An invariant analytic orthonormalization procedure with an application to coherent states

We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators $A_j$, $j=1,2,...,n$, starting from a fixed normalized vector in \Hil and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of {\em coherent-like} vectors can be produced and in which condition over the lattice spacing this can be done

Some invariant biorthogonal sets with an application to coherent states

We show how to construct, out of a certain basis invariant under the action of one or more unitary operators, a second biorthogonal set with similar properties. In particular, we discuss conditions for this new set to be also a basis of the Hilbert space, and we apply the procedure to coherent states. We conclude the paper considering a simple application of our construction to pseudo-hermitian quantum mechanics.Comment: in press in Journal of Mathematical Analysis and Applications. arXiv admin note: text overlap with arXiv:0904.088

Coordinate representation for non Hermitian position and momentum operators

In this paper we undertake an analysis of the eigenstates of two non self-adjoint operators $\hat q$ and $\hat p$ similar, in a suitable sense, to the self-adjoint position and momentum operators $\hat q_0$ and $\hat p_0$ usually adopted in ordinary quantum mechanics. In particular we discuss conditions for these eigenstates to be {\em biorthogonal distributions}, and we discuss few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with $\hat q$ and $\hat p$, based on the so-called {\em quasi *-algebras}.Comment: Accepted in Proceedings of the Royal Society

Local spectral theory for r and s satisfying rnsrn = rj

In this paper, we analyze local spectral properties of operators R, S and RS which satisfy the operator equations RnSRn = Rj and Sn RSn = Sj for same integers j 65 n 65 0. We also continue to study the relationship between the local spectral properties of an operator R and the local spectral properties of S. Thus, we investigate the transmission of some local spectral properties from R to S and we illustrate our results with an example. The theory is exemplified in some cases

Diurnal habitat suitability for a Mediterranean steppeland bird, identified by Ecological Niche Factor Analysis

Context. The negative effects of agricultural intensification and policies, use of pesticides, fertilisers and mechanised harvesting on several populations of pseudo-steppe birds have increasingly required more detailed and effective habitat suitability models. Distribution models of farmland species are prone to incur recordings of false absence data. Ecological Niche Factor Analysis (ENFA) allows identification of environmental predictors of species distribution by using presence data only. Aims. We quantified the diurnal habitat preferences and niche width of one steppe species, the stone curlew (Burhinus oedicnemus), with unfavourable conservations status in a Mediterranean area and reclassified a map with respect to habitat suitability classes according to the resulting distribution model. Methods. Ecological Niche Factor Analysis was used with GIS cartography customised with habitat and anthropogenic variables recorded during field surveys carried out in four study plots (~500 ha) and at different spatial scales. Key results. The stone curlew selected areas with low vegetation cover, such as fields following artichoke harvesting and tillage, close to rural buildings and unpaved roads. In contrast, the stone curlew avoided areas with high vegetation cover and areas highly disturbed by human-induced fires. The occurrence of natural vegetation was neither preferred nor avoided. The most robust model was based on a large-scale analysis (200mfrom the bird location points), according to which the optimal area for stone curlew distribution during its breeding season was restricted to 1% of the entire study area. Conclusions. Two uncorrelated factors, ‘marginality’ and ‘tolerance’, described the stone curlew’s niche in the area. The first index indicated selection for habitats that were marginal with respect to those available in the area, whereas the second indicated a species with a medium–wide environmental niche. In particular, the stone curlew occupied a much more restricted niche (low tolerance) in relation to individual variables. The use of customised databases at a large scale of analysis was found to more effectively reveal ecological requirements of this marginal and specialised species. Implications. Our results allowed us to indicate practical land management actions for the stone curlew, such as prevention of human-induced fires and increase of pastoral activities. Our results indicated a potentially positive role of littledisturbed service roads along rural buildings in stone curlew distribution, which warrants further research. In addition, studies are needed to verify the presence of an ecological trap in artichoke fields, their preferred habitat. As we showed for the stone curlew, niche analyses conducted at a large scale using customised databases could greatly improve habitat suitability models of farmland species

SVEP and local spectral radius formula for unbounded operators

In this paper we study the localized single valued extension property for an unbounded operator T. Moreover, we provide sufficient conditions for which the formula of the local spectral radius holds for these operators

Maximal extensions of a linear functional

Extensions of a positive hermitian linear functional ω, defined on a dense *-subalgebra A0 of a topological *-algebra A[τ] are analyzed. It turns out that their maximal extensions as linear functionals or hermitian linear functionals are everywhere defined. The situation however changes deeply if one looks for positive extensions. The case of fully positive and widely positive extensions considered in [2] is revisited from this point of view. Examples mostly taken from the theory of integration are discussed