Kneser theorem

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.

Contents

Statement of the theorem

Consider an ordinary linear homogenous differential equation of the form

y'' %2B q(x)y = 0\,

with

q: [0,%2B\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 %2B\infty} x^2 q(x) < \tfrac{1}{4}

and oscillating if

\liminf_{x \to %2B\infty} x^2 q(x) > \tfrac{1}{4}.

Example

To illustrate the theorem consider

q(x) = (\frac{1}{4} - a) x^{-2} \quad{\rm 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 %2B\infty} x^2 q(x) = \liminf_{x \to %2B\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) %2B \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} %2B \sqrt{a}} %2B 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.

Extensions

There are many extensions to this result. For a recent account see[2].

References

  1. ^ Teschl, Gerald (2011). Ordinary Differential Equations and Dynamical Systems. American Mathematical Society. http://www.mat.univie.ac.at/~gerald/ftp/book-ode/. 
  2. ^ Helge Krüger and Gerald Teschl, Effective Prüfer angles and relative oscillation criteria, J. Diff. Eq. 245 (2008), 3823-3848 [1]