Spectra of Periodic Schrödinger Operators on the Octagonal Lattice

We consider the spectrum of the Schrödinger operator on an octagonal lattice using the Floquet-Bloch transform of the Laplacian. We will first consider the spectrum of the Laplacian in detail and prove various properties thereof, including spectral-band limits and locations of singularities. In addition, we will prove that Schrödinger operators with 1-1 periodic potentials can open at most two gaps in the spectrum precisely at energies $pm1$, and that a third gap can open at 0 for 2-2 periodic potentials. We describe in detail the structure of these operators for higher periods, and motivate our expectations of their spectra.

Name thesis:degree_name

Master of Science

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2020

Author dc:creator

• Storms, Rebecah Helen

Chair dc:contributor.committeechair

• Embree, Mark P.

Committee members dc:contributor.committeemember

• Rossi, John F.
• Elgart, Alexander
• Fillman, Jacob

• Spectral Theory
• Schrödinger Operator
• Periodic Graphs

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

vt_gsexam:26691

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/99147