Monomiality principle, Sheffer-type polynomials and the normal ordering problem

Abstract

We solve the boson normal ordering problem for $(q(a^{†})a+v(a^{†}){)}^n$ with arbitrary functions $q(x)$ and $v(x)$ and integer $n$, where $a$ and $a^{†}$ are boson annihilation and creation operators, satisfying $[a,a^{†}]=1$. This consequently provides the solution for the exponential $e^{\lambda (q(a^{†})a+v(a^{†}))}$ generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures.

arXiv commentPresented at the 8'th International School of Theoretical Physics "Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005), Myczkowce, Poland. 13 pages, 31 references

References in corpus (5)

• Lie-algebraic discretization of differential equations
• Representations of Monomiality Principle with Sheffer-type Polynomials and Boson Normal Ordering
• Boson Normal Ordering via Substitutions and Sheffer-type Polynomials
• Combinatorics of boson normal ordering and some applications
• Hidden $s{l}_2$-algebra of finite-difference equations

Cited by in corpus (1)

• Combinatorics of boson normal ordering and some applications