Sturm–Liouville Theory

Sturm–Liouville theory is the spectral theory of a broad class of self-adjoint second-order differential operators. It explains, within one framework, why many separated quantum equations have real eigenvalues, weighted orthogonal eigenfunctions, ordered nodes, and useful eigenfunction expansions.

The conclusions depend on the hypotheses. A regular problem on a finite interval has a clean discrete theorem. Radial equations, infinite intervals, and coefficients that vanish at an endpoint are singular problems; they retain much of the same structure but require additional endpoint and spectral analysis.

Standard form

A Sturm–Liouville eigenvalue equation is

$$-\frac{d}{dx} \left[ p(x)\frac{dy}{dx} \right] +q(x)y =\lambda w(x)y.$$

The functions have distinct roles:

• $p(x)$ controls the derivative or flux term;
• $q(x)$ is the multiplication term;
• $w(x)$ is the weight;
• $\lambda$ is the spectral parameter.

It is convenient to divide by the weight and define

$$\mathcal L y = \frac{1}{w} \left[ -(py')'+qy \right].$$

Then the eigenvalue equation is $\mathcal L y=\lambda y$ in the weighted space $L^2_w([a,b])$. Its elements satisfy

$$\int_a^b \lvert f(x)\rvert^2w(x)\,dx <\infty,$$

with inner product

$$\langle f,g\rangle_w = \int_a^b f(x)^*g(x)w(x)\,dx.$$

The weight is not optional notation. It determines normalization, orthogonality, expansion coefficients, and the Hilbert space in which $\mathcal L$ acts.

The regular problem

A standard regular problem has:

• a finite closed interval $[a,b]$;
• real coefficients with enough continuity for the integrations below;
• $p(x)>0$ and $w(x)>0$ throughout the closed interval;
• two homogeneous boundary conditions that make the boundary form vanish;
• boundary conditions independent of $\lambda$.

One common separated choice is

$$\begin{aligned} \alpha_1y(a) +\alpha_2p(a)y'(a)&=0,\\ \beta_1y(b) +\beta_2p(b)y'(b)&=0, \end{aligned}$$

where all coefficients are real and neither pair is identically zero. Dirichlet, Neumann, and real Robin conditions are included. Periodic and twisted conditions couple the endpoints and are also self-adjoint, but some theorem statements, especially simplicity of eigenvalues, must then be modified.

The boundary choices are part of the operator. Their classification and physical interpretation are developed in Boundary Conditions.

Lagrange identity

For sufficiently regular functions $u$ and $v$,

$$\begin{aligned} u^* \left[ -(pv')'+qv \right] &- \left[ -(pu')'+qu \right]^*v\\ &= \frac{d}{dx} \left[ p\left(u'^*v-u^*v'\right) \right]. \end{aligned}$$

Integrating gives Green’s identity,

$$\begin{aligned} \langle u,\mathcal Lv\rangle_w -\langle\mathcal Lu,v\rangle_w &= \left[ p\left(u'^*v-u^*v'\right) \right]_a^b. \end{aligned}$$

The right side is the boundary form. Self-adjoint boundary conditions make it vanish for every pair in the operator domain. This identity is the engine behind reality and orthogonality.

Reality of the eigenvalues

Suppose $\mathcal Ly_n=\lambda_n y_n$ and the boundary form vanishes. Set $u=v=y_n$ in Green’s identity:

$$\left( \lambda_n-\lambda_n^* \right) \langle y_n,y_n\rangle_w =0.$$

A nonzero eigenfunction has positive weighted norm, so

$$\lambda_n=\lambda_n^*.$$

Thus the eigenvalues are real. This is not merely a consequence of real coefficients: the domain and boundary conditions are essential.

Weighted orthogonality

Let $y_m$ and $y_n$ have distinct eigenvalues. Green’s identity gives

$$\left( \lambda_n-\lambda_m \right) \langle y_m,y_n\rangle_w =0.$$

Therefore,

$$\int_a^b y_m(x)^*y_n(x)w(x)\,dx =0.$$

This holds whenever $\lambda_m\ne\lambda_n$.

If an eigenvalue is degenerate, one can choose an orthonormal basis inside its eigenspace, but orthogonality does not follow from the eigenvalue difference alone.

For a scalar regular problem with separated self-adjoint boundary conditions, each eigenvalue is simple. Coupled conditions can permit degeneracy: the periodic Laplacian has independent sine and cosine modes at the same positive eigenvalue.

The regular spectral theorem

For a real regular scalar Sturm–Liouville problem with separated self-adjoint boundary conditions, the eigenvalues can be ordered as

$$\lambda_0 <\lambda_1 <\lambda_2 <\cdots, \qquad \lambda_n\longrightarrow+\infty.$$

With indexing beginning at zero, an eigenfunction $y_n$ has exactly $n$ zeros in the open interval $(a,b)$. The eigenfunctions are mutually orthogonal in $L^2_w$ and complete there.

These conclusions should be read with their scope attached:

• simplicity and the stated node count assume separated conditions;
• regularity excludes infinite intervals and singular endpoints;
• completeness is a Hilbert-space statement, not automatic pointwise convergence of every formal series;
• a singular problem may also have continuous spectrum.

The node theorem gives a recognizable quantum pattern: the ground-state mode has no interior node, and higher modes acquire nodes in spectral order.

Completeness and expansions

Choose normalized eigenfunctions,

$$\langle y_m,y_n\rangle_w=\delta_{mn}.$$

Completeness means that a function $f\in L^2_w([a,b])$ can be approximated in weighted mean square by finite eigenfunction sums:

$$f_N(x) = \sum_{n=0}^{N} c_ny_n(x), \qquad c_n=\langle y_n,f\rangle_w.$$

More precisely,

$$\lim_{N\to\infty} \lVert f-f_N\rVert_w =0.$$

Parseval’s identity then reads

$$\lVert f\rVert_w^2 = \sum_{n=0}^{\infty} \lvert c_n\rvert^2.$$

Pointwise convergence, convergence at endpoints, uniform convergence, and term-by-term differentiation require stronger assumptions on $f$ and the problem. Those distinctions belong to Sequences, Series, and Convergence.

In quantum mechanics, this is the mathematical basis for expanding a state in stationary modes. Time dependence then acts on the coefficients through phase factors when the spectrum is discrete.

Rayleigh quotient and the lowest mode

For Dirichlet data, integration by parts gives the Rayleigh quotient

$$\mathcal R[y] = \frac{ \displaystyle \int_a^b \left( p\lvert y'\rvert^2 +q\lvert y\rvert^2 \right)dx }{ \displaystyle \int_a^b w\lvert y\rvert^2\,dx }.$$

The lowest eigenvalue satisfies

$$\lambda_0 = \min_{y\ne0} \mathcal R[y]$$

over the appropriate form domain. Higher eigenvalues follow from min–max principles with orthogonality constraints. Other boundary conditions can add boundary terms to the quadratic form, so the displayed numerator should not be transplanted unchanged.

This variational characterization explains several qualitative facts. If $q\ge0$ under Dirichlet data, then $\lambda_0>0$. Trial functions give upper bounds on $\lambda_0$, and a nodeless lowest mode avoids unnecessary derivative cost.

Example: the finite interval

The problem

$$-y''=\lambda y, \qquad y(0)=y(L)=0$$

has $p=w=1$ and $q=0$. Its normalized eigenfunctions and eigenvalues are

$$\begin{aligned} y_n(x) &= \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right),\\ \lambda_n &= \left(\frac{n\pi}{L}\right)^2, \qquad n=1,2,\ldots. \end{aligned}$$

The indexing here begins at $n=1$, so $y_n$ has $n-1$ interior zeros. These modes form the Fourier sine basis. The physical Hamiltonian multiplies $\lambda_n$ by $\hbar^2/(2m)$; see Infinite Square Well.

Example: Legendre polynomials

Legendre’s equation can be written

$$-\frac{d}{dx} \left[ (1-x^2)\frac{dP}{dx} \right] = \lambda P \qquad \text{on }[-1,1].$$

Thus

$$\begin{aligned} p(x)&=1-x^2, &q(x)&=0,\\ w(x)&=1. \end{aligned}$$

Because $p$ vanishes at both endpoints, this is not a regular problem in the strict sense. Requiring the solution to remain admissible at $x=\pm1$ selects

$$\lambda_\ell=\ell(\ell+1), \qquad P_\ell(x),$$

and the polynomials satisfy

$$\int_{-1}^{1} P_\ell(x)P_{\ell'}(x)\,dx =0 \qquad \text{for }\ell\ne\ell'.$$

Their recurrence relations and normalization are developed in Legendre Polynomials.

Example: Bessel’s equation

Bessel’s equation,

$$r^2R'' +rR' +\left(k^2r^2-\nu^2\right)R =0,$$

has Sturm–Liouville form

$$-\frac{d}{dr} \left( r\frac{dR}{dr} \right) +\frac{\nu^2}{r}R = k^2rR.$$

Here $p(r)=r$ and $w(r)=r$. On a finite disk, regularity at $r=0$ and a condition at the outer radius quantize $k$. Modes with distinct radial eigenvalues are orthogonal with measure $r\,dr$, exactly the radial part of the two-dimensional area element. The origin is singular because $p$ vanishes and $q$ diverges there.

See Bessel Functions for zeros, recurrences, and normalization formulas.

Example: the radial Schrödinger equation

For a central potential, the unreduced radial equation can be arranged as

$$\begin{aligned} -\frac{d}{dr} \left( r^2\frac{dR_\ell}{dr} \right) &+\ell(\ell+1)R_\ell\\ &+\frac{2mr^2}{\hbar^2}V(r)R_\ell\\ &= \frac{2mE}{\hbar^2} r^2R_\ell. \end{aligned}$$

The Sturm–Liouville weight is $w(r)=r^2$, matching the radial probability measure. Passing to $u_\ell=rR_\ell$ converts the problem to a one-dimensional Schrödinger form with weight one, but it also changes the endpoint condition at the origin.

The interval $(0,\infty)$ and the endpoint $r=0$ are singular, so the regular finite-interval theorem cannot simply be quoted. The physical derivation is in Radial Schrödinger Equation.

Example: Hermite polynomials

Hermite’s equation,

$$H_n''-2xH_n'+2nH_n=0,$$

becomes

$$-\frac{d}{dx} \left( e^{-x^2}H_n' \right) = 2n\,e^{-x^2}H_n.$$

Thus $p=w=e^{-x^2}$, $q=0$, and $\lambda_n=2n$ on the whole real line. The weighted orthogonality is

$$\int_{-\infty}^{\infty} H_m(x)H_n(x)e^{-x^2}\,dx =0.$$

This holds for $m\ne n$.

This is a singular infinite-interval problem, but Gaussian decay suppresses the boundary term for polynomial solutions. The relation to oscillator wavefunctions is developed in Hermite Polynomials.

Liouville transformation

Under suitable smoothness and positivity assumptions, define a new coordinate and dependent variable by

$$\begin{aligned} z(x) &= \int^x \sqrt{\frac{w(s)}{p(s)}}\,ds,\\ u(z) &= \left[p(x)w(x)\right]^{1/4}y(x). \end{aligned}$$

The Sturm–Liouville equation becomes a Schrödinger-type equation

$$-\frac{d^2u}{dz^2} +Q(z)u =\lambda u,$$

where $Q$ combines $q$, $p$, $w$, and their derivatives. This transformation clarifies why second-order spectral problems share so many features. It can also move or change singular endpoints, so boundary data must be transformed along with the equation.

Recognizing hidden Sturm–Liouville form

Suppose an equation is written as

$$a(x)y'' +b(x)y' +\left[ c(x)+\lambda r(x) \right]y =0.$$

Choose an integrating factor $\mu$ satisfying

$$\frac{\mu'}{\mu} = \frac{b-a'}{a}.$$

Then $p=\mu a$ obeys $p'=\mu b$, and multiplication by $\mu$ produces

$$-(py')' -\mu c\,y = \lambda\mu r\,y.$$

This identifies

$$q=-\mu c, \qquad w=\mu r.$$

The algebra is only the first check. One must still verify positivity of $p$ and $w$, endpoint behavior, and self-adjoint boundary conditions.

Singular problems

A Sturm–Liouville problem is singular if, for example:

• an endpoint is infinite;
• $p$ or $w$ vanishes at an endpoint;
• a coefficient is not integrable in the required sense;
• the potential term diverges.

At a singular endpoint, square-integrability may uniquely select the admissible behavior or may leave a boundary condition to be chosen. These are the limit-point and limit-circle alternatives. Depending on the problem, the spectrum can be discrete, continuous, or mixed, and a complete expansion may involve both sums and integrals.

Legendre, Bessel, Hermite, and radial Schrödinger equations are all singular in this technical sense. Calling them Sturm–Liouville problems remains useful, but it does not license every conclusion of the regular theorem without proof.

Numerical use

A self-adjoint discretization should preserve the weighted structure:

1. discretize the flux $py'$ consistently;

2. represent the weight through a mass matrix $W$ when appropriate;

3. solve the generalized eigenproblem

   $$A\mathbf y =\lambda W\mathbf y;$$

4. normalize eigenvectors with $\mathbf y_m^\dagger W\mathbf y_n=\delta_{mn}$;

5. check residuals, weighted orthogonality, node counts, and convergence under refinement.

Replacing a generalized problem by $W^{-1}A$ can destroy visible symmetry and worsen conditioning. Structure-preserving solvers usually work directly with the Hermitian pair $(A,W)$ or a symmetric factorization of $W$.

Common mistakes

• Omitting the weight from normalization or expansion coefficients.
• Assuming real coefficients alone guarantee real eigenvalues.
• Applying the regular discrete theorem to an infinite interval.
• Claiming all Sturm–Liouville eigenvalues are simple despite periodic degeneracies.
• Confusing completeness in $L^2_w$ with pointwise convergence everywhere.
• Using the Dirichlet Rayleigh quotient for Robin data without boundary terms.
• Treating regularity at a singular endpoint as an arbitrary aesthetic choice.
• Changing from $R_\ell$ to $u_\ell=rR_\ell$ without transforming the measure and endpoint condition.
• Discretizing $w$ but normalizing eigenvectors with the ordinary Euclidean dot product.
• Identifying an integrating factor and skipping the sign and positivity checks.

Exercises

1. Derive weighted orthogonality for two eigenfunctions $y_m$ and $y_n$ of a self-adjoint Sturm–Liouville problem with distinct eigenvalues.

Solution

Write

$$\begin{aligned} -(py_n')'+qy_n&=\lambda_nwy_n,\\ -(py_m')'+qy_m&=\lambda_mwy_m. \end{aligned}$$

Multiply the first equation by $y_m^*$, multiply the complex conjugate of the second by $y_n$, and subtract. Define

$$\begin{aligned} B_{mn}(x) &= p(x) \bigl[ y_m'(x)^*y_n(x)\\ &\qquad -y_m(x)^*y_n'(x) \bigr]. \end{aligned}$$

Integration gives

$$\left( \lambda_n-\lambda_m \right) \int_a^b y_m^*y_nw\,dx = \left[B_{mn}\right]_a^b.$$

The boundary form vanishes for the self-adjoint domain. Since $\lambda_n\ne\lambda_m$,

$$\int_a^b y_m(x)^*y_n(x)w(x)\,dx=0.$$

2. On $1\le x\le e$, consider

   $$\begin{gathered} -\frac{d}{dx} \left( x\frac{dy}{dx} \right) = \lambda\frac{1}{x}y,\\ y(1)=y(e)=0. \end{gathered}$$

   Find the eigenfunctions and verify their weighted orthogonality.

Solution

Set $t=\ln x$. Since

$$x\frac{dy}{dx} = \frac{dy}{dt}, \qquad \frac{d}{dx} \left( x\frac{dy}{dx} \right) = \frac{1}{x}\frac{d^2y}{dt^2},$$

the equation becomes

$$-\frac{d^2y}{dt^2} = \lambda y \qquad \text{on }0\le t\le1$$

with Dirichlet endpoints. Hence

$$y_n(x)=\sin\left(n\pi\ln x\right), \qquad \lambda_n=n^2\pi^2.$$

The weight is $w(x)=1/x$. Because $dt=dx/x$,

$$\begin{aligned} I_{mn} &= \int_1^e y_m(x)y_n(x)\frac{dx}{x} \\ &= \int_0^1 \sin(m\pi t)\sin(n\pi t)\,dt\\ &= \frac{1}{2}\delta_{mn}. \end{aligned}$$

The normalized eigenfunctions are therefore $\sqrt{2}\sin(n\pi\ln x)$.

3. For $-y''=\lambda y$ on $[0,L]$ with periodic boundary conditions, show that every positive eigenvalue is at least twofold degenerate over the complex numbers. Explain why this does not contradict the simplicity theorem stated above.

Solution

Periodic conditions require

$$y(L)=y(0), \qquad y'(L)=y'(0).$$

The allowed wave numbers are $k_n=2\pi n/L$, with modes

$$y_n(x)=e^{2\pi i n x/L}, \qquad n\in\mathbb Z.$$

The eigenvalue is

$$\lambda_n = \left(\frac{2\pi n}{L}\right)^2.$$

For each $n\ne0$, the modes $y_n$ and $y_{-n}$ are linearly independent and have the same eigenvalue. Equivalently, a real basis is given by sine and cosine. There is no contradiction because the simplicity theorem was stated for separated self-adjoint boundary conditions, whereas periodic conditions couple the two endpoints.

4. Starting from Hermite’s equation,

   $$H_n''-2xH_n'+2nH_n=0,$$

   derive its Sturm–Liouville form and identify $p$, $q$, $w$, and $\lambda$. Why does the boundary term vanish for polynomial solutions?

Solution

Multiply by $e^{-x^2}$. Since

$$\frac{d}{dx} \left( e^{-x^2}H_n' \right) = e^{-x^2} \left( H_n''-2xH_n' \right),$$

the equation becomes

$$-\frac{d}{dx} \left( e^{-x^2}H_n' \right) = 2n\,e^{-x^2}H_n.$$

Therefore,

$$\begin{aligned} p(x)&=e^{-x^2}, &q(x)&=0,\\ w(x)&=e^{-x^2}, &\lambda_n&=2n. \end{aligned}$$

The boundary form contains

$$e^{-x^2} \left( H_m'H_n-H_mH_n' \right).$$

The factor in parentheses is a polynomial, while the Gaussian decays faster than any polynomial grows. The expression therefore tends to zero at both infinities, which gives weighted orthogonality for distinct $m$ and $n$.

References

• A. Zettl, Sturm–Liouville Theory, American Mathematical Society, 2005.
• E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, 2nd ed., Oxford University Press, 1962.
• A. M. Krall, Hilbert Space, Boundary Value Problems and Orthogonal Polynomials, Birkhäuser, 2002.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.
• R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, 1989.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.