Plenary Lecture

Rayleigh Quotient Flattening Methods for the Eigenvalue Problems of Linear Operators Between Separable Hilbert Spaces

Professor N. A. Baykara
Marmara University, Mathematics Department
Istanbul, TURKEY
Email: nabaykara@gmail.com

Abstract: An algebraic Rayleigh quotient is a ratio of two quadratic forms whose kernel matrices can be considered as symmetric or Hermitian without any loss of generality. The vectors in these types of entities are taken from a Cartesian space whose dimensionality is equivalent to the number of the elements of those vectors. Cartesian spaces are structures very close to the real life spaces which can be perceived by our sensual organs. Analogues to real spaces, the distance between the points, the norm of the vectors each of which is considered to represent a point in the space, and the angle between the vectors are all defined in Cartesian spaces. Hence, in the sense of linear vector spaces theory they belong to the class of Hilbert spaces. The only restriction in Cartesian spaces is the members of the space. They are specific structures, algebraic vectors while the members of a given Hilbert space can be any kind of mathematical object as long as it fulfills the requirements to form a linear vector space. The dimensionality of a Cartesian space is generally finite as long as infinite algebraic vectors are not under consideration. Whereas the dimensionality of Hilbert spaces are generally infinite in the most widely encountered practical cases. For example, functions, univariate or multivariate, form a Hilbert space where the inner product is the integral over the product of two functions appearing as the argument of the relevant functional. These spaces which can be called function spaces are infinite dimensional since any function belonging to such a space can be uniquely expressed as a linear combination of certain linearly independent universal functions which are denumerable. The denumerability brings the concept of separability. We focus on only separable Hilbert spaces. The linear operators on the other hand, mapping from an Hilbert space to itself very frequently comes to the scene in many applications of mathematical and engineering modellings. A quadratic form for such an operator is defined as the inner product of an arbitrary function's image under the considered operator with the same arbitrary function. If a quadratic form of this typeis accompanied by a condition, like normalization, on the arbitrary function then it is better to use Rayleigh quotient which is a ratio between the quadratic form and a divisor quadratic form like the one having unit operator as the argument, instead of introducing Lagrange multipliers. The kernel of the divisor quadratic form does not need to be a unit operator and some other, but preferably positive definite, operators can be used to this end. Then the Rayleigh quotient can be called \Weighted Rayleigh Quotient". It is very well known that the extrema of a Rayleigh (or Weighted Rayleigh) quotient are the eigenvalues of the eigenvalue (or weighted eigenvalue) problem of the linear operator under consideration and these values are achieved when the arbitrary function of the Rayleigh (or Weighted) quotient becomes equal to the eigenvector(s) corresponding to the relevant eigenvalue. This implies that making a Rayleigh quotient sufficiently at around an eigenvalue of the related eigenvalue problem facilitates the solution of the spectral problem related to the considered Rayleigh quotient. The solution of the eigenvalue problem related to the extremization of a Rayleigh quotient may not be easy depending on the structure of the linear operator under consideration, the weight operator (if any) and certain accompanying conditions like boundary conditions. So certain approximation methods ay be required. Their construction can be based on certain facilitating features of the Hilbert space under consideration. However, beyond those, it is possible to consider some closure properties. For example the subspace composed of multiples of an eigenfunction must be closed under the action of the considered linear operator. Similarly if there are some conditions to define the subspace (these conditions must be linear and homogeneous for producing subspace) the image of a function from that subspace under the relevant operator should be inside the same subspace. The so-called Wronskian approaches and Space Pruning techniques are amongst the approaches are based on these types of ideas. The talk will focus on these issues within the given time limitations.

Brief Biography of the Speaker: N. A. BAYKARA was born in Istanbul,Turkey on 29th July 1948. He received a B.Sc. degree in Chemistry from Bosphorous University in 1972. He obtained his PhD from Salford University, Greater Manchester, Lancashire,U.K. in 1977 with a thesis entitled “Studies in Self Consistent Field Molecular Orbital Theory”, Between the years 1977–1981 and 1985–1990 he worked as a research scientist in the Applied Maths Department of The Scientific Research Council of Turkey. During the years 1981-1985 he did postdoctoral research in the Chemistry Department ofMontreal University, Quebec, Canada. Since 1990 he is employed as a Staff member of Marmara University. He is now an Associate Professor of Applied Mathematics mainly teaching Numerical Analysis courses and is involved in HDMR research and is a member of Group for Science and Methods of Computing in Informatics Institute of Istanbul Technical University. Other research interests for him are “Density Functional Theory” and “Fluctuationlessness Theorem and its Applications” which he is actually involved in. Most recent of his concerns is focused at efficient remainder calculations of Taylor expansion via Fluctuation–Free ?Integration, and Fluctuation–Free Expectation Value Dynamics.