Riemann-Hilbert Problems, Toeplitz Determinants and Applications

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A considerable variety of problems in mathematics, physics and engineering can be expressed in terms of Toeplitz matrices (matrices whose diagonals parallel to the main diagonal are all constants). Recently the asymptotic study of Toeplitz determinants has found important applications in random matrix theory and mathematical physics using the Riemann-Hilbert analysis and operator theoretic methods. Of particular interest are the so-called Szegö asymptotics of Toeplitz determinants generated by smooth functions and Fisher-Hartwig asymptotics generated by functions that may possess discontinuities, zeros, winding and (integrable) singularities.The asymptotic study of Toeplitz determinants with an external parameter T demonstrates the importance of double-scaling limits in mathematics and physics. For example, regarding the 2D Ising model and the XY spin chain model, the Toeplitz determinants can be used to analyze the double-scaling limit for certain correlation functions as T goes to the critical value Tc and simultaneously the separation between the spins goes to infinity.As a concrete application, the proposal includes the study of entanglement, which is a quantum phenomenon providing a primary resource for quantum computation and information processing. As a fundamental measure of "quantumness," it shows how much quantum effect we can observe and use to control one subsystem by another. In the last two decades, substantial effort has been made to understand entanglement in various quantum systems. Of particular interest is the so-called XY quantum spin chain, which can be viewed as a toy model for quantum computers. The model consists of particles ordered in a line and indexed by the positive integers with only nearest neighbor interactions. Because the XY spin chain allows for exact calculations for physically relevant quantities, they provide valuable insight into real-life physical systems where such calculations are currently impossible.The methods of the proposed research include the Riemann-Hilbert problem (RHP), which has a long and impressive history going back to Riemann's dissertation (1851) and Hilbert's related results at the beginning of the 20th century. The Riemann-Hilbert problem, which can be described as a problem of finding an analytic function in the complex plane with a prescribed jump across a given curve, is closely connected to one-dimensional singular integral operators, convolution operators, Toeplitz operators, and Wiener-Hopf operators. In addition to the Riemann-Hilbert approach, new operator-theoretic methods will be developed related to concrete operators and matrices.