Asymmetry, Embedded Eigenvalues, and Resonance for Differential Operators

Louisiana State University, Baton Rouge LA

Investigators

Abstract

This project addresses the mathematical foundations of physical principles that underlie the design of optical and electronic devices. The pervading concept is the principle of "resonance", which lies behind lasers, filters, antennas, and light-emitting diodes, etc. The technical term "embedded eigenvalue" in the title refers to a mathematical concept that is intimately intertwined with resonance; it is related to specific configurations of electromagnetic fields in a structure or device and how they interact with external inputs. The project combines theory with computer simulations. The principal investigator runs a research seminar, which integrates seasoned researchers, a post-doctoral associate, doctoral students, and undergraduate students, each of whom is involved in a specific aspect of the project. Emphasis is placed on developing methods at the interfaces of mathematics, physics, and engineering, paying special attention to bridging communication gaps between the disciplines and training a new generation of researchers who can work in an interdisciplinary setting. Classically familiar examples of embedded eigenvalues are induced by finite symmetry groups of a differential operator. Recent demonstrations of non-symmetry-induced spectrally embedded bound states in electromagnetic structures have opened the way toward a richer set of resonance phenomena. This project develops a theory of asymmetry and embedded eigenvalues. The investigations employ functional analysis and partial differential equations, complex analysis, spectral theory, and analytic geometry. These are the specific problems of the project: (1) It investigates the precise connection between asymmetry, embedded eigenvalues, and reducibility of an analytic variety called the "Fermi surface" for periodic operators, including the partial differential equations (PDE) of physical systems and mathematical graph models; (2) The principal investigator's recent work shows that bilayer quantum-graph graphene is special in that its Fermi surface is always reducible, regardless of the type of asymmetries in the potentials on the edges connecting the two sheets of graphene. The project seeks the mathematical reasons behind this and the PDE counterpart; (3) The investigator and collaborators are developing numerical algorithms for probing the very complex situation of embedded eigenvalues for the Maxwell equations of electromagnetics; (4) The existence and construction of embedded eigenvalues for the Neumann-Poincare boundary-integral operator that is fundamental to the theory of "plasmonic resonances" is also being studied. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.