Georgia Institute of Technology College of Sciences

When an object is small enough that quantum mechanics matters, many of its physical properties, such as energy levels, are determined by the eigenvalues of some linear operators. For quantum wires, waveguides, and graphs, geometry and topology show up in the operators and affect the set of eigenvalues, known as the spectrum. It turns out that the spectrum can't be just any sequence of numbers, both because of some general theorems about the eigenvalues and because of inequalities involving the shape. I'll discuss some of the extreme cases that test the theorems and inequalities and connect them to the shapes of the structures and to algebraic properties of the operators.To understand this lecture it would be helpful to know a little about PDEs and eigenvalues, but no knowledge of quantum mechanics will be needed.