Order Integrals Of Motion Research Articles

Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 1. Systems with cylindric symmetry

Cylindrically symmetric quantum mechanical systems with position dependent masses admitting at least one second order integral of motion are classified. It is proved that there exist 68 such systems which are inequivalent. Among them there are thirty superintegrable and twelve maximally superintegrable ones. The arbitrary elements of the corresponding Hamiltonians (i.e.,masses and potentials) are presented explicitly.

Superintegrable quantum mechanical systems with position dependent masses invariant with respect to two parametric Lie groups

Quantum mechanical systems with position dependent masses (PDMs) admitting two parametric Lie symmetry groups are classified. Namely, all PDM systems are specified which, in addition to their invariance w.r.t. a two parametric Lie group, admit at least one second order integral of motion. The presented classification is partially extended to the more generic systems which do not accept any Lie group.

New infinite families of Nth-order superintegrable systems separating in Cartesian coordinates

A study is presented of superintegrable quantum systems in two-dimensional Euclidean space E 2 allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian H and the second order integral of motion X, responsible for the separation of variables, they allow a third integral that is a polynomial of order N (N ⩾ 3) in the components p 1, p 2 of the linear momentum. We focus on doubly exotic potentials, i.e. potentials V(x, y) = V 1(x) + V 2(y) where neither V 1(x) nor V 2(y) satisfy any linear ordinary differential equation (ODE). We present two new infinite families of superintegrable systems in E 2 with integrals of order N for which V 1(x) and V 2(y) are given by the solution of a nonlinear ODE that passes the Painlevé test. This was verified for 3 ⩽ N ⩽ 10. We conjecture that this will hold for any doubly exotic potential and for all N, and that moreover the potentials will always actually have the Painlevé property.

Integrable and Superintegrable Systems with Higher Order Integrals of Motion: Master Function Formalism

We construct two-dimensional integrable and superintegrable systems in terms of the master function formalism and relate them to Mielnik’s and Marquette’s construction in supersymmetric quantum mechanics. For two different cases of the master functions, we obtain two different two-dimensional superintegrable systems with higher order integrals of motion.

Spherical type integrable classical systems in a magnetic field

We show that four classes of second order spherical type integrable classical systems in a magnetic field exist in the Euclidean space , and construct the Hamiltonian and two second order integrals of motion in involution for each of them. For one of the classes the Hamiltonian depends on four arbitrary functions of one variable. This class contains the magnetic monopole as a special case. Two further classes have Hamiltonians depending on one arbitrary function of one variable and four or six constants, respectively. The magnetic field in these cases is radial. The remaining system corresponds to a constant magnetic field and the Hamiltonian depends on two constants. Questions of superintegrability—i.e. the existence of further integrals—are discussed.

Modified n-level, n − 1-mode Tavis–Cummings model and algebraic Bethe ansatz

Using the quantum group technique we construct a one-parametric family of integrable modifications of the n-level, mode Tavis–Cummings Hamiltonian possessing an additional Stark-type term. We show that in the ‘quasiclassical’ limit the constructed Hamiltonian transforms into the integrable Hamiltonian of the quantum n-level, mode Tavis–Cummings model with the equal interaction strengths considered in Skrypnyk (2008 J. Phys. A: Math. Theor. 41 475202, 2009 J. Math. Phys. 50 103523). We diagonalize the constructed ‘modified’ Tavis–Cummings Hamiltonian and its second order integrals of motion using the nested Bethe ansatz.

On realizations of polynomial algebras with three generators via deformed oscillator algebras

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analogue of this construction for the most general polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.

Superintegrable systems with spin invariant with respect to the rotation group

Quantum non-relativistic systems with 2 × 2 matrix potentials are investigated. Physically, they simulate charged or neutral fermions with non-trivial dipole momenta, interacting with an external electric field. Assuming rotational invariance of the Hamiltonian, all such systems allowing second order integrals of motion are identified. It is shown that the integrals of motion can be effectively used to separate variables and to reduce the systems to decoupled ordinary differential equations. Solutions for two of the problems discussed are presented explicitly.

Change of the time for the periodic Toda lattices and natural systems on the plane with higher order integrals of motion

We discuss some special classes of canonical transformations of the time variable, which relate different integrable systems. Such dual systems have different integrals of motion, Lax equations, separated variables and bi-hamiltonian structures. As an example the two-dimensional periodic Toda lattices associated with the classical root systems and the dual natural systems on the palne are considered in detail.

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendent potentials

We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painlevé transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock-type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions.

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integrals of motion. We construct the most general cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in Cartesian coordinates with a third order integral are known. The general formalism is applied to quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E2. We also discuss these potentials from the point of view of supersymmetric and PT-symmetric quantum mechanics.

Quasiseparation of variables in the Schrödinger equation with a magnetic field

We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order integral of motion does not guarantee the separation of variables in the Schrödinger equation. We introduce the concept of “quasiseparation of variables” and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra.

Quadratic Hamiltonians in phase space and their eigenstates

The solution of the Schrödinger equation for a Hamiltonian ℋ that is a general second order polynomial in the canonical coordinates qi and momenta pi is discussed. Examples of such Hamiltonians abound in various fields of physics and, e.g., the problem of small vibrations has been solved a long time ago. In the general case we first use linear canonical transformations [the group Sp (2n, R)] to reduce ℋ to a simplified representative for ℋR. Next we imbed each ℋR into a complete set of commuting second order integrals of motion, making use of a recently obtained classification of maximal Abelian subalgebras of the algebra sp (2n, R). This imbedding is then used to separate variables in the representative Schrödinger equations and to obtain complete sets of their eigenfunctions. Finally the solutions of the representative equations can be transformed back into those of the original one making use of representations of the canonical transformations. The program is implemented fully for n=1,2 and its structure is analyzed for arbitrary n. Special cases of interest are treated in detail, e.g., Hamiltonians for a system of n particles invariant under rotations, translations and permutations.