Bernard M. de Dormale

Substrate‐related effects on the optical behavior of a granular surface: The Maxwell Garnett theory revisited

An approximate treatment, describing the influence of a dielectric substrate on the optical behavior of a granular surface, is reported. It shows that discrepancies between experimental results and predictions mainly based upon the Maxwell Garnett theory [Philos. Trans. R. Soc. London 203, 385 (1904); 205A, 237 (1906)] cannot be interpreted as substrate‐related effects. The magnitudes and locations of the multiple images of the unperturbed dipole of a small metallic sphere have been carried out in an approximate, though reliable, way in the presence of polarizing fields both parallel and perpendicular to the substrate. The resulting dipole is introduced in a long‐known optical model, describing a granular surface as a planar array of equal dipoles interacting with each other. Graphical results, showing the influence of substrates of various dielectric constants, are presented. A discussion of possible improvements of the available model is also reported.

Spectral representation and decomposition of self‐adjoint operators

This paper is an extension of results established by Jauch and Misra [Helv. Physica Acta 38, 30 (1965)] concerning finite or countable sets of commuting self‐adjoint operators. We have obtained the following results: let A={Ai}i‐I be a set of commuting self‐adjoint operators on a separable Hilbert space H. Then (i) for any I (possibly noncountable), there exists a spectral representation for A iff A″ is maximal Abelian. (ii) If I is finite or countable, ‐i‐I,n‐N D (Ani) is dense in H. As a corollary of a theorem of Maurin [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 7, 471 (1959)], this implies the existence of a common complete set of generalized eigenvectors.

Perturbation theory for relativistic Lee model

Proof of convergence of perturbation series for the S matrix elements is given for the relativistic Lee model, subject to the validity of an hypothesis explained in the paper. A proof of isometry for the wave operators is also given.

Equation radiale de Schrödinger et complétude du système {H, L2, Lz}

This paper gives the relationship between the spectral analysis of the Hamiltonian in quantum mechanics and the Schrodinger radial equation, using for this a generalization of the Riesz–Lorch lemma...