Reduced Gutzwiller formula with symmetry: case of a finite group

We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a finite group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_\chi$ to the symmetry subspace $L^2\_\chi(\mathbb{R}^d)$ of $L^2(\mathbb{R}^d)$ coming from the decomposition of Peter-Weyl. We give reduced semi-classical asymptotics of a regularised spectral density describing the spectrum of $\hat{H}\_\chi$ near a non critical energy $E\in\mathbb{R}$. If $\Sigma\_E:=\{H=E \}$ is compact, assuming that periodic orbits are non-degenerate in $\Sigma\_E/G$, we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space $\Sigma\_E/G$ appear. The method is based upon the use of coherent states, whose propagation was given in the work of M. Combescure and D. Robert.Comment: 20 page

Nanomaterials for the cleaning and pH adjustment of vegetable tanned leather

Leather artifacts in historical collections and archives are often contaminated by physical changes such as soiling, which alter their appearance and readability, and by chemical changes which occur on ageing and give rise to excessive proportion of acids that promote hydrolysis of collagen, eventually leading to gelatinization and loss of mechanical properties. However, both cleaning and pH adjustment of vegetable tanned leather pose a great challenge for conservators, owing to the sensitivity of these materials to the action of solvents, especially water-based formulations and alkaline chemicals. In this study the cleaning of historical leather samples was optimized by confining an oil-in-water (o/w) nanostructured fluid in a highly retentive chemical hydrogel, which allows the controlled release of the cleaning fluid on sensitive surfaces. The chemical gel exhibits optimal viscoelasticity, which facilitates its removal after the application without leaving residues on the object. Nanoparticles of calcium hydroxide and lactate, dispersed in 2-propanol, were used to adjust the pH up to the natural value of leather, preventing too high alkalinity which causes swelling of fibers and denaturation of the collagen. The treated samples were characterized using Scanning Electron Microscopy (FE SEM), controlled environment dynamic mechanical analysis (DMA-RH), and infrared spectroscopy (ATR-FTIR). The analytical assessment validated the use of tools derived from colloid and materials science for the preservation of collagen-based artifacts