Numerical Ranges and its applications to Physics and to Quantum Computing

This proposal concerns the study numerical ranges and numerical radii of certain operators in Krein and Hilbert spaces. The notion of numerical range (numerical radius, and its generalizations) is related to and has applications in many different branches of pure and applied science, such as operator theory, functional analysis, C*-algebras, Banach algebras, matrix norms, inequalities, numerical analysis, perturbation theory, matrix polynomials, systems theory, quantum physics, quantum computing, etc.. On the other hand, a wide range of mathematical tools including algebra, analysis, geometry, combinatorial theory, and computer programming are useful in its study. The generalization to Krein spaces is important and potentially valuable in applications, for instance in the investigation of the spectra of non-hermitian operators with PT symmetry, an area of study within the scope of this project. The project tasks focus the investigation of the properties of the numerical range (and some of its variations and generalizations) as a set, and the implementation of computational procedures to generate numerical ranges. In general, having an accurate plot of the numerical ranges would help one to get deeper insight about the theory. There exist computer programs for plotting certain kinds of generalized numerical ranges. To improve the efficiency of the existing programs and generating new ones are fields of our research. In many instances, the geometrical properties of the numerical range can be used to classify special classes of operators (self-adjoint, normal, unitary), so we will emphasize the investigation of geometrical properties of the set. In addition, the study of spectral inequalities for operators in Krein spaces using the theory will be developed. Some of the problems under consideration are motivated by problems in quantum physics.

The team is investigating the following problems

1) Classification of the algebraic curves which are boundary generating curves of the numerical range

2) Classification of the J-numerical range of J-Toeplitz and J-circulant matrices

3) Investigation of the incidence of the concept of C-numerical range in quantum-computing

4) Investigation of the spectra of certain non.hermitian operators with PT symmetry arising in quantum physics

Natália Isabel Quadros Bebiano Pinheiro da Providência e Costa

João Pinheiro da Providência e Costa

João da Providência Santarém e Costa

Maria da Graça Soares

Ricardo Emanuel Cunha Teixeira

Ana Cristina Becerra Nata dos Santos