Related papers: Modified Wave operators for the Hartree equation w…
Studying the electronic structure of defects in materials is an important subject in condensed matter physics. From a mathematical point of view, nonlinear mean-field models of localized defects in insulators are well understood. We present…
In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space \[ \partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = - |u|^{p-1} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}; \] and introduce a…
Based on the ideology of the Maslov's complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated…
For a semi-linear Schr\"{o}dinger equation of Hartree type in three spatial dimensions, various approximations of singular, point-like perturbations are considered, in the form of potentials of very small range and very large magnitude,…
Using the method of the "exact discretization" of the Schr\"odinger equation, we propose a particular discretized version of the N=2 Supersymmetric Quantum Mechanics. After defining the corresponding shape invariance condition, we show that…
One-dimensional scattering by a Coulomb potential V(x)=lambda/|x| is studied for both repulsive (c>0) and attractive (c<0) cases. Two methods of regularizing the singularity at x=0 are used, yielding the same conclusion, namely, that the…
This paper is concerned with nonlinear Schr\"odinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it is proved that the equation admits a…
The wave equation with energy critical sources and nonlinear damping defined on a 3D bounded domain is considered. It is shown that the resulting dynamical system admits a global attractor. Under the additional assumption of strong…
In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -\Delta u+\frac{\mu}{|x|^2} u=u^p\quad {\rm in}\quad \Omega\setminus\{0\},\qquad…
We consider the cubic Schrodinger equation on the line, for which the scattering theory requires modifications due to long range effects. We revisit the construction of the modified wave operator, and recall the construction of its inverse,…
In this paper, we consider the asymptotic behavior of traveling wave solutions of the degenerate nonlinear parabolic equation: $u_{t}=u^{p}(u_{xx}+u)-\delta u$ ($\delta = 0$ or $1$) for $\xi \equiv x - ct \to - \infty$ with $c>0$. We give a…
We propose a method to construct the ground state $\psi(\lambda)$ of local lattice hamiltonians with the generic form $H_0 + \lambda H_1$, where $\lambda$ is a coupling constant and $H_0$ is a hamiltonian with a non degenerate ground state…
In this paper we study a Cauchy problem for the nonlinear damped wave equations for a general positive operator with discrete spectrum. We derive the exponential in time decay of solutions to the linear problem with decay rate depending on…
In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-\epsilon})u^{2_{s}^{\sharp}-2-\epsilon} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}\Omega,\\…
The use of fractional momentum operators and fractionary kinetic energy used to model linear damping in dissipative systems such as resistive circuits and a spring-mass ensambles was extended to a quantum mechanical formalism. Three…
Exact bound state solutions and the corresponding wave functions of the Schr\"odinger equation for some non-central potentials including Makarov potential, modified-Kratzer plus a ring-shaped potential, double ring-shaped Kratzer potential,…
We consider two three-dimensional isotropic harmonic oscillators interacting with the quantum electromagnetic field in the Coulomb gauge and within dipole approximation. Using a Bogoliubov-like transformation, we can obtain transformed…
The purpose of this note is to prove the existence of a conformal scattering operator for the cubic defocusing wave equation on a non-stationary background. The proof essentially relies on solving the characteristic initial value problem by…
We consider dynamical behavior of non-autonomous wave-type evolutionary equations with nonlinear damping, critical nonlinearity, and time-dependent external forcing which is translation bounded but not translation compact (i.e., external…
Given a spatially dependent mass distribution we obtain potential functions for exactly solvable nonrelativistic problems. The energy spectrum of the bound states and their wavefunctions are written down explicitly. This is accomplished by…