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Exact Methods for ODEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Giesbrecht et al.[492] presents an algorithm for factoring partial differential operators for a large class of operators. Technically they find factorizations over many algebras, including the polynomial nth Weyl algebra (Qn). This is defined by Qn=Kx1,…,xn,∂1,…,∂n,∂ixj=xj∂iif i≠j,qixj∂i+1if i=j,0=∂i∂j−∂j∂i=xixj−xjxi,
New bound state solutions of diatomic molecules and mass spectrum of heavy quarkonium with improved modified Kratzer potential plus screened Coulomb potential model in the framework of three-dimensional nonrelativistic noncommutative phase-space symmetries
Published in Molecular Physics, 2023
The non-commutativity of the phase-space is based on the deformed Heisenberg–Weyl algebra, which is determined by the above canonical commutation relations. The two parameters and are invertible antisymmetric real constant (3 × 3) matrices that satisfied with and , here are interpreted as being new infinitesimal constants and is the effective Plank’s constant which equals . However, the operators and in (HP and IP, respectively) are dependent on the corresponding operator in SP from the following projection relations [17–22]: here , and in three representations (SP, HP, and IP) in NRQM symmetries, while the dynamics of new systems are described from the following motion equations in 3D-NRNCPS symmetries:
Estimation of nonclassical properties of multiphoton coherent states from optical tomograms
Published in Journal of Modern Optics, 2018
Pradip Laha, S. Lakshmibala, V. Balakrishnan
In models of atom optics involving two-photon processes, and (the two-photon creation and destruction operators respectively) do not satisfy the Heisenberg–Weyl commutation algebra by themselves, in contrast to the relation . In all oscillator-like settings, it is convenient to work with creation and destruction operators that satisfy the Heisenberg–Weyl algebra. Motivated by this a search for such appropriate operators has been conducted in [17].
Algebraic differentiators through orthogonal polynomials series expansions
Published in International Journal of Control, 2018
The algebraic method proposed in this paper is greatly based on the structural properties of the Weyl algebra. A major advantage in this sense is to obtain closed formulas for the differentiators. Similar approaches can be found in Ushirobira, Perruquetti, Mboup, and Fliess (2012) and Ushirobira, Perruquetti, Mboup, and Fliess (2013).