Analysis and Applied Mathematics Seminar
Hassan Babaei
University of Illinois Chicago
1-dimenstional Dirac equation on half-line with Dirichlet boundary conditions
Abstract: In this talk, I will present the construction of solutions to the one-dimensional Dirac
equation on the half-line with Dirichlet boundary conditions. While the Dirac equation is a
four-dimensional system arising in quantum field theory, I will focus on the one-
dimensional initial-boundary value problem. The primary analytical tool is the unified
transform method (or known as Fokas method), which provides an explicit representation
of the solution. To introduce the method, I will first demonstrate it in the context of the heat
equation on the half-line. I will then apply it to the Dirac equation to derive explicit solution
formulas and analyze the associated boundary behavior at the origin. Furthermore, I will
discuss the long-time dynamics of these solutions. If time permits, I will conclude with a
discussion of Sobolev-space energy estimates for the solutions, including control of both
spatial norms and time-regularity.
Monday September 29, 2025 at 4:00 PM in 636 SEO