The search for NLS ground states on a hybrid domain: Motivations, methods, and results

We discuss the problem of establishing the existence of the Ground States for the subcritical focusing Nonlinear Schr\"odinger energy on a domain made of a line and a plane intersecting at a point. The problem is physically motivated by the experimental realization of hybrid traps for Bose-Einstein Condensates, that are able to concentrate the system on structures close to the domain we consider. In fact, such a domain approximates the trap as the temperature approaches the absolute zero. The spirit of the paper is mainly pedagogical, so we focus on the formulation of the problem and on the explanation of the result, giving references for the technical points and for the proofs.Comment: Contribution for the Proceedings of the Summer School of the Puglia Trimester 2023 "Singularities, Asymptotics and Limiting Models". 18 pages, 1 figure. Keywords: hybrids, standing waves, nonlinear Schr\"odinger, ground states, delta interaction. Minor revisions have been made with respect to the previous versio

NLS ground states on a hybrid plane

We study the existence, the nonexistence, and the shape of the ground states of a Nonlinear Schr\"odinger Equation on a manifold called hybrid plane, that consists of a half-line whose origin is connected to a plane. The nonlinearity is of power type, focusing and subcritical. The energy is the sum of the Nonlinear Schr\"odinger energies with a contact interaction on the half-line and on the plane with an additional quadratic term that couples the two components. By ground state we mean every minimizer of the energy at a fixed mass. As a first result, we single out the following rule: a ground state exists if and only if the confinement near the junction is energetically more convenient than escaping at infinity along the halfline, while escaping through the plane is shown to be never convenient. The problem of existence reduces then to a competition with the one-dimensional solitons. By this criterion, we prove existence of ground states for large and small values of the mass. Moreover, we show that at given mass a ground state exists if one of the following conditions is satisfied: the interaction at the origin of the half-line is not too repulsive; the interaction at the origin of the plane is sufficiently attractive; the coupling between the half-line and the plane is strong enough. On the other hand, nonexistence holds if the contact interactions on the half-line and on the plane are repulsive enough and the coupling is not too strong. Finally, we provide qualitative features of ground states. In particular, we show that in the presence of coupling every ground state is supported both on the half-line and on the plane and each component has the shape of a ground state at its mass for the related Nonlinear Schr\"odinger energy with a suitable contact interaction. These are the first results for the Nonlinear Schr\"odinger Equation on a manifold of mixed dimensionality.Comment: 31 pages, 1 figure. Keywords: hybrids, standing waves, nonlinear Schr\"odinger, ground states, delta interaction, radially symmetric solutions, rearrangement

The search for NLS ground states on a hybrid domain: Motivations, methods, and results

We discuss the problem of establishing the existence of the Ground States for the subcritical focusing Nonlinear Schrödinger energy on a domain made of a line and a plane intersecting at a point. The problem is physically motivated by the experimental realization of hybrid traps for Bose-Einstein Condensates, that are able to concentrate the system on structures close to the domain we consider. In fact, such a domain approximates the trap as the temperature approaches the absolute zero. The spirit of the paper is mainly pedagogical, so we focus on the formulation of the problem and on the explanation of the result, giving references for the technical points and for the proofs

NLS Ground States on a Hybrid Plane

We study the existence, the nonexistence, and the shape of the ground states of a Nonlinear Schrödinger Equation on a manifold called hybrid plane, that consists of a half-line whose origin is connected to a plane. The nonlinearity is of power type, focusing and subcritical. The energy is the sum of the Nonlinear Schrödinger energies with a contact interaction on the half-line and on the plane with an additional quadratic term that couples the two components. By ground state we mean every minimizer of the energy at a fixed mass. As a first result, we single out the following rule: a ground state exists if and only if the confinement near the junction is energetically more convenient than escaping at infinity along the halfline, while escaping through the plane is shown to be never convenient. The problem of existence reduces then to a competition with the one-dimensional solitons. By this criterion, we prove existence of ground states for large and small values of the mass. Moreover, we show that at given mass a ground state exists if one of the following conditions is satisfied: the interaction at the origin of the half-line is not too repulsive; the interaction at the origin of the plane is sufficiently attractive; the coupling between the half-line and the plane is strong enough. On the other hand, nonexistence holds if the contact interactions on the half-line and on the plane are repulsive enough and the coupling is not too strong. Finally, we provide qualitative features of ground states. In particular, we show that in the presence of coupling every ground state is supported both on the half-line and on the plane and each component has the shape of a ground state at its mass for the related Nonlinear Schrödinger energy with a suitable contact interaction. These are the first results for the Nonlinear Schrödinger Equation on a manifold of mixed dimensionality

Learning from the COVID-19 pandemic in Italy to advance multi-hazard disaster risk management

COVID-19 challenged all national emergency management systems worldwide overlapping with other natural hazards. We framed a ‘parallel phases’ Disaster Risk Management (DRM) model to overcome the limitations of the existing models when dealing with complex multi-hazard risk conditions. We supported the limitations analysing Italian Red Cross data on past and ongoing emergencies including COVID-19 and we outlined three guidelines for advancing multi-hazard DRM: (i) exploiting the low emergency intensity of slow-onset hazards for preparedness actions; (ii) increasing the internal resources and making them available for international support; (iii) implementing multi-hazard seasonal impact-based forecasts to foster the planning of anticipatory actions

The Trend of CEACAM3 Blood Expression as Number Index of the CTCs in the Colorectal Cancer Perioperative Course

Pathological stage seems to be the major determinant of postoperative prognosis of solid tumors, but additional prognostic determinants need to be better investigated. The most important tumor marker for colorectal cancer (CRC) is the cell-surface antigen, Carcinoembryonic Antigen (CEA), and its assessment is considered a valuable index of circulating tumor cells (CTCs). In this paper, CEACAM3 evaluation was applied given its great specificity in the CRC. Whole blood from the basilic vein of 38 CRC patients was collected before and at various time intervals after the curative resection. Also, from 20 of them, we have obtained two additional intraoperative samples. CEACAM3 expression was evaluated in all the samples by RT-PCR. CEACAM3 duct values showed a decreasing trend from preoperative through early and later postoperative to 6th-month samples (p<0.001). The average values of CEACAM3 were related to the cancer size (T stage) (p=0.034) and WHO stage (p=0.035). A significant effect of the baseline value of CEACAM3 dCt on the temporal trend has been observed (p<0.001). In this study, we have demonstrated the CEACAM3 specificity and a perioperative trend of CTCs which is coherent with the clinical/pathological considerations and with previous experimental findings in different cancer types