scholarly journals Two commuting operators associated with a tridiagonal pair

2012 ◽  
Vol 437 (1) ◽  
pp. 242-270 ◽  
Author(s):  
Sarah Bockting-Conrad
Keyword(s):  
Commuting Operators ◽  
Tridiagonal Pair

Complete sets of commuting operators and O (3) scalars in the enveloping algebra of SU (3)

1974 ◽  
Vol 15 (10) ◽  
pp. 1787-1799 ◽  
Author(s):  
B. R. Judd ◽  
W. Miller ◽  
J. Patera ◽  
P. Winternitz
Keyword(s):  
Enveloping Algebra ◽  
Commuting Operators ◽  
Complete Sets

scholarly journals Quantized toroidal dipole eigenvalues in nano-systems

2021 ◽  
Vol 2090 (1) ◽  
pp. 012151
Author(s):  
D. V. Anghel ◽  
A. T. Preda
Keyword(s):  
Parity Violation ◽  
Nuclear Reactions ◽  
General Procedure ◽  
Quantum Systems ◽  
Electromagnetic Structure ◽  
New Class ◽  
Quantum Properties ◽  
Commuting Operators ◽  
Dipole Operator ◽  
Theoretical Considerations

Abstract The parity violation in nuclear reactions led to the discovery of the new class of toroidal multipoles. Since then, it was observed that toroidal multipoles are present in the electromagnetic structure of systems at all scales, from elementary particles, to solid state systems and metamaterials. The toroidal dipole T (the lowest order multipole) is the most common. This corresponds to the toroidal dipole operator T ^ in quantum systems, with the projections T ^ i (i = 1, 2, 3) on the coordinate axes. These operators are observables if they are self-adjoint, but, although it is commonly discussed of toroidal dipoles of both, classical and quantum systems, up to now no system has been identified in which the operators are self-adjoint. Therefore, in this paper we use what are called the “natural coordinates” of the T ^ 3 operator to give a general procedure to construct operators that commute with T ^ 3 . Using this method, we introduce the operators p ^ ( k ) , p ^ ( k 1 ) , and p ^ ( k 2 ) , which, together with T ^ 3 and L ^ 3 , form sets of commuting operators: ( p ^ ( k ) , T ^ 3 , L ^ 3 ) and ( p ^ ( k 1 ) , p ^ ( k 2 ) , T ^ 3 ) . All these theoretical considerations open up the possibility to design metamaterials that could exploit the quantization and the general quantum properties of the toroidal dipoles.

POISSON STRUCTURE AND INTEGRABILITY OF THE NEUMANN-MOSER-UHLENBECK MODEL

1990 ◽  
Vol 05 (23) ◽  
pp. 4477-4488 ◽  
Author(s):  
J. AVAN ◽  
M. TALON
Keyword(s):  
Poisson Structure ◽  
Point Of View ◽  
Conserved Quantities ◽  
Poisson Brackets ◽  
Quantum Level ◽  
Quantum Integrability ◽  
Commuting Operators ◽  
Lax Operator

Neumann’s model, describing the motion of a particle on an N-sphere under harmonic forces, is studied from the point of view of classical and quantum integrability. Classical integrability is derived from a generalized structure, “R-S couple” or “D-matrix” for the Poisson brackets of the Lax operator. The already-known set of conserved quantities for this model turns out to follow straightforwardly from this structure. It gives rise to a set of commuting operators at the quantum level, and the algebra of Lax operators directly follows from the classical one.

Representations by Commuting Operators

1995 ◽  
pp. 409-521
Author(s):  
Y. M. Berezansky ◽  
Y. G. Kondratiev
Keyword(s):  
Commuting Operators

scholarly journals DUNKL OPERATORS FOR COMPLEX REFLECTION GROUPS

2003 ◽  
Vol 86 (1) ◽  
pp. 70-108 ◽  
Author(s):  
C. F. DUNKL ◽  
E. M. OPDAM
Keyword(s):  
Reflection Groups ◽  
Dunkl Operators ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Commuting Operators ◽  
Rham Complex ◽  
Rational Cherednik Algebra ◽  
De Rham ◽  
Parameterized Family ◽  
Primary 52C35

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameterized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the ‘rational Cherednik algebra’, and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups $G(m, p, N)$, the set of singular parameters in the parameterized family of these structures is described explicitly, using the theory of non-symmetric Jack polynomials.2000 Mathematical Subject Classification: 20F55 (primary), 52C35, 05E05, 33C08 (secondary).

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