La metrica di Agmon ed il decadimento esponenziale delle soluzioni di equazioni alle derivate parziali

The aim of this master thesis is to study the exponential decay of solutions of elliptic partial equations. This work is based on the results obtained by Agmon. To this purpose, first, we define the Agmon metric, that plays an important role in the study of exponential decay, because it is related to the rate of decay. Under some assumptions on the growth of the function and on the positivity of the quadratic form associated to the operator, a first result of exponential decay is presented. This result is then applied to show the exponential decay of eigenfunctions with eigenvalues whose real part lies below the bottom of the essential spectrum. Finally, three examples are given: the harmonic oscillator, the hydrogen atom and a Schrödinger operator with purely discrete spectrum.

Late Transition Metal-Oxo complexes: Synthesis, and Biorelevant Reactivity

High-valent terminal metal-oxygen adducts are supposed to be potent oxidising intermediates in enzymatic catalyses. In contrast to those from groups 6-8, oxidants that contain late transition metals (Co, Ni, Cu) are poorly understood. Because of their high reactivity, only a few examples of these compounds have been observed. The aim of this project was to investigate the reactivity of high-valent Ni(III) complexes, containing a monodentate oxygen-donor ligands, in hydrogen atom abstraction (HAA) and oxygen atom transfer (OAT) reactions which are typical of biological high-valent metal-oxygen species. Particularly, the Ni(III) complexes were generated in situ, at low temperature, from the oxidation of the Ni(II) species.The nickel complexes studied during this work were supported by tridentate ligands, with a strong σ-donating ability and exceedingly resistant to several common degradation pathways. These complexes vary based on the monodentate group in the fourth coordination position site, which can be neutral or anionic. In particular, we prepared four different Ni(III) complexes [NiIII(pyN2Me2)(OCO2H)] (12), [NiIII(pyN2Me2)(ONO2)] (14), [NiIII(pyN2Me2)(OC(O)CH3)] (18) and [NiIII(pyN2Me2)(OC(O)H)] (25). They feature a bicarbonate (-OCO2H), nitrate (-ONO2), acetate (-OC(O)CH3) and formate (-OC(O)H) group, respectively.HAA and OAT reactions were performed by adding 2,6-di-tert-butylphenol (2,6-DTBP) at -40°C, and triphenylphosphine (PPh3) at -80°C, to the in situ generated Ni(III) complexes, respectively. These reactions were carried out by adding 7 to 500 equivalents of substrate, in order to ensure pseudo-first order conditions. Since, the reactivity of the Ni(III) complex featured by the bicarbonate group has been studied in a previous work, we only investigated that of the species bearing the nitrate, acetate and formate ligand. Finally we compared the value of the reaction rate of all the four species in the HAA and OAT reactions.