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An integrable model for SU($\nu$) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the {\it renormalized harmonic-oscillators} which…
A method to compute the bound state eigenvalues and eigenfunctions of a Schr\"{o}dinger equation or a spinless Salpeter equation with central interaction is presented. This method is the generalization to the three-dimensional case of the…
We study the following focusing intercritical nonlinear Schr\"odinger equation with partial harmonic confinement: \begin{equation*} \begin{cases} i\partial_t u+\Delta_{z}u-y^2 u =- |u|^{\alpha}u,\quad t\in \mathbb{R},\newline u(0,z)=…
We solve the stationary Schr\"odinger equation for a particle confined to a 3D spherical wedge -- the region $\{(r,\theta,\phi): 0 \leq r \leq R,\, 0 \leq \theta \leq \pi,\, 0 \leq \phi \leq \Phi\}$ with Dirichlet BCs on all surfaces. This…
The eigenstates of a real or complex cubic anharmonic oscillator are investigated using original and alternative methods. The procedure consists of determining global solutions of the Schr\"odinger equation that comply with the pertinent…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation…
The bending energy of any freely deformable closed surface is quadratic in its curvature. In the absence of constraints, it will be minimized when the surface adopts the form of a round sphere. If the surface is confined within a…
In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schr\"odinger equation with harmonic potential, $(i\partial_t + \frac {\Delta}2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free…
A class of three-dimensional models which satisfy supersymmetric intertwining relations with the simplest - oscillator-like - variant of shape invariance is constructed. It is proved that the models are not amenable to conventional…
The purpose of this document is to describe the solution and implementation of the time-independent and time-dependent Schr\"odinger using pseudospectral methods. Currently, the description is for single particle systems interacting with a…
The polynomial solution of the Schrodinger equation for the Pseudoharmonic potential is found for any arbitrary angular momentum $l$. The exact bound-state energy eigenvalues and the corresponding eigen functions are analytically…
In this paper, we propose a linearized Fourier pseudo-spectral method, which preserves the total mass and energy conservation laws, for the damped nonlinear Schr\"{o}dinger equation in three dimensions. With the aid of the semi-norm…
We study confined scattering of a quantum particle by two centers fixed on the longitudinal axis of a harmonic waveguide-like trap. The conditions of confinement-induced resonances (CIRs) appearing in these systems, when scattering cross…
In a recent paper we proposed and compared various approaches to compute the ground state and dynamics of the Schr\"{o}dinger--Poisson--Slater (SPS) system for general external potential and initial condition, concluding that the methods…
This paper introduces an efficient high-order numerical method for solving the 1D stationary Schr\"odinger equation in the highly oscillatory regime. Building upon the ideas from [Arnold, Ben Abdallah, Negulescu, SIAM J. Numer. Anal.,…
We report on the observation of confinement-induced resonances for strong three-dimensional (3D) confinement in a lattice potential. Starting from a Mott-insulator state with predominantly single-site occupancy, we detect loss and heating…
We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of…
We consider the 3D cubic nonlinear Schr\"odinger equation (NLS) with a strong toroidal trap. In the first part, we show that as the confinement is strengthened, a large class of global solutions to the time-dependent model can be described…
In this article, we study numerical approximation of eigenvalue problems of the Schr\"{o}dinger operator $\displaystyle -\Delta u + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in…