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

Characterisation of flow dynamics within and around an isolated forest, through measurements and numerical simulations

The case study of 'Bosco Fontana', a densely-vegetated forest located in the north of Italy, is analysed both experimentally and numerically to characterise the internal ventilation of a finite forest with a vertically non-homogeneous canopy. Measurements allow for the evaluation of the turbulent exchange across the forest canopy. The case study is then reproduced numerically via a two-dimensional RANS simulation, successfully validated against experimental data. The analysis of the internal ventilation leads to the identification of seven regions of motion along the predominate-wind direction, for whose definition a new in-canopy stability parameter was introduced. In the vertical direction, the non-homogeneity of the canopy leads to the separation of the canopy layer into an upper foliage layer and a lower bush layer, characterised respectively by an increasing streamwise velocity and turbulence intensity, and a weak backflow. The conclusions report an improved description of the dynamic layer and regions of motion presented in the literature