The singular Hartree equation in fractional perturbed Sobolev spaces

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schr\"odinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.Comment: Published on Journal of Nonlinear Mathematical Physics (2018

Point-like perturbed fractional Laplacians through shrinking potentials of finite range

We reconstruct the rank-one, singular (point-like) perturbations of the $d$-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schr\"{o}dinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both the possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schr\"{o}dinger operators formed by a fractional Laplacian and a regular potential

Fractional powers and singular perturbations of quantum differential Hamiltonians

We consider the fractional powers of singular (point-like) perturbations of the Laplacian, and the singular perturbations of fractional powers of the Laplacian, and we compare such two constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schr\"{o}dinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated.Comment: Published on J. Math. Phys. (2018

On fractional powers of singular perturbations of the Laplacian

We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms

The Hypothesis of Superluminal Neutrinos: comparing OPERA with other Data

The OPERA Collaboration reported evidence for muonic neutrinos traveling slightly faster than light in vacuum. While waiting further checks from the experimental community, here we aim at exploring some theoretical consequences of the hypothesis that muonic neutrinos are superluminal, considering in particular the tachyonic and the Coleman-Glashow cases. We show that a tachyonic interpretation is not only hardly reconciled with OPERA data on energy dependence, but that it clashes with neutrino production from pion and with neutrino oscillations. A Coleman-Glashow superluminal neutrino beam would also have problems with pion decay kinematics for the OPERA setup; it could be easily reconciled with SN1987a data, but then it would be very problematic to account for neutrino oscillations.Comment: v1: 10 pages, 2 figures; v2: 12 pages, 2 figures, improved discussion of CG case as for pion decay and neutrino oscillations, added reference

Geomorphological analysis of the historic landslide of Sottrù (Badia Valley, Italy) reactivated on December 13, 2012

The historic landslide of Sottrù, located in the Badia Valley (Autonomous Province of Bolzano), was reactivated on 13 December 2012 after a long period of dormancy. According to historic documents, the previous activation dates back to June 1821. At that time, the slope was mobilised after a period of persistent rain and snow melt causing the damming of the Gadera stream. After snowmelting, in June 1827 the natural dam collapsed flooding meadows and villages down valley. In December 2012, the landslide was partly reactivated, showing the first slow movements on the 13th and more rapid displacements on the 14th. A detailed geomorphological survey of the landslide was carried out during summer 2014. A surficial highly permeable level (in-situ or reworked till) has been found laying on top of a thick sequence of clays, shales and marls, both over the landslide body and the surrounding areas. From the landslide scar uphill, the hydrographic pattern is extremely irregular, made of discontinuous drainage lines indicative of higher infiltration rate than superficial run off. The comparison between the current morphology and the pre-2012 one has not evidenced any movement within the historic source area, while relevant displacements occurred at two lateral zones whose dynamics caused the partial collapse of the central part of the landslide, originating a second, lower crown. Since the previous activation (1821), the landslide has been dormant for quite a long period, i.e. 191 years, and its last activation occurred during a season where landslides are usually not frequent. Temperature and precipitation time series, recorded at the nearby meteorological station of La Villa, have been analysed for the last 27 years. It has been found out that November 2014 had many rainy days, three of which with precipitation amounts ranging from 30 to 50 mm. The rainiest day (November 29) was suddenly followed by a dramatic drop below 0°C of maximum and minimum air temperatures, lasting until the 13th of December i.e., the landslide activation day. This probably caused the freezing of the soil which at that time was snow free and completely saturated. Moreover, it is possible that the 13-day period below 0°C enabled also the freezing of a spring located in the lower part of the slope nearby the village of Sottrù thus blocking groundwater drainage, leading to the onset of a confined-aquifer condition and, ultimately, to the rapid increase of pore water pressure inside the clay-rich landslide material. This hypothesis has been made also in consideration of similar events occurred in the Dolomites (e.g., in Zoldo Valley in 1991). However, in order to confirm this hypothesis, geophysical and geotechnical analysis would be needed

Visual Field Defect after Cardiac Surgery: The Striking Role of Interdisciplinary Collaboration

Perioperative visual loss (POVL) is a potentially devastating complication that can occur following ocular or nonocular surgery. The leading causes of this disease are retinal vascular occlusions, ischemic optic neuropathies, and cortical blindness. POVL pathogenesis is strictly influenced by surgery, anesthesia, and patients’ comorbidities. We report of a 55-year-old caucasian man who presented with complaints of sudden painless loss of vision and unilateral campimetric deficit. We recorded a preserved visual acuity but at fundus examination a bilateral ischemic optic neuropathy (ION) was suspected. Our hypothesis was supported by uncommon and peculiar visual field defects and a history of cardiovascular surgery shortly before was a striking data. When we examined his medical records we found strong accordance with what is reported in literature to be risk factors for postoperative ION development. He presented intraoperative hypotension, anemia, and hypothermia, he was older than 50 years, and surgery lasted for more than five hours. We are currently monitoring his visual acuity and visual fields which remain unchanged. As there is no proved therapy for such severe adverse events, we recommend intraoperative check of blood pressure, blood loss, and body temperature, associated with repeated eye checks and patients’ interview