Kneser's theorem (differential equations)

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 homogeneous differential equation of the form

$y''+q(x)y=0$

with

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

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

$\limsup _{x\to +\infty }x^{2}q(x)<{\tfrac {1}{4}}$

and oscillating if

$\liminf _{x\to +\infty }x^{2}q(x)>{\tfrac {1}{4}}.$

Example

To illustrate the theorem consider

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

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

$\limsup _{x\to +\infty }x^{2}q(x)=\liminf _{x\to +\infty }x^{2}q(x)={\frac {1}{4}}-a$

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

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

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

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

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.