相关论文: An infinite family of solvable and integrable quan…

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly…

The general description of superintegrable systems with one polynomial integral of order $N$ in the momenta is presented for a Hamiltonian system in two-dimensional Euclidean plane. We consider classical and quantum Hamiltonian systems…

A method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly-solvable. It is shown that in two dimensional Euclidean or pseudo-Euclidean…

A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in N-dimensional Euclidean space. Two different sets of N commuting second order operators are found, overlapping in the Hamiltonian…

We present a wide class of differential systems in any dimension that are either integrable or complete integrable. In particular, our result enlarges a known family of planar integrable systems. We give an extensive list of examples that…

A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum. New…

Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite…

Superintegrable systems are a class of physical systems which possess more conserved quantities than their degrees of freedom. The study of these systems has a long history and continues to attract significant international attention. This…

A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials.…

In this paper, we introduce a family of sextic potentials that are exactly solvable, and for the first time, a family of triple-well potentials with their whole energy spectrum and wavefunctions using supersymmetry method. It was suggested…

It is shown that the 3-body trigonometric G_2 integrable system is exactly-solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the…

We introduce an exactly-solvable family of one-dimensional driven-diffusive systems defined on a discrete lattice. We find the quadratic algebra of this family which has an infinite-dimensional representation. We discuss the phase diagram…

We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate…

We develop a systematic procedure for constructing quantum many-body problems whose spectrum can be partially or totally computed by purely algebraic means. The exactly-solvable models include rational and hyperbolic potentials related to…

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic…

We find new families of shape invariant potentials depending on n>=1 parameters subject to translation by the inclusion of non-trivial invariants. New dependencies of the spectra are found, and it opens the door to the engineering of…

Local solvability is analyzed for natural families of partial differential operators having double characteristics. In some families the set of all operators that are not locally solvable is shown to have both infinite dimension and…

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general cubic algebra and we present specific…

A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the…

We discuss a new class of coordinate systems for a plane, which provide an analytical representation of arbitrary straightline, and then define the form of potential on the plane, under which the equations of motion of a mass point are…