Parikshit Upadhyaya: The Eigenvector-Dependent Nonlinear Eigenvalue Problem

Many numerical methods geared towards solving the Schrödinger equation eventually have to solve a nonlinear eigenvalue problem where the nonlinearity is present due to the dependence on the eigenvectors. This problem is usually solved using an iterative procedure called the “Self-consistent field”(SCF) iteration or one of its variants. Since it is an iterative algorithm, we can always ask the following questions:
1) Under what conditions does this algorithm converge to the actual solution?
2) What makes the rate of convergence slower/faster?

In this talk, we will begin by discussing the sources of the problem("Hartree-Fock" discretization and "Density Functional Theory") and one of many formulations of the SCF iteration. Eventually, we will look at "answers" to the two questions. No previous knowledge of numerical methods for Schrödinger equations is required to understand the talk.