Sturm-Liouville Theory

Separation of variables produces $X^{''} + λ X = 0$ 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 + λ r (x) y = 0

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

α_{1} y (0) + α_{2} y^{'} (0) = 0, β_{1} y (1) + β_{2} y^{'} (1) = 0.

It is convenient to name the differential operator

L [y] = - (p (x) y^{'})^{'} + q (x) y,

so the equation reads $L [y] = λ r (x) y$ , a continuous analogue of the matrix eigenvalue problem $A x = λ x$ . The correspondence is close enough that Sturm-Liouville theory and the theory of symmetric matrices are two faces of one subject, linear operator theory.¹

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 $μ (x) = \frac{1}{P} exp \int (Q / P) d x$ to bring it into the self-adjoint shape $(μ P y^{'})^{'} + μ R y = 0$ , the same integrating-factor idea that solved first-order linear equations. The Sturm-Liouville results therefore apply to every such equation, not only to those already in divergence form.

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.

Symmetry from Lagrange's identity

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

0 \int 1 (L [u] v - u L [v]) d x = - p (x) (u^{'} (x) v (x) - u (x) v^{'} (x))_{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

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

The proof again uses symmetry: substituting $L [ϕ_{m}] = λ_{m} r ϕ_{m}$ and $L [ϕ_{n}] = λ_{n} r ϕ_{n}$ into $(L [ϕ_{m}], ϕ_{n}) = (ϕ_{m}, L [ϕ_{n}])$ yields $(λ_{m} - λ_{n}) \int_{0}^{1} r ϕ_{m} ϕ_{n} d x = 0$ , and the factor $λ_{m} - λ_{n}$ is nonzero. The orthogonality of the sines and cosines is the case $r = 1$ , where the weight is invisible.

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 $ϕ_{n}$ has exactly $n - 1$ interior zeros, so higher modes oscillate more. Choosing the free constant so that $\int_{0}^{1} r ϕ_{n}^{2} d x = 1$ normalizes each eigenfunction; the family is then orthonormal with respect to $r$ , compactly $\int_{0}^{1} r ϕ_{m} ϕ_{n} d x = δ_{mn}$ .

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.

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) = n = 1 \sum \infty c_{n} ϕ_{n} (x) .

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

c_{m} = 0 \int 1 r (x) f (x) ϕ_{m} (x) d x = (f, r ϕ_{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.

Worked example. Take $y^{''} + λ y = 0$ 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 $ϕ_{n} (x) = k_{n} sin (λ_{n} x)$ , and the condition at $x = 1$ forces the eigenvalues to satisfy the transcendental equation

sin λ_{n} + λ_{n} cos λ_{n} = 0, i.e. λ_{n} = - tan λ_{n} .

The roots have no closed form; graphing $λ$ against $- tan λ$ locates them, with $λ_{n} \approx (2 n - 1) π /2$ for large $n$ . The first few are $λ_{1} \approx 4.116$ , $λ_{2} \approx 24.14$ , $λ_{3} \approx 63.66$ . Normalizing with $\int_{0}^{1} ϕ_{n}^{2} d x = 1$ gives $k_{n} = 2 (1 + cos^{2} λ_{n})^{- 1/2}$ . Expanding $f (x) = x$ in these eigenfunctions, $c_{n} = \int_{0}^{1} x ϕ_{n} (x) d x$ , and integration by parts with the eigenvalue relation gives

c_{n} = \frac{2 2 sin λ _{n}}{λ _{n} ( 1 + cos ^{2} λ _{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 $[0, 1]$ .

Graphical solution of

λ = - tan λ

with

s = λ

: the line

u = s

meets each descending branch of

u = - tan s

once, at roots near the odd multiples of

π /2

, fixing

λ_{n} = s_{n}^{2}

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.

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^{''} + 2 y = - x, y (0) = 0, y (1) + y^{'} (1) = 0.

Rewrite it as $- y^{''} = 2 y + x$ , matching $L [y] = μ r y + f$ with $μ = 2$ , $r = 1$ , $f (x) = x$ ; the homogeneous problem is the eigenvalue problem just solved, with eigenvalues $λ_{n}$ and normalized eigenfunctions $ϕ_{n}$ . Expand $y = \sum_{n} b_{n} ϕ_{n}$ and apply $L [ϕ_{n}] = λ_{n} ϕ_{n}$ term by term. Since $L [y] = \sum_{n} b_{n} λ_{n} ϕ_{n}$ and $μ y + f / r = \sum_{n} (μ b_{n} + c_{n}) ϕ_{n}$ , matching coefficients gives $(λ_{n} - μ) b_{n} = c_{n}$ , so

b_{n} = \frac{c _{n}}{λ _{n} - 2}, c_{n} = \frac{2 2 sin λ _{n}}{λ _{n} ( 1 + cos ^{2} λ _{n} ) ^{1/2}},

the coefficients found above. Because $μ = 2$ is not an eigenvalue, no denominator vanishes and the solution is unique:

y = 4 n = 1 \sum \infty \frac{sin λ _{n}}{λ _{n} ( λ _{n} - 2 ) ( 1 + cos ^{2} λ _{n} )} sin (λ_{n} x) .

This problem also has the elementary closed form $y = \frac{sin ( 2 x )}{sin 2 + 2 cos 2} - \frac{x}{2}$ ; by uniqueness the two expressions agree. Had $μ$ equaled some $λ_{m}$ , the problem would have no solution unless $f$ were orthogonal to $ϕ_{m}$ , 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 $r = 0$ .³ The classical special functions arise this way:

Bessel's equation $(x y^{'})^{'} + (λ x - ν^{2} / x) y = 0$ has $p (x) = x$ and weight $r (x) = x$ , both vanishing at $x = 0$ ; its bounded solutions are the Bessel functions.
Legendre's equation $((1 - x^{2}) y^{'})^{'} + λ y = 0$ 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.

Oscillation: Sturm's theorems

The claim that $ϕ_{n}$ 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 $y^{''} + P y^{'} + Q y = 0$ into the normal form $u^{''} + q (x) u = 0$ (no first-derivative term) through the substitution $y = uv$ with $v = exp (- \frac{1}{2} \int P d x)$ . Since $v$ never vanishes, this leaves the zeros, and hence the oscillation, unchanged.

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 $π /2$ , are the model example.

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.

The companion theorem compares two different equations.

Increasing $q$ presses the solution back toward the axis harder, packing its zeros closer. For example, comparing $y^{''} + 4 y = 0$ with $y^{''} + y = 0$ : $sin 2 x$ 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.

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).

Boyce, Elementary Differential Equations and Boundary Value Problems, §11.1, §11.2 — the occurrence of two-point boundary value problems, the self-adjoint operator $L [y] = - (p y^{'})^{'} + q y$ , Lagrange's identity, and the theorems on real eigenvalues and orthogonal eigenfunctions. ↩
Boyce, §11.2, §11.6 — eigenfunction expansions with the weight $r$ , the coefficient formula $c_{m} = (f, r ϕ_{m})$ , and the convergence theorem paralleling the Fourier convergence theorem. ↩
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, 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 — the Sturm comparison theorem and its application to the spacing of the zeros of Bessel functions. ↩

The self-adjoint form

Symmetry from Lagrange's identity

The three theorems

Eigenfunction expansions

Singular problems and special functions

Oscillation: Sturm's theorems

Footnotes