Earthline Journal of Mathematical Sciences

Spectral Spectral Rigidity and Geometric Quantization of Coadjoint Orbits

2026-05-12T16:13:56+00:00

Coadjoint orbits provide a fundamental link between symplectic geometry and the representation theory of Lie groups, as formalized by the orbit method of Kirillov. In this paper, we investigate the spectral properties of the Casimir operator in relation to the geometry of coadjoint orbits and their quantization.

We establish a spectral rigidity phenomenon: for compact semisimple Lie groups, the eigenvalue of the Casimir operator determines the corresponding coadjoint orbit and the associated irreducible representation. This rigidity is shown to be independent of the choice of G-invariant metric on the orbit, highlighting the intrinsic algebraic nature of the Casimir operator.

We provide explicit computations in the case of SU(2), where coadjoint orbits are 2-spheres, and analyze the relationship between the Casimir operator and the Laplace-Beltrami operator under metric variations. We further extend the discussion to real semisimple Lie groups, where rigidity persists in a weaker form through the infinitesimal character and the Harish-Chandra isomorphism.

Our results clarify the role of the Casimir operator as a bridge between geometry, spectral theory, and geometric quantization.

New Huygens Type Trigonometric Inequalities

2026-05-19T16:14:58+00:00

In this paper, some Huygens type inequalities involving trigonometric functions are refined and sharpened. We thus improve established inequalities and provide new ones.

Explicit Identities for Horadam Polynomials: Generalized Fibonacci Formulations and Special Cases

2026-05-21T17:16:02+00:00

In this paper, we investigate the generalized Fibonacci (Horadam) polynomials and concentrate on two special subclasses, which we introduce as the (r,s)-Fibonacci and (r,s)-Lucas polynomials. Our primary aim is to present and establish several identities that connect these two families, thereby extending classical relations between Fibonacci and Lucas sequences into a broader polynomial framework. The identities obtained not only highlight the structural interplay between the (r,s)-Fibonacci and (r,s)-Lucas polynomials but also enrich the theory of generalized Horadam polynomials by revealing new algebraic connections. This work is devoted exclusively to the derivation and exposition of such identities, providing a foundation for further exploration of recurrence-based polynomial structures.