentation of the theory of fermion fields in one-tetrad gravitational theories. That presentation gives a hint to the construction of a bi-tetrad theory, the two tetrad fields being denoted by h^A_k and h?^A_k. The tetrad field h^A_k. gives the Riemannian metric g_kl while the tetrad field h?h^A_k is orthonormalized with respect to the flat metric a_kl. Specializing h?^A_k in such a way that they have the form δ^A_k in the preferred coordinates of Minkowski space and using a matter Lagrangian which contains these h?^A_k only by the anholonomic components of the metric Christoffel symbols, we obtain a dynamical energy momentum tensor which is equal to the canonical one. Then we consider the relations of the bi-tetrad theory to other theories which are only covariant with respect to global Lorentz transformations from the beginning. As an example we formulate the main relations of the two-component neutrino theory.     

Surface States and Spectra

Let ℤ₊ ^{d +1}= ℤ⁺×ℤ₊, let H ₀ be the discrete Laplacian on the Hilbert space l ²(ℤ₊ ^{d +1}) with a Dirichlet boundary condition, and let V be a potential supported on the boundary ∂ℤ₊ ^{d +1}. We introduce the notions of surface states and surface spectrum of the operator H=H ₀+V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on σ(H ₀) with probability one. To prove this result we combine Aizenman–Molchanov theory with techniques of scattering theory. Received: 18 September 2000 / Accepted: 21 November 2000     

Anomalies in the scattering of gamma-rays from single crystals of HF-doped ice-I h around 100 K

Using the high energy resolution of the Mössbauer effect, the elastic and inelastic scattering of gamma-rays from single crystals of HF-doped IceI _h has been distinguished. The crystals were studied in the region of 100 K, since measurements by previous workers have suggested the possibility of an order-disorder transition at this temperature. Marked anomalies in the scattered intensity were observed between 106 and 125 K, and these were shown to depend on the thermal history of the crystals.     

Breit-Wigner formulas for the scattering phase and the total scattering cross-section in the semi-classical limit

In this paper we prove results in resonance scattering for the Schrödinger operatorP _v=–h ²+V, V being a smooth, short range potential onR ⁿ . More precisely, for energy near a trapping energy level ₀ for the classical system defined by the Hamiltonianp(x,)= ²+V(x), we prove that the scattering phase and the scattering cross sections associated to (P _v, P₀) have the Breit-Wigner form (Lorentzian line shape) in the limith0.