Related papers: On the Thomas-Fermi ground state in a harmonic pot…
A useful semiclassical method to calculate eigenfunctions of the Schroedinger equation is the mapping to a well-known ordinary differential equation, as for example Airy's equation. In this paper we generalize the mapping procedure to the…
Ground state of the energy-critical Gross-Pitaevskii equation with a harmonic potential can be constructed variationally. It exists in a finite interval of the eigenvalue parameter. The supremum norm of the ground state vanishes at one end…
We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schr\"odinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for…
We study the ground states of the extended Gross--Pitaevskii equation with the Lee--Huang--Yang correction from both theoretical and numerical perspectives. Starting from the three-dimensional model, we derive reduced one- and…
We investigate the excited stationary states of Bose-Einstein condensates trapped in harmonic potentials. We derive simple analytical approximations of the first few eigenstates of the associated time-independent one-dimensional…
We justify the Thomas--Fermi approximation for the elliptic problem with the repulsive nonlinear confinement used in the recent physical literature. The method is based on the resolvent estimates and the fixed-point iterations.
We present a suite of programs to determine the ground state of the time-independent Gross-Pitaevskii equation, used in the simulation of Bose-Einstein condensates. The calculation is based on the Optimal Damping Algorithm, ensuring a fast…
We study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground state energy and chemical potential, and obtain simple…
We note that the Thomas Fermi limit of Gross Pitaevskii equation and $N>>1$ limit of quantum mechanics, where $N$ is the dimensionality of space, are based on the same point of view. We combine these two to produce a modified Thomas Fermi…
We consider positive and spatially decaying solutions to the Gross-Pitaevskii equation with a harmonic potential. For the energy-critical case, there exists a ground state if and only if the frequency belongs to (1,3) in three dimensions…
In a previous paper we solved a countably infinite family of one-dimensional Schr\"odinger equations by showing that they were supersymmetric partner potentials of the standard quantum harmonic oscillator. In this work we extend these…
The Schroedinger equation is solved for an A-nucleon system using an expansion of the wave function in nonsymmetrized hyperspherical harmonics. Our approach is both an extension and a modification of the formalism developed by Gattobigio et…
Excited states are stationary localized solutions of the Gross--Pitaevskii equation with a harmonic potential and a repulsive nonlinear term that have zeros on a real axis. Existence and asymptotic properties of excited states are…
This article is a review of results on the nonlinear Schroedinger / Gross-Pitaevskii equation (NLS / GP). Nonlinear bound states and aspects of their stability theory are discussed from variational and bifurcation perspectives. Nonlinear…
We study trapped systems of bosons at zero temperature in three and two dimensions. Conditions are fulfilled for the application of Gross-Pitaevskii theory with a positive scattering length. Series expansions for ground-state properties are…
We consider the nonlinear Gross-Pitaevskii equation at positive density, that is, for a bounded solution not tending to 0 at infinity. We focus on infinite ground states, which are by definition minimizers of the energy under local…
We study the ground state and the first three radially excited states of a self-gravitating Bose-Einstein- Condensate with respect to the influence of two external parameters, the total mass and the strength of interactions between…
Bose-Fermi mixtures in one dimension are studied in detail on the basis of an exact solution. Corresponding to three possible choices of the referecce state in the quantum inverse scattering method, three sets of Bethe-ansatz equations are…
We examine ground state correlations for repulsive, quasi one-dimensional bosons in a harmonic trap. In particular, we focus on the few particle limit N=2,3,4,..., where exact numerical solutions of the many particle Schroedinger equation…
In this paper we prove the existence, regularity and symmetry of a ground state for a nonlinear equation in the whole space, involving a pseudo-relativistic Schr\"odinger operator.