Agnieszka Tereszkiewicz

Integrable multi-boson systems and orthogonal polynomials

The strict relation between a certain class of multi-boson Hamiltonian systems and the corresponding class of orthogonal polynomials is established. The correspondence is effectively used to integrate the systems. As an explicit example we integrate the class of multi-boson systems corresponding to q-Hahn class polynomials.

Some integrable systems in nonlinear quantum optics

In the paper we investigate the theory of quantum optical systems. As an application we integrate and describe the quantum optical systems which are generically related to the classical orthogonal polynomials. The family of coherent states related to these systems is constructed and described. Some applications are also presented.

Orthogonal Polynomials of Compact Simple Lie Groups

Recursive algebraic construction of two infinite families of polynomials in variables is proposed as a uniform method applicable to every semisimple Lie group of rank . Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type . The obtained not Laurent-type polynomials are equivalent to the partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for the Lie groups of types , , , , , , and together with lowest polynomials.

Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials

Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank ≤3 are explicitly studied. We derive the polynomials of simple Lie groups B3 and C3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.

Systems with Intensity-Dependent Conversion Integrable by Finite Orthogonal Polynomials

We present exact solutions of a class of the nonlinear models which describe the parametric conversion of photons. Hamiltonians of these models are related to the classes of finite orthogonal polynomials. The spectra and exact expressions for eigenvectors of these Hamiltonians are obtained.

Generalized tricobsthal and generalized tribonacci polynomials

Abstract In this work, we will introduce generalized tribonacci and generalized tricobsthal polynomials. We introduce definitions, formulas for both families of polynomials and the Binet formulas, generating functions. We analyze special points for considered polynomials and present some of polynomials pictorially.

Generalized Jacobsthal polynomials and special points for them

In this work we introduce a family of polynomials that satisfy the recurrence relations for Jacobsthal polynomials with generalized initial conditions by analogy to work of V.K. Gupta, Y.K. Panwar, and O. Sikhwal from 2012. Explicit closed form and the Binet formulas for the generalized Jacobsthal polynomials are presented. The generating function and other relations for them are also found. Special points for this family are analyzed and presented pictorially.

Coherent state maps related to the bounded positive operators

We show that for any bounded positive operator H with the simple spectrum, one can canonically define two coherent state maps. The algebras generated by annihilation operators defined by these coherent state maps are studied. We describe also how the Toda isospectral deformation of H deforms the corresponding coherent state maps and the related operator algebras.

Operator algebras related to quantum optical systems and integrations

We show that some quantum optical systems generate quantum algebras being the natural generalization of the Heisenberg-Weyl algebra. The importance of these algebras for the integration of the systems under consideration is discussed.

ON GENERALIZATION OF SPECIAL FUNCTIONS RELATED TO WEYL GROUPS

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.