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.     

Formalization of the “Twin Paradox” for Non-uniformly Accelerated Motions

The use of hyperbolic numbers for studying space-time geometry and trigonometry is extended for demonstrating the Frenet’s formulas in space-time. By means of this introduction the twin paradox for non-uniformly accelerated motions is formalized in a straightforward way.     

Non-linear Stark effect and molecular localization

We consider a non-linear Stark effect as a model for localization and symmetry breaking of a molecule in a gas. By a comparison method with respect to the linear Stark effect, we prove the existence of level bifurcation and symmetry breaking at a critical value of the gas pressure exponentially small for large nuclear masses. Extending the Davies results, we confirm the predictions of Claverie-Jona Lasinio for pyramidal molecules as the ammonia one.     

Exterior products of forms and the cohomology ring of a complex space

In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology.     

Luminescence Properties of Eu3+ Complexes of Tripode and Tetrapode Ligands Containing 2,2′-Bipyridine Units

The Eu³⁺ complexes of the tripode and tetrapode ligands 1 and 2 , respectively, containing 2,2′-bipyridine coordinating units have been prepared. The UV absorption and luminescence spectra, lifetimes, and quantum yields have been measured under a variety of experimental conditions. The contributions of different paths to the decay of the luminescent excited state are evaluated, and the structures of the complexes are discussed on the basis of spectroscopic and photophysical data.     

Theory of thin-film, narrowband, linear-polarization rejection filters with superlattice structure

A device to be obtained by depositing a columnar thin-film on a transparent substrate decorated periodically by an array of rectangular grooves is proposed as a narrowband, linear-polarization rejection filter. The rigorous coupled-wave approach is harnessed to calculate the reflectances and transmittances. High-quality filters of chosen materials and with less than 2 nm bandwidths can be designed with a vapor flux angle for deposition of the columnar thin-film as the controlling parameter.