Sturm's Comparison and Oscillation Theorems

Sturm's theorems provide qualitative information about the zeros and oscillation of solutions to second-order linear ODEs without solving them explicitly.

Sturm's Comparison Theorem

Theorem9.5Sturm's comparison theorem

Consider two equations on an interval $[a, b]$ :

$y'' + q_1(x)y = 0 \quad \text{and} \quad z'' + q_2(x)z = 0,$

where $q_1, q_2$ are continuous and $q_1(x) \leq q_2(x)$ on $[a, b]$ (with strict inequality on a subset of positive measure). Let $y$ be a nontrivial solution of the first equation with consecutive zeros at $\alpha, \beta \in [a,b]$ .

Then every nontrivial solution $z$ of the second equation has at least one zero in $(\alpha, \beta)$ .

Proof

Suppose for contradiction that $z(x) > 0$ on $(\alpha, \beta)$ (without loss of generality, adjusting sign if necessary). Since $y$ has consecutive zeros at $\alpha$ and $\beta$ , we may assume $y(x) > 0$ on $(\alpha, \beta)$ .

Consider the Wronskian-like expression:

$W(x) = y(x)z'(x) - y'(x)z(x).$

Differentiating: $W'(x) = yz'' - y''z = y(-q_2 z) - (-q_1 y)z = (q_1 - q_2)yz$ .

Since $q_1 \leq q_2$ and $y, z > 0$ on $(\alpha, \beta)$ : $W'(x) \leq 0$ on $(\alpha, \beta)$ , with strict inequality on a set of positive measure.

Evaluate $W$ at the endpoints:

At $x = \alpha$ : $W(\alpha) = y(\alpha)z'(\alpha) - y'(\alpha)z(\alpha) = -y'(\alpha)z(\alpha)$ . Since $y$ increases from zero at $\alpha$ , $y'(\alpha) > 0$ , and $z(\alpha) > 0$ (or $z(\alpha) = 0$ ). If $z(\alpha) > 0$ , then $W(\alpha) < 0$ ... but we need to be more careful.

Actually, $W(\alpha) = -y'(\alpha)z(\alpha) \leq 0$ (since $y'(\alpha) \geq 0$ and $z(\alpha) \geq 0$ ). Similarly, $W(\beta) = -y'(\beta)z(\beta)$ . Since $y$ decreases to zero at $\beta$ , $y'(\beta) \leq 0$ (actually $y'(\beta) < 0$ since $y > 0$ between), so $W(\beta) = -y'(\beta)z(\beta) \geq 0$ .

Thus $W(\beta) \geq 0 \geq W(\alpha)$ . But $W' \leq 0$ on $(\alpha, \beta)$ implies $W$ is non-increasing, so $W(\beta) \leq W(\alpha)$ . Combining: $W$ is constant, so $W' = 0$ a.e., hence $(q_1 - q_2)yz = 0$ a.e. Since $y, z > 0$ on $(\alpha, \beta)$ , this requires $q_1 = q_2$ a.e., contradicting the hypothesis. $\blacksquare$

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Sturm's Oscillation Theorem

Theorem9.6Sturm's separation theorem

Let $y_1$ and $y_2$ be two linearly independent solutions of $y'' + q(x)y = 0$ . Then between any two consecutive zeros of $y_1$ , there is exactly one zero of $y_2$ (and vice versa).

In other words, the zeros of $y_1$ and $y_2$ interlace.

ExampleZero interlacing

For $y'' + y = 0$ : $y_1 = \cos x$ has zeros at $\pi/2 + n\pi$ , and $y_2 = \sin x$ has zeros at $n\pi$ . The zeros alternate: $0, \pi/2, \pi, 3\pi/2, 2\pi, \ldots$ , confirming Sturm separation.

For Bessel functions $J_0(x)$ and $Y_0(x)$ (solutions of the Bessel equation of order $0$ ), their positive zeros also interlace. This is used in practice to locate zeros of Bessel functions.

RemarkApplications of Sturm's theorems

Bounding zeros: Comparing $y'' + q(x)y = 0$ with $y'' + \lambda y = 0$ (constant coefficient), Sturm's comparison theorem bounds the distance between zeros of solutions.
Oscillation criteria: If $q(x) \to +\infty$ as $x \to \infty$ , then all solutions oscillate (have infinitely many zeros). If $q(x) \leq 0$ for large $x$ , solutions are eventually monotone (non-oscillatory).
Eigenvalue counting: In Sturm-Liouville theory, the $n$ -th eigenfunction has exactly $n-1$ zeros in the interval, which follows from Sturm oscillation theory.