The Spectrum of the Cubic Oscillator

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian, $$\begin{array}{ll}H(\beta)=-\frac{d^2}{dx^2}+x^2+i\sqrt{\beta}x^3\end{array}$$ , for β in the cut plane ${\mathcal{C}_c:=\mathcal{C}\backslash \mathcal{R}_-}$ . Moreover, we prove that the spectrum consists of the perturbative eigenvalues {E _n (β)} _{n ≥ 0} labeled by the constant number n of nodes of the corresponding eigenfunctions. In addition, for all ${\beta \in \mathcal{C}_c, E_n(\beta)}$ can be computed as the Stieltjes-Padé sum of its perturbation series at β = 0. This also gives an alternative proof of the fact that the spectrum of H(β) is real when β is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case.     

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive decomposition of the set of coordinates into a dimension tree gives a hierarchy of nested subspaces and corresponding nested bases. The dimensions of these subspaces yield a notion of multilinear rank. This rank tuple, as well as quasi-optimal low-rank approximations by rank truncation, can be obtained by a hierarchical singular value decomposition. For fixed multilinear ranks, the storage and operation complexity of these hierarchical representations scale only linearly in the order of the tensor. As in the matrix case, the set of hierarchical tensors of a given multilinear rank is not a convex set, but forms an open smooth manifold. A number of techniques for the computation of hierarchical low-rank approximations have been developed, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations. This article gives a survey of these developments. We also discuss applications to problems in uncertainty quantification, to the solution of the electronic Schrödinger equation in the strongly correlated regime, and to the computation of metastable states in molecular dynamics.     

Theoretical study of the reaction 2,5-dimethylfuran + H → products

The reaction of 2,5-dimethylfuran (DMF) with H-atoms was studied using a potential energy surface calculated at the CBS-QB3 level of theory and master equation/RRKM modeling. Hydrogen abstraction by H-atom and hydrogen additions on DMF were considered. As the decomposition pathways of the initial adducts were unknown, a large number of decomposition routes was explored for these adducts. An important number of interconnected product channels were found and preliminary master equation calculations were performed to select the crucial wells and exit channels. The ipso substitution DMF + H → methylfuran (MF) + CH₃ and the formation of 1,3-butadiene and acetyl radical (CH₃CO) were found to be the major product channels in the addition process. The total calculated rate constant was found in good agreement with experimental data and is nearly pressure-independent. A small sensitivity to pressure was found for the computed branching ratios. At 1 bar, the yields of the two product channels of the addition process are maximal at 1100 K with computed branching ratios of 39% (MF + CH₃) and 27% (1,3-C₄H₆ + CH₃CO). Above 1300 K, hydrogen abstraction by H-atom becomes dominant and reaches a branching ratio of 56% at 2000 K.     

Polarization control of the nonlinear emission of semiconductor microcavities

The degree of circular polarization ( Weierstrass p ) of the nonlinear emission in semiconductor microcavities is controlled by changing the exciton-cavity detuning. The polariton relaxation towards K approximately 0 cavitylike states is governed by final-state stimulated scattering. The helicity of the emission is selected due to the lifting of the degeneracy of the +/-1 spin levels at K approximately 0. At short times after a pulsed excitation Weierstrass p reaches very large values, either positive or negative, as a result of stimulated scattering to the spin level of lowest energy (+1/-1 spin for positive/negative detuning).     

Interplay of Antiferromagnetic Coupling and Spin Crossover in Dinuclear Iron(II) Complexes

This article reports on the study of the interplay between magnetic coupling and spin transition in 2,2′-bipyrimidine (bpym)-bridged iron(II) dinuclear compounds. Coexistence of both phenomena has been observed in [Fe(bpym)(NCS)₂]₂bpym, [Fe(bpym)(NCSe)₂]₂bpym and [Fe(bt)(NCS)₂]₂bpym (bpym = 2,2′-bipyrimidine, bt = 2,2′-bithiazoline) by the action of external physical factors namely pressure or electromagnetic radiation. Competition between magnetic exchange and spin crossover has been studied in [Fe(bpym)(NCS)₂]₂bpym at 6.3 kbar. LIESST experiments carried out in [Fe(bpym)(NCSe)₂]₂bpym and [Fe(bt)(NCS)₂]₂bpym at 4.2 K have shown that is possible to achieve dinuclear molecules with different spin states in this class of compounds. This revised version was published online in August 2006 with corrections to the Cover Date.     

Nearly zero ultraflattened dispersion in photonic crystal fibers

We present a procedure for achieving photonic crystal fibers with nearly zero ultraflattened group-velocity dispersion. Systematic knowledge of the special guiding properties of these fibers permits the achievement of qualitatively novel dispersion curves. Unlike the behavior of conventional fibers, this new type of dispersion behavior permits remarkably improved suppression of third-order dispersion, particularly in the low-dispersion domain.     

MULTIQ: A Queueing Model for FMSs with Several Pallet Types

Analytical modelling of the work flow through flexible manufacturing systems (FMSs), based on closed queueing network models, has been successfully applied to the early stages of design and analysis of FMSs. This paper describes the advantages of using multiple job-class closed queueing networks for modelling realistic situations occurring in FMSs. The general modelling of FMSs by closed queueing networks is first reviewed. The way Solberg's CAN-Q—a single job-class queueing-based package—deals with several part types is clarified. A new model called MULTIQ, allowing multiple pallet types, each of which is used by several part types, is proposed. Results are derived using the data from an existing FMS. The use of the MULTIQ model for optimization purposes is suggested by some examples.     

Towards relativistic ECP / DFT description of chemical bonding in E112 compounds: spin-orbit and correlation effects in E112X versus HgX (X=H, Au)

The relativistic effective core potential (RECP) approach combined with the spin-orbit DFT electron correlation treatment was applied to the study of the bonding of eka-mercury (E112) and mercury with hydrogen and gold atoms. Highly accurate small-core shape-consistent RECPs derived from Hartree-Fock-Dirac-Breit atomic calculations with Fermi nuclear model were employed. The accuracy of the DFT correlation treatment was checked by comparing the results in the scalar-relativistic (spin-orbit-free) limit with those of high level scalar-relativistic correlation calculations within the same RECP model. E112H was predicted to be slightly more stable than its lighter homologue (HgH). The E112-Au bond energy is expected to be ca. 25–30 % weaker than that of Hg-Au. The role of correlations and magnetic (spin-dependent) interactions in E112-X and Hg-X (X=H, Au) bonding is discussed. The present computational procedure can be readily applied to much larger systems and seems to be a promising tool for simulating E112 adsorption on metal surfaces.     

The additive property of the inconsistency degree in intertemporal decision making through the generalization of psychophysical laws

Intertemporal decision making involves choices among options whose effects occur at different moments. These choices are influenced not only by the effect of reward value perception at different moments, but also by the time perception effect. One of the main difficulties that affect standard experiments involving intertemporal choices is the simultaneity of both effects on time discounting. In this paper, we unify the psychophysical laws and discount value functions using the one-parameter exponential and logarithmic functions from nonextensive statistical mechanics. Also, we propose to measure the degree of inconsistency. This quantity allow us to discriminate both effects of time and value perception on discounting process and, by integration, obtain other main quantities like impulsivity and discount functions.