Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs

Bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$ are investigated. We impose the Neumann boundary condition on the two concentric windows of the radii $a$ and $b$ located on the opposite walls and the Dirichlet boundary condition on the remaining part of the boundary of the strip. We prove that such a system exhibits discrete eigenvalues below the essential spectrum for any $a,b>0$. When $a$ and $b$ tend to the infinity, the asymptotic of the eigenvalue is derived. A comparative analysis with the one-window case reveals that due to the additional possibility of the regulating energy spectrum the anticrossing structure builds up as a function of the inner radius with its sharpness increasing for the larger outer radius. Mathematical and physical interpretation of the obtained results is presented; namely, it is derived that the anticrossings are accompanied by the drastic changes of the wave function localization. Parallels are drawn to the other structures exhibiting similar phenomena; in particular, it is proved that, contrary to the two-dimensional geometry, at the critical Neumann radii true bound states exist.Comment: 25 pages, 7 figure

Propagation of axions in a strongly magnetized medium

The polarization operator of an axion in a degenerate gas of electrons occupying the ground-state Landau level in a superstrong magnetic field $H\gg H_0=m_e^2c^3/e\hbar =4.41\cdot 10^{13}$ G is investigated in a model with a tree-level axion-electron coupling. It is shown that a dynamic axion mass, which can fall within the allowed range of values $(10^{-5} eV \lesssim m_a\lesssim 10^{-2} eV)$, is generated under the conditions of strongly magnetized neutron stars. As a result, the dispersion relation for axions is appreciably different from that in a vacuum.Comment: RevTex, no figures, 13 pages, Revised version of the paper published in J. Exp. Theor. Phys. {\bf 88}, 1 (1999

Asymptotic behaviour of the spectrum of a waveguide with distant perturbations

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator

On the motion of a heavy rigid body in an ideal fluid with circulation

Chaplygin's equations describing the planar motion of a rigid body in an unbounded volume of an ideal fluid involved in a circular flow around the body are considered. Hamiltonian structures, new integrable cases, and partial solutions are revealed, and their stability is examined. The problems of non-integrability of the equations of motion because of a chaotic behavior of the system are discussed.Comment: 25 pages, 4 figure

On the full, strongly exceptional collections on toric varieties with Picard number three

We investigate full strongly exceptional collections on smooth, com- plete toric varieties. We obtain explicit results for a large family of varieties with Picard number three, containing many of the families already known. We also describe the relations between the collections and the split of the push forward of the trivial line bundle by the toric Frobenius morphism