Bilinear and Quadratic Algebra

2019 ◽  
pp. 19-43
Author(s):  
Gaëtan Chenevier ◽  
Jean Lannes
Keyword(s):  
Quadratic Algebra

Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

2014 ◽  
Vol 29 (06) ◽  
pp. 1450028 ◽  
Author(s):  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Quadratic Algebra ◽  
Two Dimensional ◽  
Oscillator Algebra ◽  
Quadratic Algebras ◽  
Energy Eigenvalues ◽  
Vector Potentials ◽  
Algebra Approach ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

scholarly journals Quadratic algebra contractions and second-order superintegrable systems

2014 ◽  
Vol 12 (05) ◽  
pp. 583-612 ◽  
Author(s):  
Ernest G. Kalnins ◽  
W. Miller
Keyword(s):  
Constant Curvature ◽  
Two Dimensions ◽  
Second Order ◽  
Quadratic Algebra ◽  
Quadratic Algebras ◽  
Physical Systems ◽  
Special Cases ◽  
Superintegrable Systems ◽  
Wilson Polynomials ◽  
Symmetry Algebras

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

scholarly journals Stability of Asai local factors for GL(2)

2020 ◽  
pp. 1-18
Author(s):  
Yeongseong Jo ◽  
M. Krishnamurthy
Keyword(s):  
Local Field ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Quadratic Algebra ◽  
Local Factors ◽  
The Stability ◽  
Gamma Factor

Let [Formula: see text] be a non-archimedean local field of characteristic not equal to 2 and let [Formula: see text] be a quadratic algebra. We prove the stability of local factors attached to irreducible admissible (complex) representations of [Formula: see text] via the Rankin–Selberg method under highly ramified twists. This includes both the Asai as well as the Rankin–Selberg local factors attached to pairs. Our method relies on expressing the gamma factor as a Mellin transform using Bessel functions.

scholarly journals Two-Parameter Deformation of the Poincaré Algebra

1997 ◽  
Vol 12 (05) ◽  
pp. 891-901 ◽  
Author(s):  
A. Stern ◽  
I. Yakushin
Keyword(s):  
Quadratic Algebra ◽  
Commuting Operators ◽  
Poincaré Algebra ◽  
Two Parameter ◽  
Parameter Deformation

We examine a two-parameter (ℏ,λ) deformation of the Poincaré algebra which is covariant under the action of SL q(2,C). When λ → 0 it yields the Poincaré algebra, while in the ℏ → 0 limit we recover the classical quadratic algebra discussed previously in Refs. 1 and 2. The analogues of the Pauli–Lubanski vector w and Casimirs p2 and w2 are found and a set of mutually commuting operators is constructed.

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