Riemannian Submersions with Discrete Spectrum

We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the spectrum of the Laplacian of the total space is discrete if and only if the spectrum of the Laplacian of the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.

Anderson-like Transition for a Class of Random Sparse Models in d >= 2 Dimensions

We show that the Kronecker sum of d >= 2 copies of a random one-dimensional sparse model displays a spectral transition of the type predicted by Anderson, from absolutely continuous around the center of the band to pure point around the boundaries. Possible applications to physics and open problems are discussed briefly.

Progress in the mathematical theory of quantum disordered systems

We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition, including some joint work with Marchetti, the (quantum and classical) Edwards-Anderson (EA) spin-glass model and return to equilibrium for a class of spin-glass models, which includes the EA model initially in a very large transverse magnetic field. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4770066]

Coherent states and related quantizations for unbounded motions

We discuss the construction of coherent states (CS) for systems with continuous spectra. First, we propose to adopt the Malkin-Manko approach, developed for systems with discrete spectra, to the case under consideration. Following this approach, we consider two examples, a free particle and a particle in a linear potential. Second, we generalize the approach of action-angle CS to systems with continuous spectra. In the first approach we start with a well-defined quantum formulation (canonical quantization) of a physical system and the construction of CS follows from such a quantization. In the second approach, the quantization procedure is inherent to the CS construction itself.

Occurrence of bipolar spectrum disorder and comorbidities in women with eating disorders

Abstract Background Eating disorder (ED) patients often have comorbidities with other psychiatric disorders, especially with mood disorders. Although recent studies suggest an intimate relationship between ED and bipolar disorder (BD), the study on a broader bipolar spectrum definition has not been done in this population. We aimed to study the occurrence of bipolar spectrum (BS) and comorbidities in eating disorder patients of a tertiary service provider. Methods Sixty-nine female patients diagnosed with anorexia nervosa, bulimia nervosa, or eating disorder not otherwise specified were evaluated. The assessment comprised the Structured Clinical Interview for DSM-IV Axis I Disorders (SCID-I), clinical criteria for diagnosis of the Zurich bipolar spectrum. Mann–Whitney tests compared means of continuous variables. The association between categorical variables and the groups was described using contingency tables and analyzed using the chi-square or Fisher's exact test. The level of significance alpha was set at 5%. Results The results showed that 68.1% of patients had comorbidity with bipolar spectrum, and this was associated with higher family income, proportion of married people, and comorbidity with substance use. The ED with BS group showed higher rates of substance use comorbidity (40.4%) than the ED without BS group (13.6%). Discussion These results showed that the bipolar spectrum is a common comorbidity in patients with eating disorders and is associated with correlates of clinical importance, notably the comorbidity with substance use. Due to the pattern of similarity between the groups with and without comorbid bipolar spectrum in relation to various outcomes evaluated, the identification of comorbidity can be difficult. However, the precise diagnosis and careful identification of clinical correlates may contribute to future advances in treating these conditions. Further studies are necessary to evaluate the association of other clinical correlates and its possible causal association.