A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space

We solve a bitangential interpolation problem for contractive multipliers on the Arveson space with an arbitrary interpolating set in the closed unit ball . The solvability criterion is established in terms of positive kernels. The set of all solutions is parametrized by a Redheffer transform. Submitted: February 2, 2002.     

Combined X-ray-electron imaging techniques: Limitations on lateral resolution

Capabilities of the imaging techniques, in which X-rays are converted to electrons and then the emitted electrons are registered by means of an electron microscope, are analyzed, the focus being on the factors limiting lateral resolution at the stage of electron emission. Bearing in mind the tendency to use harder synchrotron X-rays for some combined X-ray-electron microscopy methods, calculations were made for two significantly different X-ray energies: E = 1.828 keV (K-edge of Si) and E = 11.923 keV (L₃-edge of Au). By using Monte Carlo simulations of the electron trajectories beneath the surface of the sample we show that the radius of the spot from which photoelectrons are emitted could be as small as 1 nm. However, when proper account is taken of an entire electron cascade associated with the re-building of electron shells after photoelectron emission, spots more than one order of magnitude larger result, limiting the best lateral resolution to 20–30 nm.     

Lieb–Thirring Inequalities for Schrödinger Operators with Complex-valued Potentials

Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential.     

Mapping properties of wave and scattering operators for two-body Schrödinger operators

The modified wave and scattering operators are shown to be bounded between weighted L ²-spaces for two-body Schrödinger operators with long range potentials.     

Existence for a degenerate diffusion problem with a nonlinear operator

We study a degenerate nonlinear variational inequality which can be reduced to a multivalued inclusion by an appropriate change of the unknown function. We establish existence, uniqueness and regularity results. An application arising in the theory of water diffusion in porous media is discussed as an example.      

Continuity of the Restriction of C 0-Semigroups to Invariant Banach Subspaces

A linear semigroup in a Banach space induces a linear semigroup on a Banach space that can be continuously embedded in the former such that its image is invariant. This restriction need not be strongly continuous, although the original semigroup is strongly continuous. We show that norm or weak compactness of partial orbits is a necessary and sufficient condition for strong continuity of the restriction of a C₀-semigroup. We then show that if the embedded Banach space is reflexive and the norms of the restricted semigroup operators are bounded near the initial time, then the restricted semigroup is strongly continuous.     

The Remainder in Ruijsenaars’ Asymptotic Expansion of Barnes Double Gamma Function

We show that the remainder in Ruijsenaars’ asymptotic expansion of the logarithm of Barnes double gamma function gives rise to a completely monotone function. Fourier expansions of the multiple Bernoulli polynomials are also obtained. Research supported by the Carlsberg Foundation.