Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions

In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for the corresponding systems defined in Euclidean space and on the two-dimensional sphere. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup Cartesian and polar coordinate cases, and the details of the structure of the quadratic algebras. We also study the polynomial eigenfunctions in elliptic coordinates of the n-dimensional isotropic quantum oscillator

Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n−1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials. ©2007 American Institute of Physic

Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n-1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multi-separability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schroedinger operator, deep connections with special functions and with QES systems. Here we announce a complete classification of nondegenerate (i.e., 4-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in 10 variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly 10 nondegenerate potentials.Comment: 35 page

Hamiltonians separable in cartesian coordinates and third-order integrals of motion

We present in this article all Hamiltonian systems in E(2) that are separable in cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-De Vries equation and the Painlev\'e transcendents.Comment: 19 pages, Will be published in J. Math. Phy

Superintegrability with third order invariants in quantum and classical mechanics

We consider here the coexistence of first- and third-order integrals of motion in two dimensional classical and quantum mechanics. We find explicitly all potentials that admit such integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e. the potentials are proportional to \hbar^2, so their classical limit is free motion.Comment: 15 page

Interband absorption edge in the topological insulators Bi-2(Te1-xSex)(3)

We have investigated the optical properties of thin films of topological insulators Bi2Te3, Bi2Se3, and their alloys Bi-2(Te1-x Se-x)(3) on BaF2 substrates by a combination of infrared ellipsometry and reflectivity in the energy range from 0.06 to 6.5 eV. For the onset of interband absorption in Bi2Se3, after the correction for the Burstein-Moss effect, we find the value of the direct band gap of 215 +/- 10 meV at 10 K. Our data support the picture that Bi2Se3 has a direct band gap located at the Gamma point in the Brillouin zone and that the valence band reaches up to the Dirac point and has the shape of a downward-oriented paraboloid, i.e., without a camel-back structure. In Bi2Te3, the onset of strong direct interband absorption at 10 K is at a similar energy of about 200 meV, with a weaker additional feature at about 170 meV. Our data support the recent GW band-structure calculations suggesting that the direct interband transition does not occur at the Gamma point but near the Z-F line of the Brillouin zone. In the Bi-2(Te1-x Se-x)(3) alloy, the energy of the onset of direct interband transitions exhibits a maximum near x = 0.3 (i.e., the composition of Bi2Te2Se), suggesting that the crossover of the direct interband transitions between the two points in the Brillouin zone occurs close to this composition

Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries

We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set

Lorentz and Galilei Invariance on Lattices

We show that the algebraic aspects of Lie symmetries and generalized symmetries in nonrelativistic and relativistic quantum mechanics can be preserved in linear lattice theories. The mathematical tool for symmetry preserving discretizations on regular lattices is the umbral calculus.Comment: 5 page