---
title: Sturm-Liouville Theory
module: PDEs, Fourier Series, and Boundary Value Problems
moduleNumber: 9
lessonNumber: 3
order: 903
summary: >
  The eigenvalue problem behind separation of variables generalizes to the
  self-adjoint Sturm-Liouville form. Lagrange's identity makes the operator
  symmetric, and from that one fact follow real eigenvalues, orthogonal
  eigenfunctions, and eigenfunction expansions that behave like Fourier series.
  Singular problems admit Bessel and Legendre functions, and Sturm's separation
  and comparison theorems describe how the eigenfunctions oscillate.
topics: [PDEs, Fourier Series, and Boundary Value Problems]
sources:
  - book: Boyce
    ref: "Ch. 11 §11.1 The Occurrence of Two-Point BVPs; §11.2 Sturm-Liouville Boundary Value Problems; §11.3 Nonhomogeneous BVPs; §11.4 Singular Sturm-Liouville Problems; §11.6 Mean Convergence"
  - book: Simmons
    ref: "Ch. 7 §43 Sturm-Liouville Problems; Ch. 4 §24 Oscillations and the Sturm Separation Theorem; §25 The Sturm Comparison Theorem"
draft: false
---

Separation of variables produces $X'' + \lambda X = 0$ with boundary conditions,
whose eigenfunctions expand the data. That equation is the simplest member of a
much larger family. Sturm and Liouville showed, in papers of 1836 and 1837, that
the whole family shares three properties: real eigenvalues, orthogonal
eigenfunctions, and convergent eigenfunction expansions. All three descend from
a single algebraic fact about the operator.

## The self-adjoint form

Heat conduction in a bar of variable material properties, with a source
proportional to temperature, separates into a more general spatial equation

$$
(p(x)y')' - q(x)y + \lambda r(x)y = 0
$$

on an interval, with **separated** boundary conditions, one at each end,

$$
\alpha_1 y(0) + \alpha_2 y'(0) = 0, \qquad \beta_1 y(1) + \beta_2 y'(1) = 0.
$$

> **Definition (Regular Sturm-Liouville problem).** The differential equation
> $(py')' - qy + \lambda ry = 0$ on $0 \le x \le 1$ with separated boundary
> conditions, where $p, p', q, r$ are continuous and $p(x) > 0$, $r(x) > 0$
> throughout. The function $r$ is the **weight**. The basic case
> $y'' + \lambda y = 0$ is recovered with $p = 1$, $q = 0$, $r = 1$.

It is convenient to name the differential operator

$$
L[y] = -(p(x)y')' + q(x)y,
$$

so the equation reads $L[y] = \lambda r(x)y$, a continuous analogue of the
[matrix eigenvalue problem](/differential-equations/systems/matrices-eigenvalues-review)
$A\mathbf{x} = \lambda\mathbf{x}$. The correspondence is close enough that
Sturm-Liouville theory and the theory of symmetric matrices are two faces of
one subject, linear operator theory.[^boyce-112]

This form is not a special case but a normal form. Any second-order linear
equation $P y'' + Q y' + R y = 0$ can be multiplied by the integrating factor
$\mu(x) = \tfrac{1}{P}\exp\!\int (Q/P)\,\d x$ to bring it into the self-adjoint
shape $(\mu P y')' + \mu R y = 0$, the same
[integrating-factor idea](/differential-equations/first-order/linear-first-order-integrating-factors)
that solved first-order linear equations. The Sturm-Liouville results therefore
apply to every such equation, not only to those already in divergence form.

$$
% caption: Any second-order linear equation is brought to self-adjoint form by
% one integrating factor; the operator $L$ is then symmetric, and the three
% Sturm-Liouville theorems follow from that symmetry alone.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  box/.style={draw, minimum width=34mm, minimum height=12mm, align=center}]
\definecolor{acc}{HTML}{4A6FA5}
\node[box] (gen) at (0,0) {$Py'' + Qy' + Ry = 0$};
\node[box, draw=acc, text=acc, thick] (sa) at (5.2,0) {self-adjoint\\$(py')' - qy = \ldots$};
\node[box] (sym) at (10.6,0) {$(L[u],v) = (u,L[v])$};
\draw[->, thick] (gen) -- node[above] {factor $\mu$} (sa);
\draw[->, thick] (sa) -- node[above] {Lagrange} (sym);
\end{tikzpicture}
$$

## Symmetry from Lagrange's identity

The fact underlying the theory comes from integrating by parts twice. For any
twice-differentiable $u, v$,

$$
\int_0^1 \bigl(L[u]\,v - u\,L[v]\bigr)\,\d x = -p(x)\bigl(u'(x)v(x) - u(x)v'(x)\bigr)\Big|_0^1,
$$

**Lagrange's identity**. When $u$ and $v$ both satisfy the separated boundary
conditions, the boundary term on the right cancels, leaving

$$
(L[u], v) = (u, L[v]),
$$

where $(u,v) = \int_0^1 uv\,\d x$ is the inner product on $[0,1]$.
This is symmetry: $L$ can be moved from one factor of the inner product to the
other. A problem is **self-adjoint** when this relation holds
for every admissible pair, and the separated conditions guarantee it. From here
the theorems are short.

## The three theorems

> **Theorem (Real eigenvalues).** Every eigenvalue of a regular
> Sturm-Liouville problem is real.

To prove it, suppose $L[\phi] = \lambda r\phi$ for a possibly complex $\lambda$
and eigenfunction $\phi$. Setting $u = v = \phi$ in the symmetry relation and
using $L[\phi] = \lambda r\phi$ gives $(\lambda - \overline\lambda)\int_0^1
r|\phi|^2\,\d x = 0$. The integral is positive because $r > 0$ and $\phi \ne 0$,
so $\lambda = \overline\lambda$. The eigenvalues are real, so the search for
them stays on the real line.

> **Theorem (Orthogonality).** Eigenfunctions belonging to different
> eigenvalues are orthogonal with respect to the weight $r$: if $\lambda_m \ne
> \lambda_n$, then $\int_0^1 r(x)\phi_m(x)\phi_n(x)\,\d x = 0$.

The proof again uses symmetry: substituting $L[\phi_m] = \lambda_m r\phi_m$ and
$L[\phi_n] = \lambda_n r\phi_n$ into $(L[\phi_m],\phi_n) = (\phi_m, L[\phi_n])$
yields $(\lambda_m - \lambda_n)\int_0^1 r\phi_m\phi_n\,\d x = 0$, and the factor
$\lambda_m - \lambda_n$ is nonzero. The orthogonality of the sines and cosines is
the case $r = 1$, where the weight is invisible.

> **Theorem (Discrete spectrum).** The eigenvalues are simple, each with one
> independent eigenfunction, and form an increasing sequence $\lambda_1 <
> \lambda_2 < \lambda_3 < \cdots$ with $\lambda_n \to \infty$.

Unlike a matrix, which has finitely many eigenvalues, a Sturm-Liouville problem
has infinitely many, marching to infinity. Ordered this way, the $n$-th
eigenfunction $\phi_n$ has exactly $n - 1$ interior zeros, so higher modes
oscillate more. Choosing the free constant so that $\int_0^1 r\phi_n^2\,\d x = 1$ **normalizes**
each eigenfunction; the family is then **orthonormal** with
respect to $r$, compactly $\int_0^1 r\phi_m\phi_n\,\d x = \delta_{mn}$.

$$
% caption: The first three eigenfunctions of a regular Sturm-Liouville
% problem, ordered by eigenvalue. Mode $n$ crosses zero $n{-}1$ times in the
% interior, so the spectrum's order is the order of increasing oscillation.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
% phi_1: one arch, no interior zero
\begin{scope}
\draw[black] (0,0) -- (4,0);
\draw[thick, domain=0:4, samples=80, smooth] plot (\x, {0.85*sin(45*\x)});
\node[font=\footnotesize] at (-0.6,0) {mode $1$};
\end{scope}
% phi_2: one interior zero
\begin{scope}[yshift=-2.2cm]
\draw[black] (0,0) -- (4,0);
\fill[black] (2,0) circle (1.3pt);
\draw[thick, domain=0:4, samples=90, smooth] plot (\x, {0.85*sin(90*\x)});
\node[font=\footnotesize] at (-0.6,0) {mode $2$};
\end{scope}
% phi_3: two interior zeros
\begin{scope}[yshift=-4.4cm]
\draw[black] (0,0) -- (4,0);
\fill[black] (1.333,0) circle (1.3pt);
\fill[black] (2.667,0) circle (1.3pt);
\draw[thick, domain=0:4, samples=120, smooth] plot (\x, {0.85*sin(135*\x)});
\node[font=\footnotesize] at (-0.6,0) {mode $3$};
\end{scope}
\node[black, font=\footnotesize] at (5.6,0) {$0$ zeros};
\node[black, font=\footnotesize] at (5.6,-2.2) {$1$ zero};
\node[black, font=\footnotesize] at (5.6,-4.4) {$2$ zeros};
\end{tikzpicture}
$$

## Eigenfunction expansions

The eigenfunction expansion of Fourier series carries over to a general weight.
Suppose $f$ can be written as a series of the normalized eigenfunctions,

$$
f(x) = \sum_{n=1}^{\infty} c_n \phi_n(x).
$$

Multiply by $r(x)\phi_m(x)$ and integrate. Orthonormality collapses the sum to
a single term, giving the coefficient as a weighted projection,

$$
c_m = \int_0^1 r(x)f(x)\phi_m(x)\,\d x = (f, r\phi_m).
$$

This is the Euler-Fourier formula with the weight $r$ inserted, and the series
is a **generalized Fourier series**. Its convergence matches that of ordinary
Fourier series.

> **Theorem (Eigenfunction convergence).** Let $\phi_1, \phi_2, \dots$ be the
> normalized eigenfunctions of a regular Sturm-Liouville problem, and let $f,
> f'$ be piecewise continuous on $[0,1]$. Then the series $\sum c_n\phi_n(x)$
> with $c_m = (f, r\phi_m)$ converges to $\tfrac12\bigl(f(x^+) + f(x^-)\bigr)$
> at each interior point, the same mean-of-limits statement as the Fourier
> convergence theorem.[^boyce-112b]

> **Worked example.** Take $y'' + \lambda y = 0$ with $y(0) = 0$ and
> $y'(1) + y(1) = 0$, a zero-temperature end and a radiating end. Here
> $p = r = 1$, $q = 0$. The condition at $x = 0$ kills the cosine, leaving
> $\phi_n(x) = k_n\sin(\sqrt{\lambda_n}\,x)$, and the condition at $x = 1$ forces
> the eigenvalues to satisfy the transcendental equation
>
> $$
> \sin\sqrt{\lambda_n} + \sqrt{\lambda_n}\cos\sqrt{\lambda_n} = 0, \quad \text{i.e.} \quad \sqrt{\lambda_n} = -\tan\sqrt{\lambda_n}.
> $$
>
> The roots have no closed form; graphing $\sqrt\lambda$ against $-\tan\sqrt\lambda$
> locates them, with $\sqrt{\lambda_n} \approx (2n-1)\pi/2$ for large $n$. The
> first few are $\lambda_1 \approx 4.116$, $\lambda_2 \approx 24.14$,
> $\lambda_3 \approx 63.66$. Normalizing with $\int_0^1\phi_n^2\,\d x = 1$ gives
> $k_n = \sqrt{2}\,(1 + \cos^2\sqrt{\lambda_n})^{-1/2}$. Expanding $f(x) = x$ in
> these eigenfunctions, $c_n = \int_0^1 x\,\phi_n(x)\,\d x$, and integration by
> parts with the eigenvalue relation gives
>
> $$
> c_n = \frac{2\sqrt2\,\sin\sqrt{\lambda_n}}{\lambda_n\,(1 + \cos^2\sqrt{\lambda_n})^{1/2}},
> $$
>
> so $f(x) = x$ is reproduced by a series of sines with irregularly spaced
> frequencies. This is an eigenfunction expansion, not a Fourier sine series, yet
> the convergence theorem guarantees it converges to $x$ on $[0,1]$.

$$
% caption: Graphical solution of $\sqrt\lambda=-\tan\sqrt\lambda$ with
% $s=\sqrt\lambda$: the line $u=s$ meets each descending branch of $u=-\tan s$
% once, at roots near the odd multiples of $\pi/2$, fixing $\lambda_n=s_n^2$.
\begin{tikzpicture}[scale=0.85, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (-0.2,0) -- (5.2,0) node[right, black!70] {$s$};
\draw[black, ->] (0,-3) -- (0,3.7) node[above, black!70] {$u$};
\draw[black, dashed] (1.571,-3) -- (1.571,3.4);
\draw[black, dashed] (4.712,-3) -- (4.712,2);
\draw[thick] (0,0) -- (3.4,3.4);
\node[font=\footnotesize] at (3.15,3.55) {$u=s$};
\draw[black, thick, smooth]
  plot coordinates {(1.9,2.93) (2.029,2.029) (2.2,1.374) (2.6,0.601) (3.1416,0) (3.7,-0.623) (4.1,-1.42) (4.3,-2.29)};
\node[black, font=\footnotesize] at (4.05,0.55) {tan branch};
\fill[acc] (2.029,2.029) circle (1.8pt);
\draw[black] (2.029,2.029) -- (1.15,2.75);
\node[acc, font=\footnotesize] at (0.85,2.9) {root $s_1$};
\end{tikzpicture}
$$

$$
% caption: Partial sums of an eigenfunction expansion closing in on a target
% function; each added mode corrects the fit, exactly as Fourier partial sums
% do for a periodic function.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (-0.2,0) -- (4.6,0) node[right, black!70] {$x$};
\draw[black, ->] (0,-0.2) -- (0,2.3) node[above, black!70] {$y$};
% target f(x)=x (scaled)
\draw[acc, thick] (0,0) -- (4,2);
\node[acc, font=\footnotesize] at (3.7,2.15) {$f$};
% one-term approx: single arch off target
\draw[black!50, thick, domain=0:4, samples=80, smooth] plot (\x, {1.15*sin(33.75*\x)});
\node[black!55, font=\footnotesize] at (2.4,1.35) {$1$ term};
% three-term approx: closer, slight waviness
\draw[thick, domain=0:4, samples=140, smooth]
  plot (\x, {0.5*\x + 0.12*sin(90*\x) - 0.06*sin(150*\x)});
\node[font=\footnotesize] at (1.1,0.15) {$3$ terms};
\end{tikzpicture}
$$

The same eigenfunctions solve a nonhomogeneous equation on the interval, with
the eigenvalue expansion converting the differential equation into one algebraic
equation per coefficient.

> **Worked example.** Solve
>
> $$
> y'' + 2y = -x, \qquad y(0) = 0, \quad y(1) + y'(1) = 0.
> $$
>
> Rewrite it as $-y'' = 2y + x$, matching $L[y] = \mu r y + f$ with $\mu = 2$,
> $r = 1$, $f(x) = x$; the homogeneous problem is the eigenvalue problem just
> solved, with eigenvalues $\lambda_n$ and normalized eigenfunctions $\phi_n$.
> Expand $y = \sum_n b_n\phi_n$ and apply $L[\phi_n] = \lambda_n\phi_n$ term by
> term. Since $L[y] = \sum_n b_n\lambda_n\phi_n$ and $\mu y + f/r = \sum_n(\mu b_n + c_n)\phi_n$,
> matching coefficients gives $(\lambda_n - \mu)\,b_n = c_n$, so
>
> $$
> b_n = \frac{c_n}{\lambda_n - 2}, \qquad c_n = \frac{2\sqrt2\,\sin\sqrt{\lambda_n}}{\lambda_n\,(1 + \cos^2\sqrt{\lambda_n})^{1/2}},
> $$
>
> the coefficients found above. Because $\mu = 2$ is not an eigenvalue, no
> denominator vanishes and the solution is unique:
>
> $$
> y = 4\sum_{n=1}^{\infty}\frac{\sin\sqrt{\lambda_n}}{\lambda_n\,(\lambda_n - 2)\,(1 + \cos^2\sqrt{\lambda_n})}\sin\!\left(\sqrt{\lambda_n}\,x\right).
> $$
>
> This problem also has the elementary closed form
> $y = \dfrac{\sin(\sqrt2\,x)}{\sin\sqrt2 + \sqrt2\cos\sqrt2} - \dfrac{x}{2}$; by
> uniqueness the two expressions agree. Had $\mu$ equaled some $\lambda_m$, the
> problem would have no solution unless $f$ were orthogonal to $\phi_m$, the
> Fredholm alternative.

## Singular problems and special functions

Regularity asked that $p > 0$ and $r > 0$ across a closed interval. Many
equations of physics violate this at an endpoint, where $p$ or $r$ vanishes or
a coefficient blows up. These are **singular Sturm-Liouville problems**, and at
the singular end the usual boundary condition is replaced by a demand that the
solution stay bounded, exactly as the disk problem required $u$ bounded at
$r = 0$.[^boyce-114] The classical special functions arise this way:

- **Bessel's equation** $(xy')' + (\lambda x - \nu^2/x)y = 0$ has $p(x) = x$ and
  weight $r(x) = x$, both vanishing at $x = 0$; its bounded solutions are the
  Bessel functions.
- **Legendre's equation** $((1-x^2)y')' + \lambda y = 0$ has $p(x) = 1 - x^2$
  vanishing at $x = \pm 1$; its bounded solutions are the Legendre polynomials.

In each case the eigenfunctions remain orthogonal with respect to their weight
and expand functions in a generalized Fourier series, the vibrating-membrane
and potential-theory analogues of the sine series. These functions are the
subject of the
[series-solutions module](/differential-equations/series-solutions/bessel-and-special-functions).

## Oscillation: Sturm's theorems

The claim that $\phi_n$ has $n - 1$ interior zeros is one instance of a broader
oscillation theory. Sturm developed it by studying the equation directly,
without solving it, first putting $y'' + Py' + Qy = 0$ into the **normal form**
$u'' + q(x)u = 0$ (no first-derivative term) through the substitution
$y = uv$ with $v = \exp(-\tfrac12\int P\,\d x)$. Since $v$ never vanishes, this
leaves the zeros, and hence the oscillation, unchanged.

> **Theorem (Sturm separation).** If $y_1$ and $y_2$ are linearly independent
> solutions of $y'' + P y' + Q y = 0$, their zeros are distinct and interlace:
> $y_1$ vanishes exactly once between any two successive zeros of $y_2$, and
> conversely.[^simmons-24]

The proof rides on the Wronskian $W(y_1, y_2)$, which is nonzero and therefore
of constant sign for independent solutions. At two successive zeros of $y_2$
the Wronskian reduces to $y_1 y_2'$, and $y_2'$ flips sign between them, so
$y_1$ must flip sign too and thus vanish once in between. The zeros of
$\sin x$ and $\cos x$, alternating at multiples of $\pi/2$, are the model
example.

$$
% caption: Sturm separation: two independent solutions (solid and dashed)
% cannot share a zero, and exactly one zero of each falls strictly between
% consecutive zeros of the other.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (-0.2,0) -- (8.4,0) node[right, black!70] {$x$};
% y1 = sin: zeros at 0,4,8
\draw[thick, domain=0:8, samples=140, smooth] plot (\x, {1.0*sin(45*\x)});
\node[font=\footnotesize] at (7.4,1.2) {$y_1$};
% y2 = cos: zeros at 2,6
\draw[black, thick, dashed, domain=0:8, samples=140, smooth] plot (\x, {1.0*cos(45*\x)});
\node[black, font=\footnotesize] at (0.35,1.2) {$y_2$};
% mark zeros
\fill[black] (0,0) circle (1.5pt); \fill[black] (4,0) circle (1.5pt); \fill[black] (8,0) circle (1.5pt);
\fill[black] (2,0) circle (1.5pt); \fill[black] (6,0) circle (1.5pt);
\draw[black, dashed] (2,0) -- (2,-0.9);
\node[black, font=\footnotesize] at (2,-1.15) {one $y_2$ zero between $y_1$ zeros};
\end{tikzpicture}
$$

The companion theorem compares two different equations.

> **Theorem (Sturm comparison).** Let $y$ and $z$ be nontrivial solutions of
> $y'' + q(x)y = 0$ and $z'' + r(x)z = 0$ with $q(x) > r(x) > 0$. Then $y$
> vanishes at least once between any two successive zeros of $z$: the equation
> with the larger coefficient oscillates faster.[^simmons-25]

Increasing $q$ presses the solution back toward the axis harder, packing its
zeros closer. For example, comparing $y'' + 4y = 0$ with $y'' + y = 0$:
$\sin 2x$ has twice the zeros of $\sin x$ on any interval. Applied to Bessel's
equation in normal form, the comparison theorem pins the spacing of the Bessel
zeros between the zeros of a plain sine, the qualitative fact that makes those
zeros usable long before the power series for the Bessel functions is in hand.

$$
% caption: Sturm comparison: with a larger coefficient (solid) the solution
% packs its zeros more tightly than the solution of the equation with the
% smaller coefficient (dashed).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (-0.2,0) -- (8.4,0) node[right, black!70] {$x$};
% larger q: sin(2x)-like, period 4 -> denser
\draw[thick, domain=0:8, samples=200, smooth] plot (\x, {1.0*sin(90*\x)});
\node[font=\footnotesize] at (7.4,1.2) {larger $q$};
% smaller q: sin(x)-like, period 8
\draw[black, thick, dashed, domain=0:8, samples=140, smooth] plot (\x, {1.0*sin(45*\x)});
\node[black, font=\footnotesize] at (5.2,-1.2) {smaller $r$};
\end{tikzpicture}
$$

[^boyce-112]: **Boyce**, _Elementary Differential Equations and Boundary Value Problems_, §11.1, §11.2 — the occurrence of two-point boundary value problems, the self-adjoint operator $L[y] = -(py')' + qy$, Lagrange's identity, and the theorems on real eigenvalues and orthogonal eigenfunctions.
[^boyce-112b]: **Boyce**, §11.2, §11.6 — eigenfunction expansions with the weight $r$, the coefficient formula $c_m = (f, r\phi_m)$, and the convergence theorem paralleling the Fourier convergence theorem.
[^boyce-114]: **Boyce**, §11.4 — singular Sturm-Liouville problems, boundedness replacing a boundary condition at a singular endpoint, and Bessel and Legendre equations as the archetypes.
[^simmons-24]: **Simmons**, _Differential Equations with Applications and Historical Notes_, §24 — the normal form $u'' + q(x)u = 0$ and the Sturm separation theorem via the constant sign of the Wronskian.
[^simmons-25]: **Simmons**, §25 — the Sturm comparison theorem and its application to the spacing of the zeros of Bessel functions.