Resonances and Gamow States in Non-Local Potentials

For a large class of non-local, non separable potentials with non-compact support, the solution of the radial integrodifferential equation may be reduced to the solution of a homogeneous linear integral equation of Fredholm type with a quadratically integrable kernel. In this way we derive expansions of the wave functions and the Green's function of the Schrödinger equation with a non-local potential in terms of bound states, resonant states and a continuum of scattering functions with complex wave number. The rules of normalization, orthogonality and completeness satisfied by the eigenstates of the Schrödinger equation belonging to complex eigenvalues with Im E_n < 0, (Gamow or resonant states) are also derived. Finally, by means of a realistic example, it is shown how to use these expansions to exhibit the resonant behaviour of the differential cross section. Explicit expressions for the transition amplitudes and the partial widths in terms of expectation values of operators computed with Gamow functions are given.     

Corrosion products on steel 20 in aerated hydrazine solutions: Mössbauer study

Composition of corrosion layers on steel 20 in aerated solutions with hydrazine concentrations less than 11 ppm was studied at 50, 60, and 80°C in dynamic conditions by transmission Mössbauer spectroscopy and X-ray diffraction as supplementary technique. Corrosion rates were determined by gravimetric method. A comparison with corrosion in water at 80°C was made. The observed layers have not any protective character. For 0.1 m/s linear velocity, they are composed by nonstoichiometric magnetite, (Fe_3?x O₄,x=0.02–0.04) with lepidocrocite (γ-FeOOH) as secondary phase at 50°C. Haematite (α-Fe₂O₃) is observed at 60 and 80°C with a 19 nm particle size. It becomes smaller for higher velocity (0.7 m/s).     

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH ⁻¹ are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92     

Extended X-ray absorption fine structure studies of thulium doped GaN epilayers

The local structure of Tm³⁺ ions incorporated into GaN epilayers was studied by means of Extended X-ray Absorption Fine Structure. The samples were doped either in situ during growth by Molecular Beam Epitaxy or by ion implantation of layers grown by Metal Organic Chemical Vapour Deposition. The implantation was done at ion energy of 300 keV and different nominal fluences of 3×10¹⁵, 4×10¹⁵ cm⁻² and 5×10¹⁵ cm⁻². The concentration of Tm in the samples studied was measured by Wavelength Dispersive X-ray analysis. For the in situ doped sample with concentration of 0.5%, and for all of the implanted samples, Tm was found on the Ga site in GaN. The ion implanted sample and an in situ doped sample with a similar concentration of Tm showed the same local structure, which suggests that the lattice site occupied by Tm does not depend on the doping method. When the average Tm concentration for in situ doped samples is increased to 1.2% and 2.0%, Tm is found to occupy the Ga substitutional site and the presence of a substantial number of Tm ions in the second coordination sphere indicates dopant clustering in the films. The formation of pure TmN clusters was found in an in situ doped sample with a dopant concentration of 3.4%.