在這篇論文，我們證明二個定理。首先，如果ρ0是第一個反週期特徵值的極小化密度函數，我們證得ρ0為hχ(a,b)+Hχ[0,π]/(a,b) a.e.，最後，我們證明，在我們討論的空間底下，第一個反週期特徵值的最小值、第二個狄利克雷（Dirichlet）和第二個諾伊曼（Neumann）的特徵值會相等。

此外，假設極小化ρ0對稱於π/2，然後導出不連續點和特徵值的非線性方程式，對於這些方程式，我們使用數學軟體（Mathematica）來求出相對於極值函數的特徵值和不連續點。

In this thesis, we prove 2 theorems. First let ρ0 be
a minimizing (or maximizing) density function for the first
antiperiodic eigenvalue λ1' in E[h,H,M], then ρ0=hχ(a,b)+Hχ[0,π]/(a,b) (or ρ0=Hχ(a,b)+hχ[0,π]/(a,b)) a.e. Finally, we prove minλ1'=minμ1=minν1 where μ1 and ν1 are the first Dirichlet and second Neumann eigenvalues, respectively. Furthermore, we determine the jump point X0 of ρ0 and the corresponding eigenvalue λ1', assuming that ρ0 is symmetric about π/2 We derive the nonlinear equations for this jump point X0 and λ1',then use Mathematica to solve the equations numerically.

1.Introduction
2.Density functions with extremal antiperiodic eigenvalues
3.Determination of eigenvalues and jump points of extremal functions
4.Appendix A: Proof of theorem 1.1-ordering of spectral values
5.Appendix B: Computation of eigenvalue gaps and jump points for Schrodinger operators

1.M. S. Ashbaugh and R. D. Benguria, Eigenvalue ratios for
Sturm-Liouville operators, J. Diff. Eqns (1993)
205-219.
2. M. S. Ashbaugh and R. Svirsky, Perodic potentials with
minimal energy bands, Proc. Amer. Math. Soc (1992) 69-77
3.H. H. Chern and C. L. Shen, On the maximum and
minimum of some functionals for the eigenvalue problem of
Sturm-Liouville type, J. Diff. Eqns (1994)
68-79.
4.E. A. Coddinton and N. Levinson, Theory of Ordinary
Differential Equations, McGraw Hill (1984)
5.H. K. Huang, Optimal estimates of the eigenvalue gap
and eigenvalue ratio with variational analysis, Unpublished
master thesis, National Sun Yat-sen University, Kaohsiung, Taiwan,
R.O.C.
6.M. J. Huang, On the eigenvalue ratio with vibrating strings,
Proc. Amer. Soc.(1999) 1805-1813
7.M. Horvath, On the first two eigenvalues of
Sturm-Liouville operators, Proc. Amer. Math. Soc(2002) 1215-1224.
8.J. B. Keller, The minimum ratio of two eigenvalues,
SIAM J. Appl. Math(1976) 485-491
9.R. Lavine, The eigenvalue gap for one-dimensional
convex potentials, Proc. Amer. Math. Soc., 121, (1994) 815-821
10.T. Mahar and B. Willner, An extremal eigenvalue problem, Comm.
Pure Appl. Math(1976) 517--529.
11. W. Magnus and S. Winkler, Hill's equation, Dover, New
York. (1979)