Alexander Moroz

Quasi-periodic Green's functions of the Helmholtz and Laplace equations

A classical problem of free-space Greens function G0Λ representations of the Helmholtz equation is studied in various quasi-periodic cases, i.e., when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space. Exponentially convergent series for the free-space quasi-periodic G0Λ and for the expansion coefficients DL of G0Λ in the basis of regular (cylindrical in two dimensions and spherical in three dimension (3D)) waves, or lattice sums, are reviewed and new results for the case of a one-dimensional (1D) periodicity in 3D are derived. From a mathematical point of view, a derivation of exponentially convergent representations for Schlomilch series of cylindrical and spherical Hankel functions of any integer order is accomplished. Exponentially convergent series for G0Λ and lattice sums DL hold for any value of the Bloch momentum and allow G0Λ to be efficiently evaluated also in the periodicity plane. The quasi-periodic Greens functions of the Laplace equation are obtained from the corresponding representations of G0Λ of the Helmholtz equation by taking the limit of the wave vector magnitude going to zero. The derivation of relevant results in the case of a 1D periodicity in 3D highlights the common part which is universally applicable to any of remaining quasi-periodic cases. The results obtained can be useful for the numerical solution of boundary integral equations for potential flows in fluid mechanics, remote sensing of periodic surfaces, periodic gratings, and infinite arrays of resonators coupled to a waveguide, in many contexts of simulating systems of charged particles, in molecular dynamics, for the description of quasi-periodic arrays of point interactions in quantum mechanics, and in various ab initio first-principle multiple-scattering theories for the analysis of diffraction of classical and quantum waves.

Single-particle density of states, bound states, phase-shift flip, and a resonance in the presence of an Aharonov-Bohm potential

Both the nonrelativistic scattering and the spectrum in the presence of the Aharonov-Bohm potential are analyzed and the single-particle density of states for different self-adjoint extensions is calculated. The single-particle density of states is shown to be a symmetric and periodic function of the flux, which depends only on the distance from the nearest integer. The Krein-Friedel formula for this long-range potential is shown to be valid when regularized with the ensuremath{zeta} function. The limit when the radius R of the flux tube shrinks to zero is discussed. For Rensuremath{ne}0 and in the case of an anomalous magnetic moment

The Aharonov-Casher theorem and the axial anomaly in the Aharonov-Bohm potential

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Novel summability methods generalizing the Borel method

ensuremath{gtrsim}2 (note, e.g., that

Aspect ratio analysis for ground states of bosons in anisotropic traps

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Strong asymptotic conditions (Short guide to using summability methods)

=2.002 32 for the electron) the coupling for spin-down electrons is enhanced and bound states occur in the spectrum. Their number does depend on a regularization and generically does not match with the number of zero modes in a given field that occur when

Quantum models with spectrum generated by the flows of polynomial zeros

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Critical exponent of the localization length for the symplectic case

=2. Provided the coupling with the interior of the flux tube is not renormalized to a critical one, neither bound states nor zero modes survive the limit Rensuremath{rightarrow}0. The Aharonov-Casher theorem on the number of zero modes is corrected for the singular field configuration.

A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

Abstract The spectral properties of the Dirac Hamiltonian in the the Aharonov-Bohm potential are discussed. By using the Krein-Friedel formula, the density of states (DOS) for different self-adjoint extensions is calculated. As in the nonrelativistic case, whenever a bound state is present in the spectrum it is always accompanied by a (anti) resonance at the energy. The Aharonov-Casher theorem must be corrected for singular field configurations. There are no zero (threshold) modes in the Aharonov-Bohm potential. For our choice of the 2D Dirac Hamiltonian, the phase-shift flip is shown to occur at only positive energies. This flip gives rise to a surplus of the DOS at the lower threshold coming entirely from the continuous part of the spectrum. The results are applied to several physical quantities: the total energy, induced fermion-number, and the axial anomaly.

On uniqueness of Heine–Stieltjes polynomials for second order finite-difference equations

The paper deals with moment constant summability methods. A method is constructed which provides an analytic continuation of a function regular at the origin onto its Mittag-Leffler (principal) star. In this sense the method is optimal in contrast to the Borel one. In the next a one parameter family of such methods is introduced. The methods may be useful both in field theory and in statistical physics. Applications to the Nevanlinna theorem, the Rayleigh-Schrö-dinger perturbation theory and the dispersion-like integral are given. The proofs of theorems can be easily adapted to the study of the Mellin transform of some entire functions and a simpler proof of asymptotic properties of the gamma function can be obtained.