Reducibility of Schrödinger operators on multilayer graphs - Villalobos Alvarado, Jorge 1993- (2024)
A local defect in an atomic structure can engender embedded eigenvalues when the associated Schrödinger operator is either block reducible or Fermi reducible, and having multilayer structures appears to be typically necessary for obtaining such types of reducibility. Discrete and quantum graph models are commonly used in this context as they often capture the relevant features of the physical system in consideration. This dissertation lays out the framework for studying different types of multilayer discrete and quantum graphs that enjoy block or Fermi reducibility. Schrödinger operators with both electric and magnetic potentials are considered. We go on to construct a discrete model of AA-stacked bilayer graphene with embedded eigenvalues and prove that the corresponding bound states can exist stably within the region of continuous spectrum with respect to variations of a perpendicular magnetic field. This is accomplished by creating a defect that is compatible with the interlayer coupling, thereby shielding the bound states from the effects of the continuous spectrum, which varies erratically in a pattern known as the Hofstadter butterfly.