The Fundamental Gap Conjecture on Polygonal Domains

Details The Fundamental Gap Conjecture on Polygonal Domains The Fundamental Gap Conjecture on Polygonal Domains

2008-10-27

arxiv.org/abs/0810.4937v2

Mathematics

Spectral Theory

18 pages, 8 figures. Corrected and included more references and made minor corrections

Scientific paper

We study the fundamental gap on convex planar polygonal domains. Our main result is a compactness theorem when the domain is a triangle which demonstrates that for any triangles which collapse to the unit interval, the gap is unbounded. Consequently, there exists a triangle which minimizes the fundamental gap; like other authors, we conjecture this is the equilateral triangle. Our second theorem generalizes the compactness result to convex polygonal domains. Perhaps surprisingly, this theorem shows that generically the gap is unbounded on convex polygonal domains which collapse to the unit interval. This work initiates a general program to prove the fundamental gap conjecture on convex planar domains by studying the gap on convex polygonal domains.

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