Estimates of the stability intervals for Hill’s equation
Estimates of the stability intervals for Hill’s equation
复制标题
希尔方程稳定区间的估计
DOI：
10.1090/s0002-9939-1963-0156023-0
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发表时间：
1963
期刊：
影响因子：
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通讯作者：
H. Hochstadt
中科院分区：
文献类型：
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作者：
H. Hochstadt
HARRY HOCHSTADT 本注释的目的是陈述并证明以下定理。定理。考虑希尔方程 (1) y" +[a + q(t)]y = 0，其中 q(t+ir) =q(t)，并且 q(t) 是有界的。设 X, 表示 (1) 的解的周期为 t 的第 i 个特征值，X' 表示 (1) 的解的周期为 2ir 的对应的第 i 个特征值。众所周知 [l] 对于这些特征值，Xo < Ai" Ú \l < Xi ^ X2 < X8' Û \í < X3 á X4 < • • • 对于区间 (2) (Xo, X/), (Xa, XO, (X2, XO, (X/, X3)) 内的所有 X， • • • 仅有有界解。对于这些区间外的 X，出现无界解。这些区间 (— », Xo), (X/, X2), (Xi, X2), (X/, X/) • • • 称为不稳定区间，则如果 q(t) 有 m 个连续导数，则有：
HARRY HOCHSTADT The purpose of this note is to state and prove the following theorem. Theorem. Consider the Hill's equation (1) y" +[a + q(t)]y = 0, where q(t+ir) =q(t), and q(t) is bounded. Let X, denote the ith eigenvalue corresponding to which a solution of (1) has period t, and X' those corresponding to which (1) has a solution of period 2ir. It is well known [l] that for these eigenvalues Xo < Ai" Ú \l < Xi ^ X2 < X8' Û \í < X3 á X4 < • • • and for all X in the intervals (2) (Xo, X/), (Xa, XO, (X2, XO, (X/, X3), • • • has only bounded solutions. For X outside those intervals unbounded solutions occur. These intervals (— », Xo), (X/, X2), (Xi, X2), (X/, X/) • • • are known as the instability intervals. Then if q(t) has m continuous derivatives it follows that