Lyapunov-type Inequalities
Lyapunov-type Inequalities
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DOI：
10.1007/978-1-4614-8523-0
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
J. P. Pinasco
中科院分区：
文献类型：
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作者：
J. P. Pinasco
我曾经认为斯特姆-刘维二阶常微分方程理论是数学中最美丽的领域之一。它的简单性，再加上比较定理和振荡定理的力量，使二阶常微分方程有了不同的认识。然而，当我阅读GC Rota的一个演讲稿时，我意识到：有许多有趣的问题，无论是理论上的还是应用上的，都不能用Sturmian工具来分析。以RN中的单位球为例：仅仅简单地简化到极坐标就引入了系数rN− 1，它在原点处为零，并且对所有N都以1为上界。此外，贝塞尔，厄米，勒让德，.，几乎所有特殊的函数族，作为二阶常微分算子的本征函数出现，确实是奇异或退化算子的本征函数，Sturmian论点失败。我们现在能做什么？
I used to think that the Sturm–Liouville theory of second-order ordinary differential equations was one of the most beautiful areas of mathematics. Its simplicity, together with the power of the comparison and oscillation theorems, shed a different light on second-order ordinary differential equations. However, while reading a transcription of a talk of GC Rota, I realized something: there are many interesting problems, both of theoretical and applied origin, that cannot be analyzed with the Sturmian tools.Take the unit ball in RN: just the simple reduction to polar coordinates introduces the coefficient rN− 1, which vanishes at the origin and is bounded above by 1, for all N. Moreover, Bessel, Hermite, Legendre,..., almost all the special families of functions that appear as eigenfunctions of second-order ordinary differential operators, are indeed eigenfunctions of singular or degenerate operators, and the Sturmian arguments fail. What can we do now?