**Plenary Lecture
Kronecker Power Series in Quantum Mechanical Probabilistic Evolution Approach: Managing Arbitrariness in Spectral Issues of the Propagation Superoperator**

**Professor Metin Demiralp**

Resigned Principal Member of Turkish Academy of Sciences

Istanbul Technical University

Informatics Institute Istanbul

TURKEY

E-mail: metin.demiralp@gmail.com

**Abstract: ** Recently we have started to use Kronecker power series instead of the multivariate Taylor series, in the formulation of the “Probabilistic Evolution Approach (PEA)” to ODEs, Quantum Expectation Value Dynamics, and, Classical Statistical Mechanics within the perspective of Liouville equation, density and partition functions. The basic idea has been to expand the unknown entities in terms of Kronecker powers of a vector describing the system under consideration. This system vector is either composed of the temporally varying unknowns in the case of ODEs or certain operators’ expectation values varying in time for the other cases. The Kronecker powers of the state vector (or their expec tation values in the case of Quantum Mechanics or Classical Statistical Mechanics) have been considered as the basis set elements and certain ODEs for each of them have been constructed. The result in all cases (for ODEs, Quantum Expectation Values, Statistical Mechanical Expectation Values) was a first order linear homogeneous infinite vector ODE with generally initial value impositions, such that the coefficient matrix (we call “Evolution Matrix) for this infinite explicit vector ODE was a constant infinite square matrix. The formal analytical solution of this infinite vector ODE can be obtained and requires the evaluation of an exponential matrix varying in time whose proportionality coefficient is the Evolution Matrix. This evaluation is facilitated when the evolution matrix (which is in upper block Hessenberg form most generally) becomes block triangular because of certain limitations in the system. Triangularity makes the spectral analyses quite simple. Beyond that, the case of multinomiality where the Evolution Matrix has the main diagonal and its few upper neighbor diagonals as the nonvanishing substructures, enables to use the finite order block recursions to get solution to PEA equations.

The case of conicality is the simplest form of the multinomiality and corresponds to two term block recursions whose solutions can be analytically constructed as infinite series of the initial values of the Kronecker powers of the state vector or its expectation values. In fact all multinomial cases can be converted to two term block recursions via appropriate order reductive manipulations.

What we have told above is somehow the review of the last year developments of the “Probabilistic Evolution Theory” and it will be kept sufficiently comprehensive but, at the same time, sufficiently short during the presentation. The remaining part is completely new and based on recently developed issues. The Kronecker powers of the state vector(s) contain certain number of identicalities or linear dependences as the price of the brevity in the relevant multivariate representation. These can be in fact reflected to the Kronecker power series coefficients as certain level of arbitrarinesses. These arbitrarinesses can be expressed in terms of certain flexible parameters which can be determined as what we want to obtain, of course, within certain limitations.

A special emphasis will be given on the commutator algebra over the state vector’s Kronecker powers. The propagation superoperator acting on an operator to give the time variant exponential function image of the operator’s commutator with the Hamiltonian. The construction of certain eigenoperators of the propagation superoperator will be explained in the perspective of the Kronecker power series and the management of the arbitrariness appearing there.

**Brief Biography of the Speaker: ** Metin Demiralp was born in T¨urkiye (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, ÿIstanbul 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 ÿIstanbul 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 90 papers in well known and prestigious scientific journals, and, more than 200 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.