Kneser theorem

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In mathematics, in the field of ordinary differential equations, the Kneser theorem, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not.

[edit] Statement of the theorem

Consider an ordinary linear homogenous differential equation of the form

− y'' + q(x)y = 0

with

q: [0,+\infty] \to \mathbb{R}

continuous and q(x) > 0.
We say this equation is oscillating if it has a solution y with infinitely many zeros, and non-oscillating otherwise.
The theorem states that the equation is non-oscillating if

\liminf_{x \to +\infty} x^2 q(x) < -\frac{1}{4}

and oscillating if

\limsup_{x \to +\infty} x^2 q(x) > -\frac{1}{4}.
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