The proposers intend to continue their study of meromorphic functions using the methods of geometric theory of functions and topology, the approach which already brought significant results in their previous research. All problems in this proposal arise naturally from the results of the previous research of the proposers funded by NSF. The proposers intend to concentrate on the following specific problems: geometry and topology of real and complex spectral loci of the one-dimensional Schrödinger operators with polynomial and rational potentials; singularities of implicit analytic functions defined by entire relations; general properties of certain classes of meromorphic functions occurring in holomorphic dynamics and in spectral theory of the Schrödinger operators; polynomial approximation of discontinuous functions on systems of intervals. Proposed research will expand our understanding of solutions of transcendental equations emerging in analysis and mathematical physics.
Meromorphic functions constitute the most basic class of functions used in mathematics and most of its applications. In addition to elementary functions like the exponent, cosine and tangent, this class includes higher transcendental functions, indispensable in physics and engineering. Previous work of the authors on this subject already found important applications in control theory, material science, computer science, signal processing, mathematical physics and astrophysics. This proposal contains several problems of the function theory motivated by quantum mechanics, and the proposers expect their results to yield deeper understanding of this fundamental physical theory. The PI and coPI will also work with graduate students on research related to the project.
