A Numerical Study on an Eigenvalue Conjecture Open Access

He, Junying (2014)

Permanent URL: https://etd.library.emory.edu/concern/etds/f7623c94w?locale=en


This paper provides a brief introduction about spectral problem in mathematical study, and presents several important results related to spectral problem. Starting from Kac's question that whether we can use the full spectrum to determine the associated shape, the paper shows that the answer to Kac's question is "no" in general, but "yes" in the class of Euclidean triangles according to some related study. In the class of triangles, the paper produces a nice formula to calculate Dirichlet eigenvalues for equilateral triangles. For general triangles, the paper introduces a numerical method called finite-element method, to approximate the true eigenvalues. Moreover, the paper provides a numerical evidence for the eigenvalue conjecture on isosceles triangles.

Table of Contents

Table of Contents

Introduction P 1-9

Spectral Problem P 1-5

The Class of Triangles P 5-9

The Eigenvalue Problems on Triangles P 9-35

Eigenvalues for Equilateral Triangles P 9-20

Numerical Approach P 20-35

A Numerical Study on Isosceles Triangles P 35-44

Conclusion P 45

Reference P 46-47

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