The investigator and his colleagues develop numerical methods for eigenvalue problems in infinite-dimensional spaces. At the heart of the effort is the recently developed hybrid diagonalization/Monte Carlo approach, which they feel has led to significant advances in the computational treatment of the general low energy eigenvalue problem. They study the many unresolved mathematical issues related to approximate methods for low energy eigenvalues and eigenvectors in infinite-dimensional spaces. Among other things, the collaborators rigorously analyze the accuracy of the diagonalization/Monte Carlo approach, study the relation between eigenvalue distribution and eigenvector structure, generalize the approach to non-Hermitian matrices, and improve the computational treatment of clustered eigenvalues. In addition to studying the underlying mathematics, the investigators also begin production on a public domain library for infinite-dimensional eigenvalue problems. Eigenvalue problems play a crucial role in many diverse branches of 21st-century science and engineering. This includes for example the electronic properties of nanoscale structures and new materials; the interactions of protons and neutrons in heavy nuclei; and gene similarity database studies in bioinformatics. The eigenvalue problem consists of finding certain attributes of a given square matrix, M. Specifically one is seeking column vectors, v, such that M times v is again proportional to v. The proportionality constant is called an eigenvalue of M and v is the corresponding eigenvector. In this project computational experts in computer science, applied mathematics, and physics collaborate to investigate large-dimensional matrix eigenvalue problems. The collaborators study the underlying mathematical issues related to this problem and develop a public domain library that can be applied to matrices of infinite dimension. The extension to infinite dimensions is crucial for many applications, particularly those in quantum physics and chemistry, but the existence of this library will impact many other areas that feature large-dimensional eigenvalue problems and are currently intractable with existing methods.
