Spherical Tuples of Hilbert Space Operators

We introduce and study a class of operator tuples in complex Hilbert spaces, which we call spherical tuples. In particular, we characterize spherical multi-shifts, and more generally, multiplication tuples on RKHS. We further use these characterizations to describe various spectral parts including the Taylor spectrum. We also find a criterion for the Schatten $S_p$-class membership of cross-commutators of spherical $m$-shifts. We show, in particular, that cross-commutators of non-compact spherical $m$-shifts cannot belong to $S_p$ for $p \le m$. We specialize our results to some well-studied classes of multi-shifts. We prove that the cross-commutators of a spherical joint $m$-shift, which is a $q$-isometry or a $2$-expansion, belongs to $S_p$ if and only if $p > m$. We further give an example of a spherical jointly hyponormal $2$-shift, for which the cross-commutators are compact but not in $S_p$ for any $p <\infty$.Comment: a version close to final on

Grating-coupled excitation of multiple surface plasmon-polariton waves

The excitation of multiple surface-plasmon-polariton (SPP) waves of different linear polarization states and phase speeds by a surface-relief grating formed by a metal and a rugate filter, both of finite thickness, was studied theoretically, using rigorous coupled-wave-analysis. The incident plane wave can be either p or s polarized. The excitation of SPP waves is indicated by the presence of those peaks in the plots of absorbance vs. the incidence angle that are independent of the thickness of the rugate filter. The absorbance peaks representing the excitation of s-polarized SPP waves are narrower than those representing p-polarized SPP waves. Two incident plane waves propagating in different directions may excite the same SPP wave. A line source could excite several SPP waves simultaneously

Suboptimal quantum-error-correcting procedure based on semidefinite programming

In this paper, we consider a simplified error-correcting problem: for a fixed encoding process, to find a cascade connected quantum channel such that the worst fidelity between the input and the output becomes maximum. With the use of the one-to-one parametrization of quantum channels, a procedure finding a suboptimal error-correcting channel based on a semidefinite programming is proposed. The effectiveness of our method is verified by an example of the bit-flip channel decoding.Comment: 6 pages, no figure, Some notations differ from those in the PRA versio

Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations

We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure

Ab initio study of alanine polypeptide chains twisting

We have investigated the potential energy surfaces for alanine chains consisting of three and six amino acids. For these molecules we have calculated potential energy surfaces as a function of the Ramachandran angles Phi and Psi, which are widely used for the characterization of the polypeptide chains. These particular degrees of freedom are essential for the characterization of proteins folding process. Calculations have been carried out within ab initio theoretical framework based on the density functional theory and accounting for all the electrons in the system. We have determined stable conformations and calculated the energy barriers for transitions between them. Using a thermodynamic approach, we have estimated the times of characteristic transitions between these conformations. The results of our calculations have been compared with those obtained by other theoretical methods and with the available experimental data extracted from the Protein Data Base. This comparison demonstrates a reasonable correspondence of the most prominent minima on the calculated potential energy surfaces to the experimentally measured angles Phi and Psi for alanine chains appearing in native proteins. We have also investigated the influence of the secondary structure of polypeptide chains on the formation of the potential energy landscape. This analysis has been performed for the sheet and the helix conformations of chains of six amino acids.Comment: 24 pages, 10 figure

Surface-plasmon-polariton wave propagation guided by a metal slab in a sculptured nematic thin film

Surface-plasmon-polariton~(SPP) wave propagation guided by a metal slab in a periodically nonhomogeneous sculptured nematic thin film~(SNTF) was studied theoretically. The morphologically significant planes of the SNTF on both sides of the metal slab could either be aligned or twisted with respect to each other. The canonical boundary-value problem was formulated, solved for SPP-wave propagation, and examined to determine the effect of slab thickness on the multiplicity and the spatial profiles of SPP waves. Decrease in slab thickness was found to result in more intense coupling of two metal/SNTF interfaces. But when the metal slab becomes thicker, the coupling between the two interfaces reduces and SPP waves localize to one of the two interfaces. The greater the coupling between the two metal/SNTF interfaces, the smaller is the phase speed.Comment: 17 page

Central factorials under the Kontorovich-Lebedev transform of polynomials

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.Comment: also available at http://cmup.fc.up.pt/cmup/ since the 2nd August 201

Upper bounds and asymptotic expansion for Macdonald's function and the summability of the Kontorovich-Lebedev integrals

Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function $K_{i\tau}(x)$. The results can be applied, for instance, to study the summability of the divergent Kontorovich-Lebedev integrals in the sense of Jones. Namely, we answer affirmatively a question (cf. [6]) whether these integrals converge for even entire functions of the exponential type in a weak sense