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^[1] 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.^[2]

References

↑ Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
↑ Helge Krüger and Gerald Teschl, Effective Prüfer angles and relative oscillation criteria, J. Diff. Eq. 245 (2008), 3823–3848

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