Keynote Lecture

Gamma Function Expansions For Analytic Solutions of Infinite Linear Recursions: Polynomial Coefficient Cases

Professor Metin Demiralp
Istanbul Technical University
Informatics Institute

Abstract: Infinite linear recursions arise in many branches of sciences and engineering. The constant coefficients case can be handled by using a very powerful theory while the other cases where the variable coefficients are on the scene may necessitate case-specific methods to get analyticality. Amongst these variable coefficient cases, the linear recur- sions having polynomial coefficients play an important role in the handling of cases truely encountered in practice. In literature, factorial series take important places for such prob- lems. However, it is better to extend the case from factorial series to more amenable tools. To this end a series of certain Gamma functions can be used quite efficiently. This series representation takes out the rapid growth of the unknown as the recursion index tends to go to infinity and converts the problem to another linear recursion whose characteristics are milder than the original one. Thus it becomes possible even to numerically solve the resulting recursion especially by using certain truncating algorithms.
In this presentation the exemplification of the cases will be focused on the cases appearing in quantum mechanics. We have quite recently shown that a very useful formula over the expectation value of an algebraic function multiplication operator can be used to evaluate the eigenstates of an autonomous quantum system. To this end, the utilization of an appropriately chosen basis set enabled to get an infinite linear recursion. We have shown that this recursion can give not infinite series but finite sum of Gamma functions for quantum hydrogen-like systems and quantum harmonic oscillator.
The quantum states of an harmonic oscillator are composed of only discrete energy eigenvalues and there is no continuos spectrum corresponding to scattering phenomena. We have proven that the use of Gamma function expansion produces all of these possible states. It gives the energies of the system and also the expectation value of the position operator. These expectation values however reveal the eigenfunctions of the system Hamiltonian.
The quantum states of a hydrogen-like system is different. The energy spectrum of the system is composed of both discrete and continuous states. Discrete spectrum corresponds to the cases where the two particles composing the system move in a bounded manner. Whereas, the continous cases break down the boundedness of the particles by leading us to scattering phenomena. The abovementioned gamma function expansion again finds the energy values correctly and reveals the true eigenfunctions via the position integer power expectation values.
The presentation will discuss these types of issues as the time period permits.

Brief Biography of the Speaker: Metin Demiralp was born in Türkiye (Turkey) on 4 May 1948. His education from elementary school to university was entirely in Turkey. He got his BS, MS degrees and PhD from the same institution, ˙Istanbul Technical University. He was originally chemical engineer, however, through theoretical chemistry, applied mathematics, and computational science years he was mostly working on methodology for computational sciences and he is continuing to do so. He has a group (Group for Science and Methods of Computing) in Informatics Institute of ˙Istanbul Technical University (he is the founder of this institute). He collaborated with the Prof. Herschel A. Rabitz’s group at Princeton University (NJ, USA) at summer and winter semester breaks during the period 1985-2003 after his 14 month long postdoctoral visit to the same group in 1979-1980. He was also (and still is) in collaboration with a neuroscience group at the Psychology Department in the University of Michigan at Ann Arbour in last three years (with certain publications in journals and proceedings).
Metin Demiralp has more than 100 papers in well known and prestigious scientific journals, and, more than 230 contributions together with various keynote, plenary, and, tutorial talks to the proceedings of various international conferences. He gave many invited talks in various prestigious scientific meetings and academic institutions. He has a good scientific reputation in his country and he was one of the principal members of Turkish Academy of Sciences since 1994. He has resigned on June 2012 because of the governmental decree changing the structure of the academy and putting politicial influence possibility by bringing a member assignation system. Metin Demiralp is also a member of European Mathematical Society. He has also two important awards of turkish scientific establishments.
The important recent foci in research areas of Metin Demiralp can be roughly listed as follows: Probabilistic Evolution Method in Explicit ODE Solutions and in Quantum and Liouville Mechanics, Fluctuation Expansions in Matrix Representations, High Dimensional Model Representations, Space Extension Methods, Data Processing via Multivariate Analytical Tools, Multivariate Numerical Integration via New Efficient Approaches, Matrix Decompositions, Multiway Array Decompositions, Enhanced Multivariate Product Representations, Quantum Optimal Control.