Related papers: On the Thomas-Fermi ground state in a harmonic pot…
The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein-Weiss inequality, also known as a weighted…
We study analytically the existence and uniqueness of the ground state of the nonlinear Schr\"{o}dinger equation (NLSE) with a general power nonlinearity described by the power index $\sigma\ge0$. For the NLSE under a box or a harmonic…
We study the families of nonlinear modes described by the nonlinear Schr\"odinger equation with the PT-symmetric harmonic potential $x^2-2i\alpha x$. The found nonlinear modes display a number of interesting features. In particular, we have…
We study the ground-state of a Fermi gas with short range attrative interactions in one or two dimensions. N fermions are placed in a confining potential, and interact with each other through a negative potential, whose range is larger than…
We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy…
By incorporating the zero-point energy contribution we derive simple and accurate extensions of the usual Thomas-Fermi (TF) expressions for the ground-state properties of trapped Bose-Einstein condensates that remain valid for an arbitrary…
We prove existence and qualitative properties of ground state solutions to a generalized nonlocal 3rd-4th order Gross-Pitaevskii equation. Using a mountain pass argument on spheres and constructing appropriately localized Palais-Smale…
We mathematically and numerically study the ground states of unitary Fermi gases. Starting from the three-dimensional nonlinear Schr\"{o}dinger equation that contains a quantum pressure term and an angular momentum rotation term, we first…
We review the basic concepts of a non-equilibrium kinetic theory of a trapped bosonic gas. By extending the successful mean-field concept of the Gross-Pitaevskii equation with the effects of non-local, two particle quantum correlations, one…
We study the ground-state properties of a two-dimensional spin-polarized fluid of dipolar fermions within the Euler-Lagrange Fermi-hypernetted-chain approximation. Our method is based on the solution of a scattering Schr\"odinger equation…
We carry out a detailed examination of the ground state property of few-boson system in a one-dimensional hard wall potential with a $\delta -$ split in the center. In the Tonks-Girardeau limit with infinite repulsion between particles, we…
Nonlinear Schr\"odinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of…
We consider the nonlinear Schr{\"o}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a…
We establish the uniqueness of ground states of some coupled nonlinear Schrodinger systems in the whole space. We firstly use Schwartz symmetrization to obtain the existence of ground states for a more general case. To prove the uniqueness…
We are concerned with a system of coupled Schr\"odinger equations $$-\Delta u_i + V_i(x)u_i = \partial_{u_i}F(x,u)\hbox{ on }\mathbb{R}^N,\,i=1,2,...,K,$$ where $F$ and $V_i$ are periodic in $x$ and $0\notin \sigma(-\Delta+V_i)$ for…
In this paper we consider a non-linear Schroedinger equation with a cubic nonlinearity and a multi-dimensional double well potential. In the semiclassical limit the problem of the existence of stationary solutions simply reduces to the…
We investigate the existence of ground states for the focusing Nonlinear Schr\"odinger Equation on the infinite three-dimensional cubic grid. We extend the result found for the analogous two-dimensional grid by proving an appropriate…
This paper is devoted to studying the following nonlinear biharmonic Schr\"odinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} \Delta^{2}u-\lambda u=\mu|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\…
A path-integral method effective beyond the perturbation expansion approach is suggested to consider the quartic anharmonicity in different spatial dimensions. Due to an optimal representation of the partition function, the leading term has…
The bi-partite Gaussian state, corresponding to an anisotropic harmonic oscillator in a noncommutative-space, is investigated with the help of the Simon's separability condition (generalized Peres-Horodecki criterion). It turns out that, in…