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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.

Statement of the theorem

Consider an ordinary linear homogenous differential equation of the form

$y^{″}+q(x)y=0$

with

$q:[0,+\mathrm{\infty })\to \mathbb{ℝ}$

continuous. 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

${\mathrm{lim\; sup}}_{x\to +\mathrm{\infty }}{x}^{2}q(x)<\frac{1}{4}$

and oscillating if

${\mathrm{lim\; inf}}_{x\to +\mathrm{\infty }}{x}^{2}q(x)>\frac{1}{4}.$

Example

To illustrate the theorem consider

$q(x)=(\frac{1}{4}-a)x^{-2}\text{for}x>0$

where $a$ is real and non-zero. According to the theorem, solutions will be oscillating or not depending on whether $a$ is positive (non-oscillating) or negative (oscillating) because

${\mathrm{lim\; sup}}_{x\to +\mathrm{\infty }}{x}^{2}q(x)={\mathrm{lim\; inf}}_{x\to +\mathrm{\infty }}{x}^{2}q(x)=\frac{1}{4}-a$

To find the solutions for this choice of $q(x)$, and verify the theorem for this example, substitute the 'Ansatz'

$y(x)=x^n$

which gives

$n(n-1)+\frac{1}{4}-a={(n-\frac{1}{2})}^2-a=0$

This means that (for non-zero $a$) the general solution is

$y(x)=A{x}^{\frac{1}{2}+\sqrt{a}}+B{x}^{\frac{1}{2}-\sqrt{a}}$

where $A$ and $B$ are arbitrary constants.

It is not hard to see that for positive $a$ the solutions do not oscillate while for negative $a=-{\omega }^2$ the identity

$x^{\frac{1}{2}\pm i\omega }=\sqrt{x}{e}^{\pm (i\omega )\ln x}=\sqrt{x}(\cos (\omega \ln x)\pm i\sin (\omega \ln x))$

shows that they do.

The general result follows from this example by the Sturm–Picone comparison theorem.

Extensions

There are many extensions to this result. For a recent account see.

References

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