Related papers: Algebraic Structures and Eigenstates for Integrabl…
We give a generalization of the Hodge operator to spaces $(V,h)$ endowed with a hermitian or symmetric bilinear form $h$ over arbitrary fields, including the characteristic two case. Suitable exterior powers of $V$ become free modules over…
Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= \lambda_n x^n + \nu_n x^{n-2}$ for $\lambda_n$ and $\nu_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = \lambda_n…
We outline the construction of type II superstring field theory leading to a geometric and algebraic BV master equation, analogous to Zwiebach's construction for the bosonic string. The construction uses the small Hilbert space. Elementary…
Polynomial relations for generators of $su(2)$ Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modified Holstein-Primakoff transformations in…
We examine applications of polynomial Lie algebras $sl_{pd}(2)$ to solve physical tasks in $G_{inv}$-invariant models of coupled subsystems in quantum physics. A general operator formalism is given to solve spectral problems using…
Polynomial Lie (super)algebras $g_{pd}$ are introduced via $G_{i}$-invariant polynomial Jordan maps in quantum composite models with Hamiltonians $H$ having invariance groups $G_{i}$. Algebras $g_{pd}$ have polynomial structure functions in…
We give a general method to construct a complete set of linearly independent Casimir operators of a Lie algebra with rank N. For a Casimir operator of degree p, this will be provided by an explicit calculation of its symmetric coefficients…
We present a method to obtain higher order integrals and polynomial algebras for two-dimensional superintegrable systems from creation and annihilation operators. All potentials with a second and a third order integrals of motion separable…
A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. We define a related notion of a true-pair of a linear operator, and then show that each linear operator on a finite dimensional nonzero real…
For a field of characteristic $\ne 2$ we study vector spaces that are graded by the weight lattice of a root system, and are endowed with linear operators in each simple root direction. We show that these data extend to a graded semisimple…
We prove that the scalar and $2\times 2$ matrix differential operators which preserve the simplest scalar and vector-valued polynomial modules in two variables have a fundamental Lie algebraic structure. Our approach is based on a general…
Potential algebras can be used effectively in the analysis of the quantum systems. In the article, we focus on the systems described by a separable, 2x2 matrix Hamiltonian of the first order in derivatives. We find integrals of motion of…
The powerful group theoretical formalism of potential algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by complexifying so(2,1), thereby getting the complex algebra sl(2,\C) or $A_1$. This leads to new types of both…
We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which…
Let $G$ be a finitely generated group with polynomial growth, and let $\om$ be a weight, i.e. a sub-multiplicative function on $G$ with positive values. We study when the weighted group algebra $\ell^1(G,\om)$ is isomorphic to an operator…
We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different…
Linear operators preserving the direct sum of polynomial rings P(m)\oplus P(n) are constructed. In the case |m-n|=1 they correspond to atypical representations of the superalgebra osp(2,2). For |m-n|=2 the generic, finite dimensional…
Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with…
We consider the spectrum of a two-dimensional Pauli operator with a compactly supported electric potential and a variable magnetic field with a positive mean value. The rate of accumulation of eigenvalues to zero is described in terms of…
The relation between certain Hamiltonians, known as dual, or partner Hamiltonians, under the transformation $x{\rightarrow}\bar{x}^{\bar{\alpha}}$ has long been used as a method of simplifying spectral problems in quantum mechanics. This…