George P. Flessas

Exact solutions for a doubly anharmonic oscillator

Abstract Two classes of exact solutions and eigenvalues for an oscillator with both quartic and sextic anharmonicity as well as the conditions under which these solutions can occur are given.

On the Schrödinger equation for the x2 + λx2(1 + gx2) interaction

Abstract We present an infinite set of exact solutions and eigenvalues for the one-dimensional Schrodinger equation involving the potential x 2 + λx 2 (1 + gx 2 ) . Comparison with numerical methods is made.

Exact solutions of the Schrodinger equation (-d/dx2 + x2 + λx2/(1 + gx2))ψ(x) = Eψ(x)

The authors prove the existence of a class of exact eigenvalues and eigenfunctions of the Schrodinger equation for the potential x2 + λx2/(1 + gx2) when certain algebraic relations between λ and g hold. Some of the properties of these solutions are discussed. It is shown that in a certain sense they may be regarded as Sturmians for the Schrodinger equation with the potential x2 - λ/(g + g2x2).

On the three-dimensional anharmonic oscillator

Abstract The radial Schrodinger equation of a three-dimensional oscillator with a general anharmonicity is shown to possess simple exact solutions. The conditions for the occurrence of these solutions are given.

Exact solutions for the doubly anharmonic oscillator (II)

Abstract We present exact solutions in the form of definite integrals for the one-dimensional doubly anharmonic oscillator, the corresponding eigenvalues as well as the conditions for the validity of these solutions. Comparison with perturbative methods is made.

Symmetry, Singularities and Integrability in Complex Dynamics I: The Reduction Problem

Abstract Quadratic systems generated using Yang-Baxter equations are integrable in a sense, but we display a deterioration in the possession of the Painlevé property as the number of equations in each ‘integrable system’ increases. Certain intermediate systems are constructed and also tested for the Painlevé property. The Lie symmetries are also computed for completeness.

Generalisations of the Laplace–Runge–Lenz Vector

Abstract The characteristic feature of the Kepler Problem is the existence of the so-called Laplace–Runge–Lenz vector which enables a very simple discussion of the properties of the orbit for the problem. It is found that there are many classes of problems, some closely related to the Kepler Problem and others somewhat remote, which share the possession of a conserved vector which plays a significant rôle in the analysis of these problems.

An exact solution of the Schrodinger equation for a multiterm potential

The authors show that the Schrodinger equation with the potential Bx+Cx2+Dx3+Ex4 is exactly solvable on the half-line x>or=0, provided two simple relations between B,C,D and E hold. Some remarks concerning the B=D=0 case are made.

Definite integrals as solutions for the x2+λx2/(1+gx2) potential

Presents exact solutions and eigenvalues for the Schrodinger equation with the x2 + λx2/(1 + gx2) interaction on the half axis x >or= 0. The solutions are given in the form of definite integrals and the eigenvalues by means of a well defined limiting procedure.

Generalized cosmic no-hair theorems

We show that the quasi-exponential solution exp (Bt−At2) of R+αR2 gravity is an attractor for all homogeneous and isotropic solutions of higher order gravity theories derived from a lagrangian that is an arbitrary analytic function of the scalar curvature R. This indicates that a generalized form of the cosmic no-hair conjecture is valid in the framework of higher order gravity theories.