In this paper, we study the Chebyshev property of the 3-dimentional vector space $E =langle I_0, I_1, I_2rangle$, where $I_k(h)=int_{H=h}x^ky,dx$ and $H(x,y)=frac{1}{2}y^2+frac{1}{2}(e^{-2x}+1)-e^{-x}$ is a non-algebraic Hamiltonian function. Our main result asserts that $E$ is an extended complete Chebyshev space for $hin(0,frac{1}{2})$. To this end, we use the criterion and tools developed by...

It is shown how to introduce a geometric description of the algebraic approach to the non-relativistic quantum mechanics. It turns out that the GNS representation provides not only symplectic but also Hermitian realization of a ‘quantum Poisson algebra’. We discuss alternative Hamiltonian structures emerging out of different GNS representations which provide a natural setting for quantum bi-Ham...

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X . We study the quotient morphism μG,X//G : X//G → g//G of the moment map μG,X : X → g. We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μG,X//G have the same dimens...

We apply the algebraic method to Bateman Hamiltonian and obtain its natural frequencies ladder operators from adjoint or regular matrix representation of that operator. Present analysis shows eigenfunctions compatible with complex eigenvalues obtained earlier by other authors are not square integrable. In addition this, annihilate an infinite number such eigenfunctions.

We present an explicit computation of matrix elements of the hamiltonian constraint operator in non-perturbative quantum gravity. In particular, we consider the euclidean term of Thiemann’s version of the constraint and compute its action on trivalent states, for all its natural orderings. The calculation is performed using graphical techniques from the recoupling theory of colored knots and li...

We consider the symplectic Euler method for Hamiltonian systems with holonomic constraints and its generalization to a class of overdetermined differential-algebraic equations (ODAEs). It is shown that a natural extension of the symplectic Euler method as a SPARK method is inconsistent for ODAEs which are nonlinear in the algebraic variables. A different non-trivial extension is given and shown...

We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degenera...