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We consider ground states of $L^2$-subcritical nonlinear Schr\"{o}dinger equation (1.1), which can be described equivalently by minimizers of the following constraint minimization problem $$ e(\rho):=\inf\{E_{\rho}(u):u\in \mathcal{H}(\mathbb{R}^d),\|u\|_2^2=1\}.$$ The energy functional $E_{\rho}(u)$ is defined by $$ E_{\rho}(u):=\frac{1}{2}\int_{\mathbb{R}^d}|\nabla u|^2dx +\frac{1}{2}\int_{\mathbb{R}^d}V(x)|u|^2dx-\frac{\rho^{p-1}}{p+1}\int_{\mathbb{R}^d}|u|^{p+1}dx,$$ where $d\geq1$, $\rho>0$, $p\in\big(1, 1+\frac{4}{d}\big)$ and $0\leq V(x)\to\infty$ as $|x| \to\infty$. We present a detailed analysis on the concentration behavior of ground states as $\rho\to\infty$, which extends the concentration results shown in [22]. Moreover, the uniqueness of nonnegative ground states is also proved when $\rho$ is large enough.

Electron beam probe (EBP) is a new principle detector, which makes use of a low-intensity and low-energy electron beam to measure the transverse profile, bunch shape, beam neutralization and beam wake field of an intense beam with small dimensions. While can be applied to many aspects, we limit our analysis to beam distribution reconstruction. This kind of detector is almost non-interceptive for all of the beam and does not disturb the machine environment. In this paper, we present the theoretical aspects behind this technique for beam distribution measurement and some simulation results of the detector involved. First, a method to obtain parallel electron beam is introduced and a simulation code is developed. And then, EBP as a profile monitor for dense beam is simulated using fast scan method under various target beam profile, such as KV distribution, waterbag distribution, parabolic distribution, Gaussian distribution and halo distribution. Profile reconstruction from the deflected electron beam trajectory is implemented and compared with the actual one, and an expected agreement is achieved. Furthermore, Instead of fast scan, a slow scan, i.e. step-by-step scan, is considered, which lows the requirement for hardware, i.e. Radio Frequency deflector. we calculate the three dimensional electric field of Gaussian distribution and simulate the electron motion under this field. In addition, fast scan along the target beam direction and slow scan across the beam is also presented, and can provide a measurement of longitudinal distribution as well as transverse profile simultaneously. Final, simulation results for China Accelerator Driven Sub-critical System (CADS) and High Intensity Heavy Ion Accelerator Facility (HIAF) are given to investigate the quantitative behavior of EBP., Comment: 13 pages, 21 figures