The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit

In the present paper we study the following scaled nonlinear Schr\"odinger equation (NLS) in one space dimension: $$i\frac{d}{dt} \psi^{\varepsilon}(t) =-\Delta\psi^{\varepsilon}(t) + \frac{1}{\epsilon}V\left(\frac{x}{\epsilon}\right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t) \quad \quad \epsilon>0\ ,\quad V\in L^1(\mathbb{R},(1+|x|)dx) \cap L^\infty(\mathbb{R}) \ .$$ This equation represents a nonlinear Schr\"odinger equation with a spatially concentrated nonlinearity. We show that in the limit $\epsilon\to 0$, the weak (integral) dynamics converges in $H^1(\mathbb{R})$ to the weak dynamics of the NLS with point-concentrated nonlinearity: $$i\frac{d}{dt} \psi(t) =H_{\alpha}\psi(t) .$$ where $H_{\alpha}$ is the laplacian with the nonlinear boundary condition at the origin $\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)$ and $\alpha=\int_{\mathbb{R}}Vdx$. The convergence occurs for every $\mu\in \mathbb{R}^+$ if $V \geq 0$ and for every $\mu\in (0,1)$ otherwise. The same result holds true for a nonlinearity with an arbitrary number $N$ of concentration pointsComment: 10 page

Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on S2 . Precisely, local well-posedness is proved for any C 2 power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas

A Quantum Model of Feshbach Resonances

We consider a quantum model of two-channel scattering to describe the mechanism of a Feshbach resonance. We perform a rigorous analysis in order to count and localize the energy resonances in the perturbative regime, i.e., for small inter-channel coupling, and in the non-perturbative one. We provide an expansion of the effective scattering length near the resonances, via a detailed study of an effective Lippmann-Schwinger equation with energy-dependent potential.Comment: 29 pages, pdfLaTe

Exploring the role of blockchain technology in modern high-value food supply chains: global trends and future research directions

Trust, safety, and quality are among the most important factors in the agri-food supply chains. Traceability is a powerful tool to ensure them, but implementing a transparent and effective system is a complex operation. As a result, innovative systems, like blockchain, could be introduced. Although research on its impacts in the agri-food is recent, the literature appears fragmented. The objective is to investigate the studied aspects of the blockchain adoption in agri-food, with the purpose of retrieving meaningful considerations about the current state of the art about strategic high-value supply chains, such as wine and olive oil, particularly subjected to fraudulent behaviors. A productivity measurement was applied to retrieve the evolution of the number of documents through the years, the most productive countries, the sources, the research areas, and the most significant papers in terms of number of citations received. To understand the research trends, a co-occurrence analysis was employed. Results show that most of the existing studies focus on the role of blockchain in the resolution of some critical issues as food safety and frauds. While wine is currently an emerging sector in which this approach can be implemented, olive oil still needs more attention. In both cases, blockchain could potentially help to support the profitability and sustainability of the production. The research underlines the importance of focusing on the environmental and social dimension of the blockchain phenomenon and the use of technology to improve the efficiency of agri-food chains and reduce waste and resource use