Laplace Subspectrality

TL;DR

This paper investigates how the spectral properties of the Laplacian relate to the geometry of shapes, focusing on subspectrality and using variational and analytical methods to understand these relationships.

Contribution

It develops new analytical tools to study Laplace subspectrality, explores subspectrality in rectangles, constructs counterexamples, and relates eigenvalues to geometric functions.

Findings

Established variational methods for subspectrality analysis

Analyzed Laplace subspectrality in rectangles and other domains

Connected length subspectrality to Laplace subspectrality via heat trace

Abstract

Exploring the relationship between geometry and the resonant frequencies of a shape is of interest to pure and applied mathematicians. These resonant frequencies are related to the spectrum of the Laplacian, a partial differential operator. A long-standing research program asks: What geometric information can one deduce from these harmonics? This dissertation explores a related question. Say one shape is subspectral to another provided each successive resonant frequency of the one is less than the corresponding frequency of the other. What information can be deduced about the relationship between the shapes' geometries? We use variational arguments to study subspectrality of self-adjoint operators. We develop analytical tools to study Laplace subspectrality. We then study subspectrality in rectangles, construct counterexamples in more general classes of domains, and use the heat trace to relate length subspectrality to Laplace subspectrality. Finally, we discuss a more general question relating eigenvalues to functions, stemming from a 1961 conjecture of Polya.