Eigenvalue Ratios for Sturm-Liouville Operators
Eigenvalue Ratios for Sturm-Liouville Operators
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DOI：
10.1006/jdeq.1993.1047
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发表时间：
1993-05
影响因子：
2.4
通讯作者：
M. Ashbaugh;R. Benguria
中科院分区：
文献类型：
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作者：
M. Ashbaugh;R. Benguria
本文证明了Sturm-Liouville方程−[p(X)y‘]’+q(X)y=λw(X)y的特征值比的各种最优界和某些特化。我们的结果主要涉及Dirichlet边界条件的正则情形，但也可以对其他情形进行各种推广和推广。本文的结果推广了前文关于一维薛定谔方程λy“+q(X)y=λy在具有Dirichlet边界条件和非负势(Q≥0)的有限区间上的结果。特别地，我们得到了λm/λ1≤km2/k，其中常数k，K对所有x满足0<；k≤p(X)w(X)≤K.如果q≡0，也可以得到下界.我们的方法涉及到对薛定谔案例中使用的普鲁弗变量技术的轻微修改。我们还检验了我们最近在一维(Sturm-Liouville)背景下证明Payne-Polya-Weinberger猜想的结果。最后，对于非均匀拉伸弦的特殊情况，我们将我们的一般界与Keller以及Mahar和Willner的详细分析相比较。
In this paper we prove various optimal bounds for eigenvalue ratios for the Sturm-Liouville equation − [p(x) y′]′ + q(x)y = λw(x)y and certain specializations. Our results primarily concern the regular case with Dirichlet boundary conditions though various extensions and generalizations to other situations are possible. Our results here extend the result λm/λ1 ≤ m2 obtained in a previous paper for the one-dimensional Schrodinger equation, − y″ + q(x)y = λy, on a finite interval with Dirichlet boundary conditions and nonnegative potential (q ≥ 0). In particular, we obtain λm/λ1 ≤ Km2/k, where the constants k, K satisfy 0 < k ≤ p(x) w(x) ≤ K for all x. If q ≡ 0, lower bounds can also be obtained. Our methods involve a slight modification of the Prufer variable techniques employed in the Schrodinger case. We also examine the consequences of our recent proof of the Payne-Polya-Weinberger conjecture in the one-dimensional (Sturm-Liouville) setting. Finally, we compare our general bounds to the detailed analyses of Keller and of Mahar and Willner for the special case of the inhomogeneous stretched string.