Operator Calculus: Edification and Utilization

The operational calculus had the merit of introducing the notion of derivative operator but didn't help so much mathematics. Later, quantum mechanics introduced the position operator q linked to the derivative operator by a commutation relation. But q is not well adopted by mathematical analysis. This work proposes to reedify the operational calculus based on both derivative, position operators and a rule for the permutation of non commutative operators. Applcations for resolutions of differential equations , for obtaining operators that create by acting on monomials the polynomials most utilized in sciences and engineering. Their generating finctions, orthogonality and properties are then obtained in a coherent and very close way. This work shows also how, only from translation and Gaussian transform, to obtain the dilatation, the transform of x into real powers of x, the partial Fourier, Fourier, canonical transforms and how to get by them the transforms of arbitrary operators and functions, the related integral transform. The Laplace transform is also treated by the present method. Via the creation and the destruction operators applications in quantum mechanics are promising

Born in 1942 at Bien Hoa, Vietnam; studied at Lycee Petrus Ky and DaLat, Saigon Univ , Licence in Sci Math (1964); Dr Sci Phys at Univ libre de Bruxelles (1971) researcher in quantum mechanics and operational calculus in Belgium. Returning back to Vietnam (1996) lecturer at the HoChiMinh city University of Natural Sciences. Now retired at Bien Hoa