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We propose a generalization of Heisenberg picture quantum mechanics in which a Lagrangian and Hamiltonian dynamics is formulated directly for dynamical systems on a manifold with non--commuting coordinates, which act as operators on an…
The model of the position-dependent noncommutativety in quantum mechanics is proposed. We start with a given commutation relations between the operators of coordinates [x^{i},x^{j}]=\omega^{ij}(x), and construct the complete algebra of…
The problem of d-dimensional Schrodinger equations with a position-dependent mass is analyzed in the framework of first-order intertwining operators. With the pair (H, H_1) of intertwined Hamiltonians one can associate another pair of…
The Jacobian conjecture is a well-known open problem in affine algebraic geometry that asks if any polynomial endomorphism of the affine space $\mathbb{A}_{\mathbb{C}}^{n}$ ($n\geq2$) with jacobian $1$ is an automorphism. We present a…
We discuss the simultaneous diagonalization of a family of commuting difference operators by Koornwinder's multivariable generalization of the Askey-Wilson polynomials. The operators constitute a complete set of quantum integrals for a…
Replacing operators with continuous operator-valued functions, we prove time-dependent versions of well-known results on compressions and diagonals of bounded operators. The setting of smooth functions is also addressed. Our results have no…
The matrix normed structure of the unitization of a (non-selfadjoint) operator algebra is determined by that of the original operator algebra. This yields a classification up to completely isometric isomorphism of two-dimensional unital…
In the mid 80's it was conjectured that every bispectral meromorphic function $\psi(x,y)$ gives rise to an integral operator $K_{\psi}(x,y)$ which possesses a commuting differential operator. This has been verified by a direct computation…
In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in $[B(\cH)^n]_1$ associated with positive regular free holomorphic functions in $n$ noncommuting variables and with…
We show that the higher-order Weyl algebras over a field of characteristic zero, which are formally rigid as associative algebras, can be formally deformed in a nontrivial way as hom-associative algebras. We also show that these…
An example due to Pisier shows that two commuting, completely polynomially bounded Hilbert space operators may not be simultaneously similar to contractions. Thus, while each operator is individually similar to a contraction, the pair is…
We give two characterizations of Jacobians of curves with involution having fixed points in the framework of two particular cases of Welter's trisecant conjecture. The geometric form of each of these characterizations is the statement that…
This paper verifies a conjecture posed in a pair of papers on the fixed point sets for a class of quantum operations. Specifically, it is proved that if a quantum operation has mutually commuting operation elements that are effects forming…
Inspired by an old idea of von Neumann, we seek a pair of commuting operators X,P which are, in a specific sense, "close" to the canonical non-commuting position and momentum operators, x,p. The construction of such operators is related to…
This paper outlines a covariant theory of operators defined on groups and homogeneous spaces. A systematic use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is…
Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…
Our goal is to settle the following faded problem: The Jacobian Conjecture (JC_n): If f_1,..,f_n are elements in a polynomial ring k[X_1,..,X_n] over a field k of characteristic 0 such that det(\partial f_i/ \partial X_j) is a nonzero…
We prove the existence and uniqueness of a *projectively equivariant symbol map*, which is an isomorphism between the space of bidifferential operators acting on tensor densities over $R^n$ and that of their symbols, when both are…
We construct a family of pairwise commuting operators such that the Jack symmetric functions of infinitely many variables $x_1,x_2,...$ are their eigenfunctions. These operators are defined as limits at $N\to\infty$ of renormalised…
In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two…