Speciality:
01.04.02 (Theoretical physics)
Birth date:
05.02.1940
Phone: +7 (495) 137 63 71, +7 (0872) 34 73 83
Fax: +7 (495) 938 20 54
E-mail: Keywords: Simple and multilpe hypergeometric series; special functions; operator factorization of functional series;symbolic computer analysis of hypergeometric series;particular formula classes (transformation theory; reduction formulas; recurrence relations; generating functions; addition formulas;linearization theorems; positivity proofs; coefficients of functional expansions); quantum theory of molecular electronic structure; variational methods; multicenter integrals; regularization of divergent integrals; expansions of irreducible spherical tensors in terms of translated functions; Fourier-transforms of atomic orbitals; higher order derivatives of composite functions.
UDC: 517.521.5, 517.584, 517.586, 517.588, 517.583, 519.6, 519.677, 519.671, 519.68 MSC: 33C05, 33C10, 33C15, 33C20, 33C45, 33C50, 33C55, 33C65, 33C70, 33F05, 33F10, 65D20, 42A38, 35Q40, 81V55, 90-08, 68T35, 68W30
Subject:
I came to mathematics from theoretical physics having gone a long way through theoretical (quantum) chemistry. It exerted a strong influence on my scientific interests greatly extending their range and having exorcised the "abstactions" typical for pure mathematics. The earlier papers (with A. B. Alexandrov and Yu. A. Kukharenko) connected with kinetic equation and Green function formalism for non-ideal Fermi-systems can hardly present any interest at present. A useful parametrization of iterative mapping in the variational Hartree–Fock–Roothaan (HFR) approach to solving Schrodinger equation was introduced in 1974 with the aim to acceletate the convergence of the method and to eliminate divergences. An adiabatical principle for elimination of bifurcation problem in the HFR-method was put forward. A new class of basis functions (the so called "mosaic orbitals") was introduced to eliminate the problem of three- and four-center integrals in the HFR-method. A simple procedure for regularization of divergent "molecular" integrals was elaborated. It was shown that, like the Taylor expansion, the coefficients of the expansion of an arbitrary atomic orbital (AO), having the structure of an irreducible spherical tensor, in terms of translated AO's have the form of differential operators acting on the initial AO. There were given explicit expressions for theу differential operators which, in themselves, turn to be again irreducible tensors. In passing, an explicit form of differential operators generating the atomic orbitals from simplest orbitals of the same class was found. A general class of functions containing all AO's of practical interest was introduced. Furier-transform of a general AO was expressed through the Appell function $F_2$. The $F_2$ can be represented as a finite sum of four-dimensional harmonics. This result is a partial generalization of the known V. A. Fock theorem. It was shown that the finite sum reduces to a single four-demensional harmonic in the case of hydrogen-like function. Apparently this result was first obtained in twenties by Linus Pauling. V. A. Fock re-derived this formal result by integral equation method and used it as a groundwork for his famous physical theory giving an explanation to the degeneration of hydrogen states. It was shown (in my paper) that there is yet another case where the sum reduces to a single four-dimensional harmonic. It is the case of the Slater functions widely used in the theory of molecular electronic states. In passing, a class of functions having simplest Fourier-transforms was established. The most interesting results are connected with the operator factorization principle introduced by the author in 1983. A new analytical operation over power series has been defined. This operation allows any hypergeometric series to be expressed through simpler series (closure property) and then use the known properties of the simpler series to establish, in the most direct manner, any properties of the initial complicated series. The necessary techniques has been elaborated in detail. New theoretical concepts have been introduced including canonical form of hypergeometric series, $\Omega$-equivalent relations, $\Omega$-equivalent operators, $\Omega$-identical transformations, $\Omega$-biorthogonality relations, etc. New unified theory of simple and multiple hypergeometric series of an arbitrary type has been constructed. The main results include a new theory of transformation and reduction formulas, a new algorithm of finding reducible cases, application of the algorithm to computer-aided derivation of new reduction formulas for Gelfand's functions, new important special transformations of the $F_4, G_2$ and $H_4$ series, a new technique for finding coefficients of functional expansions, duality relations between addition coefficients and linearization coefficients, a diagram technique for multiple series. To better comprehend, from an analytical point of view, an essence of the operator factorization method one should have familiarized oneself with the paper (V. I. Perevozchikov and V. A. Lurie, co-authors) appeared in an international journal "Fractional Calculus and Applied Analysis" published in Bulgaria [2000, 3 (2), 119–132]. Earlier this paper had been rejected by the British "Journal of Physics A: Math. Gen." on the ground that reader could have taken the paper for a wisecrack. Have fun! The operator factorization method shows considerable promise in the field of computer-aided symbolic treatment of multiple hypergeometric series. The developed programs (with O. S. Paramonova) realizing linear and quadratic transformations combined with reduction finding algorithm already showed a great efficiency and productivity. None of the symbolic manipulation programs existent in this field can compete with these programs including an old fashioned approach used as a foundation of the NIST Digital Library of Mathematical Functions (USA). The program aspect of the method can be clearly seen from computer treatment of Gelfand functions and a simple case of computer processing of the Appell function $F_4[a_1,a_2,a_1,b_2;x_1,x_2]$ (O. S. Paramonova and A. W. Niukkanen, Programmirovanie, 2002, no. 2, 24–29, in Russian). These program developments colud have become a part and one of the origins of further advancement of a powerful RAS mathematical server.
Main publications:
Niukkanen A. W. Generating differential operators for the basis functions in the variational LCAO-type methods // J.Math.Phys., 1984, 25 (3), 698–705 (see also: ibid. 1983, 24 (8), 1989–1991; ibid. 1985, 26 (6), 1540–1546).
Niukkanen A. W. Fourier-transforms of atomic orbitals. I. Reduction to four-dimensional harmonics and quadratic transformations // International J. of Quantum Chemistry, 1984, 25, 941–955 (see also: ibid. 1984, 25, 957–964).
Niukkanen A. W. Operator factorization method and addition formulas for hypergeometric functions // An international journal Integral Transforms and Special Functions, 2001, 11, 25–48.
