A General Approach of Quasi-Exactly Solvable Schroedinger Equations with Three Known Eigenstates

We propose a general method for constructing quasi-exactly solvable potentials with three analytic eigenstates. These potentials can be real or complex functions but the spectrum is real. A comparison with other methods is also performed

The AppliedMathematical SimulationModeling Algorithm for a Multi Aircraft Landing Dynamic System at Bujumbura International Airport Mathematics and Innovative Technologies in Africa

The aim of this paper is to set up an efficient nonlinear application algorithm simulation model for a multi aircraft landing dynamic system in one Runway when considering Bujumbura International Airport. The mathematical modelization of the solved problem is a non-convex optimal control governed by ordinary non-linear differential equations.The dynamic programming technic is applied because it is a sufficiently high order and it does-not require computation of the partial derivatives of the aircraft dynamic. This application is be coded with Linux operating system, but it can also be run on the windows system. High runing performance are obtained with results giving feasible trajectories with a robust optimizing of the objective function. The user interfaces designed in Glade are saved as XML, and by using the GtkBuilder GTK+ object these can be loaded by applications dynamically as needed. By using GtkBuilder, Glade XML files can be used in numerous programming languages including C, C++, C#, Java, Perl, Python,AMPL,etc.. Glade is Free Software released under the GNU GPL License. The algorithm is implemented when considering discrete mathematics while using Bujumbura International Airport Geographic Information System

A General Approach of Quasi-Exactly Solvable Schroedinger Equations

We construct a general algorithm generating the analytic eigenfunctions as well as eigenvalues of one-dimensional stationary Schroedinger Hamiltonians. Both exact and quasi-exact Hamiltonians enter our formalism but we focus on quasi-exact interactions for which no such general approach has been considered before. In particular we concentrate on a generalized sextic oscillator but also on the Lame and the screened Coulomb potentials.Comment: 23 pages, no figur

QES systems, invariant spaces and polynomials recursions

peer reviewedLet us denote ${cal V}$, the finite dimensional vector spaces of functions of the form $psi(x) = p_n(x) + f(x) p_m(x)$ where $p_n(x)$ and $p_m(x)$ are arbitrary polynomials of degree at most $n$ and $m$ in the variable $x$ while $f(x)$ represents a fixed function of $x$. Conditions on $m,n$ and $f(x)$ are found such that families of linear differential operators exist which preserve ${cal V}$. A special emphasis is accorded to the cases where the set of differential operators represents the envelopping algebra of some abstract algebra. These operators can be transformed into linear matrix valued differential operators. In the second part, such types of operators are considered and a connection is established between their solutions and series of polynomials-valued vectors obeying three terms recurence relations. When the operator is quasi exactly solvable, it possesses a finite dimensional invariant vector space. We study how this property leads to the truncation of the polynomials series

Extended Jaynes-Cummings models and (quasi)-exact solvability

The original Jaynes-Cummings model is described by a Hamiltonian which is exactly solvable. Here we extend this model by several types of interactions leading to a nonhermitian operator which doesn't satisfy the physical condition of space-time reflection symmetry (PT symmetry). However the new Hamiltonians are either exactly solvable admitting an entirely real spectrum or quasi exactly solvable with a real algebraic part of their spectrum.Comment: 16 pages, 3 figures, discussion extended, one section adde

PT-Symmetric, Quasi-Exactly Solvable matrix Hamiltonians

Matrix quasi exactly solvable operators are considered and new conditions are determined to test whether a matrix differential operator possesses one or several finite dimensional invariant vector spaces. New examples of $2\times 2$-matrix quasi exactly solvable operators are constructed with the emphasis set on PT-symmetric Hamiltonians.Comment: 14 pages, 1 figure, one equation corrected, results adde

The

The Dirac oscillator is discussed in connection with the theory of quantum deformations. We point out the so-called κ-Dirac oscillator and we analyze the resulting eigenvalue problem