In this letter, we consider the second Hamiltonian structure of the constrained modified KP hierarchy. After mapping the Lax operator to a pure differential operator the second structure becomes the sum of the second and the third Gelfand-Dickey brackets defined by this differential operator. We simplify this Hamiltonian structure by factorizing the Lax operator into linear terms.

It is proved that any one-dimensional, first order Hamiltonian differential operator can be put into constant coefficient form by a suitable change of variables. Consequently, there exist canonical variables for any such Hamiltonian operator. In the course of the proof, a complete characterization of all first order Hamiltonian differential operators, as well as the general formula for the beha...

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 underlying Hilbert space. This is accomplished by defining the Lagrangian and Hamiltonian as the real part of a graded total trace over the underlying Hilbert spa...

Quantum speed limits (QSLs) provide lower bounds on the minimum time required for a process to unfold by using distance between quantum states and identifying of evolution or an upper bound it. We introduce generalization QSL characterize general operator when conjugated unitary. The resulting (OQSL) admits geometric interpretation, is shown be tight, holds flows induced arbitrary unitaries, i....

We show the natural relation between the Wigner Hamiltonian and the conformal Hamiltonian. It is presented a model in (super)conformal quantum mechanics with (super)conformal symmetry in the Wigner-Heisenberg algebra picture [x, px] = i(1 + cP) (P being the parity operator). In this context, the energy spectrum, the Casimir operator, creation and annihilation operators are defined. This superco...

The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair A1, A2, where A1 is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equivalent to finding a generalized vector field τ such that A2 = LτA1. We use this result ...