On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

AbstractThe main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz–Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved

On an Interpolation Problem for J-Potapov Functions

Let, J, be an m-by-m-signature matrix and let D be the open unit disk in the complex plane. Denote by P{J,0}(D) the class of all meromorphic m-by-m-matrix-valued functions, f, in D which are holomorphic at 0 and take J-contractive values at all points of D at which f is holomorphic. The central theme of this paper is the study of the following interpolation problem: Let n be a nonnegative integer, and let A_0, A_1, ..., A_n be a sequence of complex m-by-m-matrices. Describe the set of all matrix-valued functions, f, belonging to the class P{J,0}(D), such that the first n+1 Taylor coefficients of f coincide with A_0, A_1, ..., A_n. In particular, we characterize the case that this set is non-empty. In this paper, we will solve this problem in the most general case. Moreover, in the non-degenerate case we will give a description of the corresponding Weyl matrix balls. Furthermore, we will investigate the limit behaviour of the Weyl matrix balls associated with the functions belonging to some particular subclass of P{J,0}(D).Comment: 44 page

The truncated Hamburger matrix moment problems in the nondegenerate and degenerate cases, and matrix continued fractions

AbstractThe present paper deals simultaneously with the nondegenerate and degenerate truncated Hamburger matrix moment problems in a unified way based on the use of the Schur algorithm involving matrix continued fractions. A full analysis of them together with a relative matrix moment problem on the real axis is given. With the help of the correspondence between the moment problem on the real axis and the Nevanlinna-Pick (NP) interpolation, the solutions of the nontangential NP interpolation in the Nevanlinna class are derived as an application

Quantum Control Landscapes

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrodinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry, Vol. 26, Iss. 4, pp. 671-735 (2007

Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces

I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by rigid subvarieties in a nontrivial way). These rigidity results have a number of applications: First, they prove that many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot be smoothed (i.e., are not homologous to a smooth subvariety). Second, they provide characterizations of holomorphic bundles over compact Kahler manifolds that are generated by their global sections but that have certain polynomials in their Chern classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3 = 0, etc.).Comment: 113 pages, 6 figures, latex2e with packages hyperref, amsart, graphicx. For Version 2: Many typos corrected, important references added (esp. to Maria Walters' thesis), several proofs or statements improved and/or correcte