Keynote Lecture

Space Pruning Approach to the Solution of Eigenvalue Problems for Singular Ordinary Differential Operators

Professor Metin Demiralp
Istanbul Technical University
Informatics Institute

Abstract: Ordinary Differential Operators (ODOs) are encountered in many branches of sciences and engineering. Especially quantum mechanics have many systems described by Hamiltonians which are of this type. The eigenfunction space of such systems are mostly described by analytic basis functions and it is very well–known that the images of any eigenfunction under any nonnegative integer power of considered ODO remains in the very same space where eigenfunctions lie. However, the singular nature of the considered ODO causes the appearences of certain nonanalytic function components which do not belong to the space spanned by the basis set of analytic functions. To remove these undesired appearences, one can work with an arbitrary infinite linear combination over the entire basis set spanning the space where eigenfunctions lie and try to specifically choose the linear combination coefficients such that the undesired nonanalytic function components vanish. This, in fact, imposes at least one or maybe more than one linear vanishing conditions over the combination coefficients mentioned above. This action prunes the space spanned by the originally chosen basis set. We use the word “prune” since the dimensionality reduction via abovementioned linear vanishing equation impositions is finite against the infinite dimensionality of the original space.
The linear vanishing equation impositions are done not only for a single positive integer power of the ODO under consideration, but for all possible nonnegative integer powers of that ODO. This results in an infinite number of linear equations at the end. Because of the practicality, we do not go to infinity, but instead, until a particularly chosen positive integer for the ODO’s power. When these finite number of linear equations are used to eliminate some coefficients of the infinite linear combinations we have mentioned at the beginning, a new basis function set is obtained such that the first elements of this set approximate the eigenfunctions of the ODO under consideration.
In the space pruning approach, it is very important how the initial basis set is con- structed. The eigenvalue problem of an ODO is generally accompanied by the boundary conditions given at the end of the interval of the ODO’s independent variable, in other words, most widely encountered problems of ODOs are two point boundary value prob- lems. In addition to all these, one or both of the end points of the independent variable interval correspond(s) to singularities of the ODO. We prefer to focus on problems dealing with the cases with semi infinite interval [ 0, 1) Then, the zero endpoint is considered as the space pruning point as long as the singularity at this point is a regular one. This enable us to use the nonnegative powers of the independent variable as basis set. How- ever these basis set functions grow unboundedly at infinity urging us to use a common factor sufficiently rapid vanishing at infinity to suppress all of these types infinite grows. This common factor can be constructed by solving the asymptotic equivalent of the ODE corresponding to the eigenvalue problem of the considered ODO.
This presentation will focus on these type of issues and rather simple systems will be chosen as the target applications for the easy grasping of the basic philosophy.

Brief Biography of the Speaker: Metin Demiralp was born in Turkiye (Turkey) on 4 May 1948. His education including the university undergraduate level was completely in T¨urkiye. His BS, MS degrees and PhD are from ?Istanbul Technical University. After his theoretical chemistry, applied mathematics, and, numerical analysis years he is now basically working on computational sciences. 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 in collaboration with a neuroscience group at the Psychology Department in the University of Michigan at Ann Arbour in three years around 2010 (with certain publications in journals and proceedings).
Metin Demiralp has more than 100 papers in well known and prestigious scientific journals, and, more than 260 contributions 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.
Metin Demiralp has been officially retired on May 4th 2015 due to age limitation in Turkey state universities. However, he is still an active and productive scientist and occupying a post-retirement position like emeritus professor in his same institution at the moment. He has same group as before, PhD and MS students in the computational science and engineering program and teaching sophisticated graduate courses presently.

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