A Trotter-Kato product formula for a class of non-autonomous evolution equations

In this article dedicated to Professor V. Lakshmikantham on the occasion of the celebration of his 84th birthday, we announce new results concerning the existence and various properties of an evolution system UA+B(t, s)(0 <= s <= t <= T) generated by the sum -(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing G(B) for the algebra of all linear bounded operators on B, we can express UA+B(t, s)(0 <= s <= t <= T) as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by -A(t) and -B(t), thereby getting a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D subset of boolean AND D-t epsilon[0,D-T](A(t)+B(t)) everywhere dense in B. We then mention several possible applications of our product formula to various classes of non-autonomous parabolic initial-boundary value problems, as well as to evolution problems of Schrodinger type related to the theory of time-dependent singular perturbations of self-adjoint operators in quantum mechanics. We defer all the proofs and all the details of the applications to a separate publication. (C) 2008 Elsevier Ltd. All rights reserved.

Dinuclear Azide-Bridged Copper(II) Complex as Building Block for the Assembly of a 2D-Supramolecular Array

A novel Schiff base-copper(II) complex [Cu(2)L(2)(N(3))(2)](ClO(4))(2) 1, where L = (4-imidazolyl)ethylene-2-amino-1-ethylpyridine (apyhist), containing azide-bridges between adjacent copper ions in a dinuclear arrangement was isolated and characterized both in the solid state and in solution by X-ray crystallography and different spectroscopic techniques. Azide binding constants were estimated from titrations of the precursor [CuL(H(2)O)(2)](2+) solutions with sodium azide, giving rise to the azido-bridged species, [Cu(2)L(2)(N(3))(2)](2+). Raman spectra showed asymmetric stretching band at 2060 cm(-1), indicating the presence of azido ligands with a symmetric mu(1,) (1) binding geometry. EPA spectra, in frozen methanol/water solutions at 77 K, exhibited characteristic features of copper centers in tetragonal pyramidal coordination geometry, exhibiting magnetic interactions between them. Further, in solid state, two different values for magnetic coupling in this species were obtained, J/k = -(5.14 +/- 0.02) cm(-1) attributed to the mu(1, 1) azide-bridge mode, and J`z`/k = -(2.94 +/- 0.11) cm(-1) for the interaction between dinuclear moieties via water/perchorate bridges. Finally, an attempt was made to correlate structure and magnetic data for this dinuclear asymmetric end-on azido bridged-copper(II) 1 complex with those of another correlated dinuclear system, complex [Cu(2)L(2)Cl(2)](ClO(4))(2) 2, containing the same tridentate diimine ligand, but with chloro-bridged groups between the copper centres.