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We consider the relativistic generalization of the harmonic oscillator problem by addressing different questions regarding its classical aspects. We treat the problem using the formalism of Hamiltonian mechanics. A Lie algebraic technique…
We apply the asymptotic iteration method (AIM) to obtain the solutions of Schrodinger equation in the presence of Poschl-Teller (PT) potential. We also obtain the solutions of Dirac equation for the same potential under the condition of…
In this work we apply the Adomian decomposition method combined with the Laplace transform (LADM) in order to solve the 1-dimensional nonlinear Schrodinger equation whose nonlinear term presents a nonlinear defocusing strength that varies…
In a seminal work, S.A.R. Horsley and collaborators [S.A.R. Horsley {\em et al.}, Nature Photon. {\bf 9}, 436 (2015)] have shown that, in the framework of non-Hermitian extensions of the Schr\"odinger and Helmholtz equations, a localized…
The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space $\mathbb{E}_{3}$ where the potential is singular but invariant under…
We have developed a simple method to solve anharmonic oscillators equations. The idea of our method is mainly based on the partitioning of the potential curve into (n+1) small intervals, solving the Schr\"odinger equation in each…
The hyperspherical harmonic basis is used to describe bound states in an $A$--body system. The approach presented here is based on the representation of the potential energy in terms of hyperspherical harmonic functions. Using this…
We studied two possible approaches to one-dimensional Levinson's theorem for Sch\"odinger equation. The first one, we restrict the 3-dimensional theorem. The other one, the theorem proposed by Dong, Ma and Klauss \cite{Dong}. We find…
We analyze the classical equations of motion for a particle moving in the presence of a static magnetic field applied in the $ z $ direction, which varies as $ {1\over{x^2}} $. We find the symmetries through Lie's method of group analysis.…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
We study the dynamics of an infinite regular lattice of classical charged oscillators. Each individual oscillator is described as a point particle subject to a harmonic restoring potential, to the retarded electromagnetic field generated by…
Employing a particularly suitable higher order symplectic integration algorithm, we integrate the 1-$d$ nonlinear Schr\"odinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an…
The time-dependent one-dimensional nonlinear Schr\"odinger equation (NLSE) is solved numerically by a hybrid pseudospectral-variational quantum algorithm that connects a pseudospectral step for the Hamiltonian term with a variational step…
We consider a simple model of lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. Internal dynamics of the atom is governed by the laser field…
We solve, by separation of variables, the problem of three anyons with a harmonic oscillator potential. The anyonic symmetry conditions from cyclic permutations are separable in our coordinates. The conditions from two-particle…
The exact solution of the asymmetric exclusion problem and several of its generalizations is obtained by a matrix product {\it ansatz}. Due to the similarity of the master equation and the Schr\"odinger equation at imaginary times the…
A system of two biased, mutually exclusive random walkers on an infinite 1D lattice is studied whereby the intrinsic bias of one particle is equal and opposite to that of the other. The propogator for this system is solved exactly and…
With a number of special Hamiltonians, solutions of the Schr\"{o}dinger equation may be found by separation of variables in more than one coordinate system. The class of potentials involved includes a number of important examples, including…
We overview recent theoretical studies of the dynamics of one- and two-dimensional spatial dissipative solitons in models based on the complex Ginzburg-Landau equations with the cubic-quintic combination of loss and gain terms, which…
Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose…