The Steklov spectrum of convex polygonal domains II: investigating spectral determination

TL;DR

This paper explores how the Steklov spectrum can uniquely identify convex polygons, proving spectral determination for most triangles, certain quadrilaterals, and regular polygons, and distinguishing polygons from smooth domains.

Contribution

It demonstrates that the Steklov spectrum can often uniquely determine convex polygons and distinguish them from smooth domains, advancing spectral geometry understanding.

Findings

Almost all triangles are uniquely determined by their Steklov spectra.

Spectral determination results for rectangles, parallelograms, and kites.

Triangles and quadrilaterals are distinguished from smooth domains by their spectra.

Abstract

The extent to which the geometry of an object is determined by some associated spectral data is a longstanding problem. We investigate this problem in the context of the Steklov spectrum, focusing on convex polygons. We prove that almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles; further results depending on the types of angles in the triangles are given. We examine three special classes of convex quadrilaterals--rectangles, parallelograms, and kites--and obtain results ranging from unique spectral determination to determination up to three possibilities. For regular $n$-gons, we are again able to prove spectral determination within certain classes of polygons. More generally, we investigate the extent to which the Steklov spectrum distinguishes convex polygons from simply-connected domains with smooth boundary; that is, does the Steklov spectrum detect corners? We prove that triangles and quadrilaterals are spectrally distinguished from such smoothly bounded domains; moreover, we show that having the same Steklov spectrum as such a domain imposes substantial restrictions on the edge lengths of higher-order $n$-gons. Throughout, our main tool is the characteristic polynomial developed in works by Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher.