Sturm-Liouville Eigenvalues and Oscillation Theorems

A Sturm–Liouville problem is what a linear second-order eigenvalue ODE becomes once you insist it be self-adjoint. Write any such equation in the form $-(p\,u')'+q\,u=\lambda\,w\,u$ on an interval $[a,b]$ , attach boundary conditions that kill the boundary flux, and the differential operator $\mathcal{L}$ inherits every property of a symmetric matrix — only now on an infinite-dimensional function space. This is the ODE shadow of the spectral theorem: the eigenvalues are real, the eigenfunctions are orthogonal in the weighted inner product $\int u\,v\,w\,\mathrm{d}x$ , and together they form a complete basis in which any reasonable function can be expanded, exactly as Fourier's sines and cosines expand a periodic signal. Almost every classical special function you will meet — Legendre, Bessel, Hermite, Laguerre, Chebyshev — is the eigenfunction family of one singular Sturm–Liouville problem, which is precisely why separation of variables works across mathematical physics. Layered on top of these algebraic facts is a sharper, geometric statement about where solutions cross zero, discovered by Sturm in 1836. The comparison theorem says that raising the potential, or the eigenvalue, forces solutions to oscillate faster and pack more zeros into the same interval. The eigenvalue problem lines this up exactly: ordering the spectrum $\lambda_0<\lambda_1<\lambda_2<\cdots$ orders the eigenfunctions by node count $0,1,2,\dots$ , the ground state $u_0$ never vanishes inside $(a,b)$ , and consecutive eigenfunctions strictly interlace their zeros. Two computational engines make this concrete: the variational Rayleigh–Ritz picture, which writes each $\lambda_n$ as a constrained minimum and hands you every monotonicity law for free, and the dynamical Prüfer picture, which turns oscillation into the winding of a phase angle. This article assembles both.

Key takeaways

A regular Sturm–Liouville problem $-(p\,u')'+q\,u=\lambda\,w\,u$ on $[a,b]$ with separated boundary conditions has a real, discrete, simple spectrum $\lambda_0<\lambda_1<\lambda_2<\cdots\to+\infty$ .
The $n$ -th eigenfunction $u_n$ has exactly $n$ interior zeros, the ground state $u_0$ has none, and the zeros of $u_n$ and $u_{n+1}$ strictly interlace — the oscillation theorems.
Eigenfunctions of distinct eigenvalues are orthogonal with weight $w$ , and they are complete in $L^2([a,b],w\,\mathrm{d}x)$ : any square-integrable function expands in them.
Weyl's law fixes the growth $\lambda_n\sim\big((n+1)\pi/\!\int_a^b\!\sqrt{w/p}\,\mathrm{d}x\big)^2$ , so the high modes behave like a uniform string of rescaled length.
The Rayleigh quotient and the Courant–Fischer minimax principle characterize each eigenvalue variationally and prove monotonicity under changes of $q$ , $p$ , $w$ , and the domain.
Singular problems (Legendre, Bessel, Hermite, Laguerre) trade boundary data for regularity or integrability, and Weyl's limit-point/limit-circle test decides what data, if any, are still needed.
Key idea: putting a second-order ODE in self-adjoint form $-(p\,u')'+q\,u=\lambda\,w\,u$ turns the eigenvalue problem into a self-adjoint operator on $L^2(w)$ — the spectrum is real, discrete, ordered and simple, the $n$ -th eigenfunction has exactly $n$ interior zeros, and consecutive eigenfunctions interlace, captured at once by the Rayleigh minimax principle and the Prüfer phase.

The core insight

Putting a second-order ODE in self-adjoint form $-(p\,u')'+q\,u=\lambda\,w\,u$ turns the eigenvalue problem into a self-adjoint operator on $L^2(w)$ : the spectrum is real, discrete, ordered and simple, the $n$ -th eigenfunction has exactly $n$ interior zeros, and consecutive eigenfunctions interlace — facts captured simultaneously by the Rayleigh minimax principle and the Prüfer phase.

Where this lives today

Sturm–Liouville theory is quietly load-bearing across modern computation and physics. Production eigenvalue solvers such as SLEIGN2 and the MATSLISE package integrate the Prüfer phase precisely because node-counting gives a guaranteed index: you can ask for the 37th eigenvalue of a singular problem and know the algorithm has landed on the right one, not a neighbour, because it counts phase crossings rather than hunting blindly. In quantum device engineering — semiconductor quantum wells, cold-atom traps, photonic-crystal waveguides — the design problem is a Sturm–Liouville eigenproblem whose oscillation theorem fixes which mode is the ground state and how many bound states a well supports. The theory also runs in reverse: inverse spectral theory, built on the Gelfand–Levitan and Marchenko equations, reconstructs an unknown potential from its spectrum and is the spine of the inverse-scattering transform that solves the Korteweg–de Vries and other soliton equations. And Weyl's law $\lambda_n\sim n^2$ is the one-dimensional ancestor of "can you hear the shape of a drum?" — whether the spectrum alone fixes the geometry — answered (no, not always) only in 1992 and still driving spectral geometry.

The physical picture

Picture the equation $-(p\,u')'+q\,u=\lambda\,w\,u$ as a stretched, non-uniform string: $p(x)$ is the local stiffness, $w(x)$ the mass density, $q(x)$ a restoring potential glued to each point, and $\lambda$ the square of the vibration frequency. Pluck the string in its lowest mode and it bows in a single arch with no interior node; drive it harder — raise $\lambda$ — and the shape must wiggle faster, sprouting one new node each time $\lambda$ climbs to the next eigenvalue. That single sentence is the oscillation theorem, and the first figure shows the first three modes with their node counts $0,1,2$ and the interlacing of their zeros.

Figure

The first three modes of the string −u″ = λu on the interval from 0 to π, with u_n(x) = sin((n+1)x) and eigenvalues λ_0 = 1, λ_1 = 4, λ_2 = 9. Mode u_n has exactly n interior zeros marked in red, and the zeros of consecutive modes interlace.

Why do the nodes appear so regularly? The Prüfer angle $\theta(x)$ answers it. Rewrite the solution in polar form, $u=r\sin\theta$ , and let $\theta$ tick like the hand of a clock as $x$ runs from $a$ to $b$ : every time $\theta$ sweeps through a multiple of $\pi$ , the factor $\sin\theta$ passes through zero and $u$ has a node. A larger eigenvalue makes the clock run faster, so $u_n$ accumulates exactly $n$ extra half-turns and $n$ extra zeros. The second figure tracks that phase, showing $\theta(x)$ climbing monotonically and crossing $\pi$ and $2\pi$ at the two interior zeros of the mode $u_2$ .

Figure

The Prüfer phase θ(x) for the two-node mode climbs monotonically because θ′ = cos²θ/p + (λw − q)sin²θ is positive at every crossing. A zero of u occurs precisely where θ passes a multiple of π, so counting crossings of π and 2π counts the two interior nodes.

Two more ideas round out the picture. Orthogonality is geometry: eigenfunctions of different eigenvalues are perpendicular in the inner product weighted by $w$ , and that weight encodes the non-uniform density, so the correct expansion and completeness live in $L^2(w)$ , never in plain $L^2$ . And the Rayleigh quotient says each eigenfunction is the shape that minimizes stored energy per unit mass subject to orthogonality with the lower modes; squeezing the interval or raising the potential can only push every minimum up, which is why the monotonicity laws are one-line corollaries of minimax.

The core equations

The starting object is the self-adjoint, or Sturm–Liouville, form. Any linear second-order eigenvalue ODE can be brought to it, and only in this form does the operator behave like a symmetric matrix. The three data functions $p,q,w$ carry all the physics of the problem.

\mathcal{L}u\equiv-\frac{\mathrm{d}}{\mathrm{d}x}\!\left(p(x)\,\frac{\mathrm{d}u}{\mathrm{d}x}\right)+q(x)\,u=\lambda\,w(x)\,u \qquad (1)

where: $p(x)$ — coefficient with $p>0$ on $[a,b]$ (stiffness); $q(x)$ — real potential coefficient; $w(x)$ — weight with $w>0$ (density), setting the inner product; $\lambda$ — eigenvalue; $u(x)$ — eigenfunction. Read it as an eigenvalue equation $\mathcal{L}u=\lambda w u$ for the operator $\mathcal{L}$ against the weight $w$ . A generic equation $a_2 u''+a_1 u'+a_0 u=\lambda u$ is not self-adjoint as written, but an integrating factor fixes that. Multiplying by $p/a_2$ with $p$ chosen below recasts the first two terms as a single perfect derivative $(p\,u')'$ , and reading off the rest gives the weight and potential.

p=\exp\!\int\frac{a_1}{a_2}\,\mathrm{d}x,\qquad w=\frac{p}{a_2},\qquad q=-\frac{a_0}{a_2}\,p \qquad (2)

where: $a_0,a_1,a_2$ — coefficients of the raw ODE; $p,q,w$ — the resulting Sturm–Liouville data. Read it as a recipe: the exponential integrating factor symmetrizes any second-order operator, and everything downstream follows. Self-adjointness is a statement about an inner product. Define the weighted $L^2(w)$ pairing; the operator $\mathcal{L}$ is symmetric on the functions that satisfy the boundary conditions, meaning it can be moved from one slot to the other for free.

\langle u,v\rangle_w=\int_a^b \overline{u}\,v\,w\,\mathrm{d}x,\qquad \langle \mathcal{L}u,v\rangle-\langle u,\mathcal{L}v\rangle=\big[\,p\,(\overline{u}\,v'-\overline{u}'\,v)\,\big]_a^b \qquad (3)

where: $\langle\cdot,\cdot\rangle_w$ — inner product weighted by $w$ ; the bracket on the right is the boundary flux, the ODE analogue of the $p$ -weighted Wronskian. Read it as: separated boundary conditions kill this flux at both ends, so $\mathcal{L}$ is symmetric and its spectrum is real with orthogonal eigenfunctions. Once $\mathcal{L}$ is self-adjoint with weight $w>0$ and the interval is regular, the spectral theorem for compact resolvents delivers the whole spectrum at once: a real, ordered, non-degenerate sequence marching to infinity, with orthogonal eigenfunctions.

\lambda_0<\lambda_1<\lambda_2<\cdots\to+\infty,\qquad \langle u_m,u_n\rangle_w=0\ \ (m\neq n) \qquad (4)

where: $\lambda_n$ — the $n$ -th eigenvalue, strictly increasing and simple; $u_n$ — its eigenfunction, forming an orthogonal (and, normalized, orthonormal) family. Read it as: the eigenfunctions are a complete weighted-orthogonal basis of $L^2(w)$ . The variational face writes the lowest eigenvalue as a global minimum of the energy-per-mass functional. Its stationary points are exactly the eigenfunctions, and the minimum value is $\lambda_0$ — the fastest route to bounds and to the ground state.

R[u]=\frac{\displaystyle\int_a^b\big(p\,|u'|^2+q\,|u|^2\big)\,\mathrm{d}x}{\displaystyle\int_a^b w\,|u|^2\,\mathrm{d}x},\qquad \lambda_0=\min_{u\neq 0}R[u] \qquad (5)

where: $R[u]$ — Rayleigh quotient, stored energy over stored mass; $\lambda_0$ — ground eigenvalue as the unconstrained minimum. Read it as: any admissible trial $u$ gives $R[u]\ge\lambda_0$ , so trial functions bound the ground state from above. Higher eigenvalues need the minimax generalization, which avoids having to know the lower eigenfunctions first. Over all $(n{+}1)$ -dimensional trial subspaces, take the worst-case Rayleigh quotient, then minimize over subspaces.

\lambda_n=\min_{\substack{V\subset H\\ \dim V=n+1}}\ \max_{\substack{u\in V\\ u\neq 0}}\,R[u] \qquad (6)

where: $V$ — an $(n{+}1)$ -dimensional trial subspace; $H$ — the admissible (Sobolev) space of functions meeting the boundary conditions. Read it as: enlarging the domain, raising $q$ , or raising $w^{-1}$ can only raise every $\lambda_n$ — monotonicity in one line. The dynamical face is the Prüfer transformation, polar coordinates for the pair $(u,p\,u')$ . It converts the second-order equation into a single first-order equation for the phase $\theta$ , decoupled from the amplitude $r$ , and this phase equation is the engine of every oscillation result.

u=r\sin\theta,\quad p\,u'=r\cos\theta,\qquad \theta'=\frac{\cos^2\theta}{p}+\big(\lambda w-q\big)\sin^2\theta \qquad (7)

where: $r(x)>0$ — Prüfer radius (positive because the zeros of $u$ are simple); $\theta(x)$ — Prüfer phase; a zero of $u$ occurs at every $\theta=k\pi$ . Read it as: node-counting is now literally counting how many multiples of $\pi$ the phase sweeps. Finally, the large- $n$ behaviour is universal. High modes oscillate so fast that they ignore the potential $q$ and see only the effective length $\int\sqrt{w/p}$ , giving Weyl's asymptotic law.

\lambda_n\ \sim\ \left(\frac{(n+1)\,\pi}{\displaystyle\int_a^b\sqrt{\tfrac{w}{p}}\,\mathrm{d}x}\right)^{2}\qquad (n\to\infty) \qquad (8)

where: $n$ — mode index; $\int_a^b\sqrt{w/p}\,\mathrm{d}x$ — the effective acoustic length. Read it as: asymptotic spacing $\lambda_{n+1}-\lambda_n\propto n$ , the hallmark of a one-dimensional wave operator.

Derivation AFrom a bare ODE to a self-adjoint operator

The goal is to show that a generic linear second-order eigenvalue equation, which as written has no symmetry at all, can always be repackaged as a self-adjoint operator with a positive weight, and that this alone forces the two facts every application leans on: real eigenvalues and weighted-orthogonal eigenfunctions. We build the integrating factor, run the Lagrange integration by parts, and read off reality and orthogonality.

Step 1. Start from $a_2 u''+a_1 u'+a_0 u=\lambda u$ and demand an integrating factor $\mu(x)$ so that $\mu(a_2 u''+a_1 u')=(p\,u')'$ for some $p$ . Expanding $(p\,u')'=p\,u''+p'\,u'$ and matching coefficients gives $p=\mu a_2$ and $p'=\mu a_1$ , whose ratio is a first-order equation $\mu'/\mu=(a_1-a_2')/a_2$ . Integrating yields $p=\exp\!\int(a_1/a_2)\,\mathrm{d}x$ , then $w=p/a_2$ and $q=-(a_0/a_2)p$ . Why: the exponential of a coefficient ratio is precisely the classical integrating factor, now applied to an operator instead of a scalar equation — it is what makes the leading terms a perfect derivative.

Step 2. With $\mathcal{L}u=-(p\,u')'+qu$ , compute $\langle\mathcal{L}u,v\rangle-\langle u,\mathcal{L}v\rangle$ . The $q$ -terms and any $\lambda w$ -terms are symmetric and cancel, leaving second-derivative pieces that two integrations by parts collapse to a pure boundary term $\big[p(\overline{u}v'-\overline{u}'v)\big]_a^b$ . Why: integration by parts is the continuum version of transposing a matrix; the leftover boundary term is the only obstruction to symmetry.

Step 3. Impose separated boundary conditions $\alpha u(a)+\alpha' u'(a)=0$ and $\beta u(b)+\beta' u'(b)=0$ . At each endpoint the vectors $(u,u')$ for any two domain functions are proportional, so the $2\times 2$ determinant $\overline{u}v'-\overline{u}'v$ vanishes there, killing the boundary term and making $\mathcal{L}$ symmetric. Why: a separated condition pins the ratio $u'/u$ at the endpoint to a single number, and equal ratios make the Wronskian-like bracket collapse.

Step 4. Feed in two eigenpairs $\mathcal{L}u_m=\lambda_m w u_m$ and $\mathcal{L}u_n=\lambda_n w u_n$ . Symmetry gives $\overline{\lambda_m}\langle u_m,u_n\rangle_w=\lambda_n\langle u_m,u_n\rangle_w$ . Setting $m=n$ with $\|u_n\|_w^2>0$ forces $\lambda_n=\overline{\lambda_n}$ , so every eigenvalue is real; then $m\neq n$ with $\lambda_m\neq\lambda_n$ forces $\langle u_m,u_n\rangle_w=0$ . Why: this is the identical algebra proving a Hermitian matrix has real eigenvalues and orthogonal eigenvectors, transplanted to the operator.

Go deeper · Derivation B · 2 minSturm comparison and the oscillation theorem

Derivation BSturm comparison and the oscillation theorem

The goal is to prove the geometric heart of the theory: if one Sturm–Liouville equation has a larger effective potential than another, its solutions oscillate faster, so between any two consecutive zeros of the slower solution the faster one must vanish at least once. From this single comparison the node count — $u_n$ has exactly $n$ interior zeros — and the interlacing of consecutive eigenfunctions follow by induction. We build the cross-bracket, integrate it between zeros, and extract a sign contradiction.

Step 1. Let $u$ solve $-(p\,u')'+q_1 u=0$ and $v$ solve $-(p\,v')'+q_2 v=0$ on a subinterval, with $q_1\ge q_2$ pointwise. Differentiate the cross-bracket $p(u'v-uv')$ and substitute both equations; the terms with $p''$ and $p'$ cancel cleanly, leaving $\big(p(u'v-uv')\big)'=(q_1-q_2)\,u\,v$ . Why: the cross-bracket is the Wronskian of the two solutions weighted by $p$ , and its derivative isolates exactly the difference of potentials — the quantity driving the extra oscillation.

Step 2. Take $x_1<x_2$ to be two consecutive zeros of $u$ with $u>0$ in between, so $u'(x_1)>0$ and $u'(x_2)<0$ . Integrate the identity across $[x_1,x_2]$ : the left side becomes $p u'(x_2)v(x_2)-p u'(x_1)v(x_1)$ and the right side becomes $\int_{x_1}^{x_2}(q_1-q_2)uv\,\mathrm{d}x$ . Why: choosing the integration limits to be zeros of $u$ throws away the $uv'$ contributions at the endpoints, leaving a clean relation between endpoint slopes and the bulk integral.

Step 3. Suppose, for contradiction, that $v>0$ throughout $(x_1,x_2)$ . Then the right side is $\ge 0$ because $q_1-q_2\ge 0$ and $uv>0$ , while the left side is $\le 0$ because $u'(x_2)<0$ and $u'(x_1)>0$ with $v\ge 0$ at the endpoints. Both cannot hold unless everything is zero, so $v$ must vanish somewhere in $(x_1,x_2)$ . Why: a strictly larger potential cannot leave a solution non-vanishing across a full arch of the smaller-potential solution — it is forced to turn over.

Step 4. Apply the comparison with $q_1=q-\lambda_{n+1}w$ and $q_2=q-\lambda_n w$ , where $\lambda_{n+1}>\lambda_n$ makes $q_1<q_2$ , i.e. the higher eigenfunction is the faster one. Each arch of $u_n$ then contains a zero of $u_{n+1}$ , so $u_{n+1}$ has at least one more interior zero than $u_n$ ; running the argument the other way caps the gain at exactly one. Starting from the nodeless ground state $u_0$ , induction gives $u_n$ exactly $n$ interior zeros, and the same bracket applied between consecutive zeros of $u_{n+1}$ shows one zero of $u_n$ lies between them — strict interlacing. Why: the comparison is a difference engine; iterating it turns one extra zero per step into an exact node count and a clean alternation of zeros.

Go deeper · Derivation C · 2 minThe Prüfer phase counts the eigenvalues

Derivation CThe Prüfer phase counts the eigenvalues

The goal is to re-derive discreteness, ordering and node-counting at once, dynamically, by showing that the Prüfer phase at the right endpoint is a strictly increasing function of $\lambda$ . That single monotonicity turns the boundary condition into a quantization rule: as $\lambda$ sweeps upward, $\theta(b)$ climbs through $\beta,\beta+\pi,\beta+2\pi,\dots$ , hitting each once, so eigenvalues are isolated, ordered and labelled by how many multiples of $\pi$ the phase has crossed. We derive the phase equation, locate the zeros, and prove the monotonicity.

Step 1. Substitute $u=r\sin\theta$ and $p\,u'=r\cos\theta$ into the Sturm–Liouville equation. Differentiating both relations and eliminating $r$ by cross-multiplication yields the closed phase equation $\theta'=p^{-1}\cos^2\theta+(\lambda w-q)\sin^2\theta$ , in which the amplitude $r$ has completely dropped out. Why: writing the solution vector in polar form separates its rotation from its length, and only the rotation controls where $u$ crosses zero.

Step 2. Because $r>0$ , the zeros of $u$ are exactly the points where $\sin\theta=0$ , i.e. $\theta\in\pi\mathbb{Z}$ . At such a point $\sin\theta=0$ so the phase equation reduces to $\theta'=1/p>0$ : the phase crosses every multiple of $\pi$ strictly upward and never slides back through one. Why: an upward-only crossing means zeros can be created as $\lambda$ grows but never destroyed, which is what makes the node count a reliable label.

Step 3. Differentiate the phase equation with respect to $\lambda$ : $\partial_\lambda(\theta')=w\sin^2\theta+(\text{terms}\propto\partial_\lambda\theta)$ , whose inhomogeneous part $w\sin^2\theta\ge 0$ is non-negative and positive on average since $w>0$ . A Gronwall comparison then gives $\partial\theta(b;\lambda)/\partial\lambda>0$ strictly. Why: raising $\lambda$ adds positive drive to the phase everywhere the solution is nonzero, so the accumulated phase at the far end can only increase.

Step 4. The boundary condition fixes $\theta(a)=\alpha$ and demands $\theta(b)=\beta+n\pi$ for integer $n$ . As $\lambda$ increases from $-\infty$ , $\theta(b;\lambda)$ rises monotonically from near $\alpha$ through each target value $\beta+n\pi$ exactly once, producing the ordered simple spectrum $\lambda_0<\lambda_1<\cdots$ ; at $\lambda=\lambda_n$ the phase has swept past $\pi,2\pi,\dots,n\pi$ inside $(a,b)$ , so $u_n$ has exactly $n$ interior zeros. Why: strict monotonicity converts a transcendental boundary condition into a counting statement, delivering discreteness, ordering and the oscillation theorem in a single stroke.

Limiting cases

The general theory specializes to the elementary string and to every classical special function once the data $p,q,w$ and the interval are chosen.

Case	Result	Meaning
Vibrating string	$-u''=\lambda u,\ u(0)=u(\pi)=0$	$p=w=1,\ q=0$ : $\lambda_n=(n+1)^2$ , $u_n\propto\sin((n+1)x)$ with exactly $n$ interior zeros — the prototype that confirms the oscillation theorem exactly.
Legendre (singular)	$-\big((1-x^2)P'\big)'=\ell(\ell+1)P,\ x\in[-1,1]$	$p=1-x^2$ vanishes at both ends; regularity replaces boundary data, $\lambda_\ell=\ell(\ell+1)$ , and $P_\ell$ has $\ell$ zeros in $(-1,1)$ .
Bessel (singular at 0)	$-(xR')'+\tfrac{m^2}{x}R=\lambda x R,\ x\in[0,a]$	$p=w=x$ vanish at $x=0$ ; eigenvalues $\lambda_{m,k}=(j_{m,k}/a)^2$ come from Bessel zeros, eigenfunctions $J_m(\sqrt{\lambda}\,x)$ .
Hermite (whole line)	$-u''+x^2 u=\lambda u,\ x\in(-\infty,\infty)$	Quantum harmonic oscillator: discrete spectrum $\lambda_n=2n+1$ , $u_n\propto H_n(x)e^{-x^2/2}$ with $n$ nodes, despite the unbounded interval.
Large eigenvalue (WKB)	$\lambda_n\sim\big((n+1)\pi/\!\int_a^b\!\sqrt{w/p}\,\mathrm{d}x\big)^2$	High modes ignore $q$ and behave like a uniform string of rescaled length $\int\sqrt{w/p}\,\mathrm{d}x$ — Weyl's law.

Common trapThe oscillation and interlacing theorems are not universal — they belong to one self-adjoint operator with a positive weight

The node count $n$ , the nodeless ground state, and the strict interlacing are theorems about a single regular Sturm–Liouville operator with weight $w>0$ and separated boundary conditions. They break the moment those hypotheses do. Forget to convert to self-adjoint form and the eigenvalues can come out complex and the modes non-orthogonal in the unweighted inner product. Let $w$ change sign and the spectrum can run off to $-\infty$ as well, so there is no lowest mode to anchor the counting. Replace separated conditions with periodic ones — the natural choice on a circle — and the operator acquires degenerate eigenvalues: $\cos(nx)$ and $\sin(nx)$ share $\lambda=n^2$ , simplicity fails, and the clean one-to-one match between eigenvalue rank and node count collapses. At a singular endpoint the phrase "number of interior zeros" can even become infinite. Every clean statement below assumes the regular, separated, positive-weight setting.

Worked examples

1 · The vibrating string, worked end to end (confidence builder)

Take the regular problem $-u''=\lambda u$ on $[0,\pi]$ with $u(0)=u(\pi)=0$ . Find the spectrum, verify the node count and orthogonality on the lowest modes, and check Weyl's law against the exact answer.

Step 1. Here $p=w=1$ and $q=0$ . The general solution of $u''+\lambda u=0$ satisfying $u(0)=0$ is $u=\sin(\sqrt{\lambda}\,x)$ ; the right condition $u(\pi)=0$ needs $\sqrt{\lambda}\,\pi=(n{+}1)\pi$ , so $\sqrt{\lambda}=n{+}1$ for $n=0,1,2,\dots$ . The spectrum is $\lambda_n=(n+1)^2$ , giving $\lambda_0=1,\ \lambda_1=4,\ \lambda_2=9,\ \lambda_3=16$ , and the eigenfunctions are $u_n=\sin((n{+}1)x)$ .

Step 2. Node count: $u_n=\sin((n{+}1)x)$ vanishes at $x=j\pi/(n{+}1)$ for $j=1,\dots,n$ inside $(0,\pi)$ , which is exactly $n$ interior zeros — the oscillation theorem verbatim. The single node of $u_1$ sits at $x=\pi/2$ ; the two nodes of $u_2$ sit at $\pi/3$ and $2\pi/3$ , one on either side of $\pi/2$ , so the zeros of $u_1$ and $u_2$ interlace, as the theorem demands.

Step 3. Orthogonality: with weight $w=1$ , $\displaystyle\int_0^\pi\sin(x)\sin(2x)\,\mathrm{d}x=\tfrac12\int_0^\pi[\cos x-\cos 3x]\,\mathrm{d}x=\tfrac12\big[\sin x-\tfrac13\sin 3x\big]_0^\pi=\mathbf{0}$ . The lowest two modes are orthogonal, as required, and the same collapse happens for every distinct pair.

Step 4. Weyl's law: $\int_0^\pi\sqrt{w/p}\,\mathrm{d}x=\pi$ , so it predicts $\lambda_n\sim((n{+}1)\pi/\pi)^2=(n+1)^2$ — exact for every $n$ , because a uniform string is the case in which WKB carries no error. Spectrum, nodes and asymptotics agree on the nose.

2 · Reducing the Chebyshev equation to self-adjoint form

The Chebyshev equation $(1-x^2)u''-x\,u'+n^2 u=0$ on $(-1,1)$ is not self-adjoint as written. Find the integrating factor $p$ , the weight $w$ , the eigenvalue, and identify the orthogonality it enforces.

Step 1. Match to $a_2 u''+a_1 u'+a_0 u=\lambda u$ : here $a_2=1-x^2$ , $a_1=-x$ , and the eigenvalue sits as $\lambda=n^2$ with $a_0=0$ absorbed into $\lambda$ . The integrating factor is $p=\exp\!\int(a_1/a_2)\,\mathrm{d}x=\exp\!\int\frac{-x}{1-x^2}\,\mathrm{d}x$ .

Step 2. The integral is $\int\frac{-x}{1-x^2}\,\mathrm{d}x=\tfrac12\ln(1-x^2)$ , so $p=\sqrt{1-x^2}$ , which duly vanishes at both endpoints $x=\pm 1$ — the problem is singular, and no boundary data are needed there, only boundedness.

Step 3. The weight is $w=p/a_2=\sqrt{1-x^2}/(1-x^2)=\mathbf{1/\sqrt{1-x^2}}$ , and $q=0$ . The self-adjoint form is therefore $-\big(\sqrt{1-x^2}\,u'\big)'=n^2\,(1-x^2)^{-1/2}\,u$ , with eigenvalue $\lambda_n=n^2$ .

Step 4. Consequence: the eigenfunctions are the Chebyshev polynomials $T_n(x)=\cos(n\arccos x)$ , and self-adjointness guarantees $\displaystyle\int_{-1}^{1}\frac{T_m(x)\,T_n(x)}{\sqrt{1-x^2}}\,\mathrm{d}x=0$ for $m\neq n$ — the famous Chebyshev orthogonality, now seen as nothing but Sturm–Liouville orthogonality in the weight the integrating factor produced.

3 · A variational bound on the ground state

Estimate the lowest eigenvalue of $-u''=\lambda u$ on $[0,\pi]$ with $u(0)=u(\pi)=0$ using the simplest admissible trial function, and compare to the exact $\lambda_0=1$ .

Step 1. Pick the smoothest function that vanishes at both ends and has no interior node: the parabola $u(x)=x(\pi-x)$ . It is admissible (satisfies the boundary conditions) and, being nodeless, is a good proxy for the ground state.

Step 2. Numerator of the Rayleigh quotient, with $p=1$ , $q=0$ : $u'=\pi-2x$ , so $\int_0^\pi(\pi-2x)^2\,\mathrm{d}x=\int_0^\pi(\pi^2-4\pi x+4x^2)\,\mathrm{d}x=\pi^3-2\pi^3+\tfrac43\pi^3=\tfrac13\pi^3$ .

Step 3. Denominator, with $w=1$ : $\int_0^\pi x^2(\pi-x)^2\,\mathrm{d}x=\int_0^\pi(\pi^2x^2-2\pi x^3+x^4)\,\mathrm{d}x=\tfrac13\pi^5-\tfrac12\pi^5+\tfrac15\pi^5=\tfrac{1}{30}\pi^5$ .

Step 4. The quotient is $R[u]=\dfrac{\pi^3/3}{\pi^5/30}=\dfrac{10}{\pi^2}=\mathbf{1.0132}$ , an upper bound lying just 1.32 % above the exact $\lambda_0=1$ . A single crude parabola, with no fitting, pins the ground-state eigenvalue to better than two percent — the practical power of the minimax principle, and the basis of the Rayleigh–Ritz method.

Physics of the portrait

The subject carries two names because it was born from two collaborators who, in a burst of papers between 1829 and 1837, turned a class of differential equations into the first infinite-dimensional spectral theory — decades before Hilbert gave it a home.

Origins · 1 minJacques Charles François Sturm and how this was discovered

Jacques Charles François Sturm

1803–1855 · founder of the oscillation theory of second-order differential equations

Born in Geneva, Sturm made his name young: his 1829 theorem for counting the real roots of a polynomial in an interval is still taught, and it planted the seed of everything that followed — the idea that you can count solutions by watching sign changes, without ever solving. Moving to Paris, he produced in 1836 the memoir that founded oscillation theory, proving that solutions of $-(p\,u')'+q\,u=0$ oscillate faster when the potential is raised and that their zeros interlace, by the cross-bracket comparison argument that still bears his name. His friend Joseph Liouville supplied the complementary half: that the eigenvalues form a discrete increasing sequence, that the eigenfunctions are orthogonal, and that an arbitrary function expands in them — the convergence question Liouville pursued for years. Their joint papers in the 1836–37 volumes of Liouville's own journal are the charter of the field. The story did not end there: Hermann Weyl in 1910 extended the theory to singular endpoints with his limit-point/limit-circle classification, and Heinz Prüfer in 1926 recast oscillation theory through the phase-amplitude transformation that makes node-counting a matter of watching an angle wind. What Sturm and Liouville built by hand, the twentieth century recognized as the spectral theorem for a self-adjoint operator — the same mathematics that underwrites quantum mechanics.

History

1829

— Sturm publishes his theorem for counting the real roots of a polynomial

1836

— Sturm's memoir founds the comparison and oscillation theorems for second-order equations

1837

— Sturm and Liouville jointly establish eigenfunction orthogonality and expansion

1910

— Weyl introduces the limit-point and limit-circle theory for singular endpoints

1926

— Prüfer recasts oscillation theory through the phase-amplitude transformation

Going further

Texts. Stone and Goldbart, Mathematics for Physics, ch. 8 (self-adjoint operators) and ch. 9 · Arfken and Weber, Mathematical Methods for Physicists, the Sturm–Liouville chapter · Riley, Hobson and Bence, Mathematical Methods for Physics and Engineering, ch. 17 · Courant and Hilbert, Methods of Mathematical Physics, vol. 1, ch. V–VI · Coddington and Levinson, Theory of Ordinary Differential Equations, ch. 7–8 · Hassani, Mathematical Physics, the Sturm–Liouville and special-functions chapters. All results here are classical Sturm–Liouville theory. The key anchor values are: the integrating-factor formulas $p=\exp\!\int(a_1/a_2)$ , $w=p/a_2$ , $q=-(a_0/a_2)p$ ; the string spectrum $\lambda_n=(n+1)^2$ with $u_n=\sin((n+1)x)$ on $[0,\pi]$ , its node counts, and $\int_0^\pi\sin x\sin 2x\,\mathrm{d}x=0$ ; the Chebyshev reduction $p=\sqrt{1-x^2}$ , $w=(1-x^2)^{-1/2}$ , $\lambda_n=n^2$ ; the Rayleigh bound $10/\pi^2=1.0132$ against $\lambda_0=1$ ; and the classical eigenvalues Legendre $\lambda_\ell=\ell(\ell+1)$ , oscillator $\lambda_n=2n+1$ , Bessel $\lambda_{m,k}=(j_{m,k}/a)^2$ (with $j_{0,1}=2.405$ ).

Builds on. Linear Second-Order ODEs · The Wronskian and Linear Independence · Two-Point Boundary-Value Problems · Existence and Uniqueness Theorems for ODEs · Ordinary and Singular Points. Leads to. The Method of Frobenius · Power Series Solutions · Variation of Parameters.

Ask the tutor: ask me to walk you through reducing your own second-order ODE to self-adjoint form and reading off its spectrum, nodes, and Rayleigh bound step by step.

What actually makes an operator "self-adjoint" here, as opposed to just symmetric?

The working requirement is symmetry with a killed boundary term: $\langle\mathcal{L}u,v\rangle=\langle u,\mathcal{L}v\rangle$ for all functions in the domain, which separated boundary conditions guarantee. True self-adjointness is the subtler statement that the operator and its adjoint share the same domain, and for singular problems the two can differ — exactly what Weyl's limit-point/limit-circle theory diagnoses. On a regular interval with separated conditions the distinction evaporates: the symmetric operator is essentially self-adjoint and every spectral consequence holds verbatim. The care is needed only at singular endpoints, where you must ask whether extra boundary data are required at all.

Why is the weight w so important — can't I just use the ordinary L² inner product?

No, and using plain $L^2$ is one of the most common errors. The operator $\mathcal{L}=-(p\,u')'+q$ is symmetric with respect to the weighted inner product $\int u\,v\,w\,\mathrm{d}x$ , not the unweighted one, because the eigenvalue equation is $\mathcal{L}u=\lambda w u$ with $w$ multiplying the eigenvalue. Orthogonality therefore reads $\int u_m u_n\,w\,\mathrm{d}x=0$ , and the expansion coefficients in an eigenfunction series carry that same $w$ . Drop the weight and the eigenfunctions will generally not look orthogonal, your coefficient formulas will be wrong, and completeness will appear to fail. The weight is the fingerprint of the non-uniform density in the physical problem, and it belongs in every inner product you write.

How can Hermite or Bessel functions be "eigenfunctions" on an infinite or singular interval?

Because singularity does not destroy discreteness — it only changes what plays the role of the boundary condition. For the harmonic oscillator on the whole line, " $u$ must decay to zero at $\pm\infty$ " replaces a fixed endpoint value, and only $\lambda_n=2n+1$ admit a decaying solution, so the spectrum is still discrete. For Bessel and Legendre the coefficient $p$ vanishes at an endpoint and "boundedness there" replaces the data. Weyl's limit-point/limit-circle test tells you, endpoint by endpoint, whether such a condition is needed and whether the spectrum stays discrete or picks up a continuous part, as the radial hydrogen problem does above threshold. The oscillation theorem survives: $H_n$ still has $n$ zeros, $P_\ell$ still has $\ell$ .

Does the ground state really never have an interior node, and why does that matter physically?

Yes — for a regular Sturm–Liouville problem the lowest eigenfunction $u_0$ has zero interior zeros, so it keeps one sign throughout $(a,b)$ . This is decisive in quantum mechanics, where the time-independent Schrödinger equation is a Sturm–Liouville problem with $p=w=1$ , $q=V$ , $\lambda=E$ : the nodeless theorem lets the ground-state wavefunction be taken everywhere positive, so it can serve as a probability amplitude of definite sign, and it underlies variational energy estimates. It also fixes the ordering — the $n$ -th bound state has exactly $n$ nodes, so counting nodes tells you which excited state you have without computing a single energy.

How is the Rayleigh–Ritz method connected to all of this, and why does it always overestimate?

The Rayleigh quotient $R[u]$ equals $\lambda_0$ only at the true ground state and is strictly larger for any other admissible function, because a trial function is a mixture of the true eigenfunctions and the higher modes drag the average energy up. So $R[u]\ge\lambda_0$ for all trial $u$ — every estimate is an upper bound, as Worked Example 3 shows with $10/\pi^2>1$ . Rayleigh–Ritz systematizes this by minimizing $R$ over a finite trial space; minimax guarantees the Ritz values bound the true $\lambda_n$ from above and converge downward as the space grows. That is why it is a workhorse for Schrödinger operators, plates and molecules — it never lies about the sign of its error.

When exactly do the oscillation and interlacing theorems fail?

Whenever the regular, separated, positive-weight hypotheses are violated. Non-separated boundary conditions — periodic ones on a ring — allow degenerate eigenvalues, and once two independent eigenfunctions share an eigenvalue the one-to-one map from rank to node count is gone. A weight $w$ that changes sign lets the spectrum run to $-\infty$ too, so there is no ground state to start the induction. Skipping the self-adjoint form can produce complex "eigenvalues" for which node-counting is meaningless, and at a genuinely singular endpoint the number of interior zeros can be infinite. The clean counting is a reward for staying inside the regular self-adjoint setting.