Sturm–Liouville Problems and Eigenfunction Expansions

A regular Sturm–Liouville problem becomes a spectral problem only after its weight and boundary domain are specified. On a finite interval, real coefficients with positive leading coefficient and weight, together with self-adjoint boundary conditions, define an operator with real discrete spectrum and a complete weighted-orthonormal set of eigenfunctions. That is what licenses a mode expansion. Formal symmetry of the differential expression alone is not enough.

This page treats the clean regular theorem and its Green-kernel consequence. Singular endpoints, continuous spectrum, and the general theory of self-adjoint extensions are marked as later handoffs.

Required background. Linear ODEs, Evolution Operators, and Wronskians supplies fundamental matrices, boundary data, and Wronskian identities.

Helpful background. Bilinear and Hermitian Forms, Adjoints, and Isometries supplies adjoints, orthogonality, and the complex inner-product convention.

The regular problem and its weighted space

Let $a<b$ be finite and consider

$$\ell y = -\frac{\mathrm d}{\mathrm dx} \left( p(x)\frac{\mathrm dy}{\mathrm dx} \right) +q(x)y.$$

For the theorem below, assume

$$p\in C^1([a,b]), \qquad q,w\in C([a,b]), \qquad p(x)>0, \quad w(x)>0,$$

with $p,q,w$ real. These assumptions are stronger than necessary, but they make both endpoints regular and keep the boundary data classical. The eigenvalue equation is

$$\ell y=\lambda w y.$$

The weight belongs in the Hilbert space:

$$\mathcal H = L^2([a,b],w(x)\,\mathrm dx), \qquad \langle f|g\rangle_w = \int_a^b \overline{f(x)}\,g(x)\,w(x)\,\mathrm dx.$$

As elsewhere on the site, the first slot is conjugate-linear. Define the operator

$$L=w^{-1}\ell.$$

An operator is an action together with a domain. A convenient maximal regular domain is

$$\begin{aligned} \mathcal D_{\max} = \bigl\{ y\in\mathcal H:\;& y,\;py'\in AC([a,b]),\\ & Ly\in\mathcal H \bigr\}. \end{aligned}$$

Absolute continuity makes $y(a)$, $y(b)$, $(py')(a)$, and $(py')(b)$ well-defined. A realization of $L$ is obtained by restricting $\mathcal D_{\max}$ with two independent boundary conditions. Which restriction is chosen changes the operator and usually changes its spectrum.

If a coefficient is merely piecewise regular, the interval can be split and the equation integrated across each interface. Continuity of $y$ and of the flux $py'$ is then typical when no singular source is present. That is an interface problem, not automatically a singular-endpoint problem.

Lagrange identity and the boundary form

For $f,g\in\mathcal D_{\max}$, direct differentiation gives

$$\overline f\,\ell g - \overline{\ell f}\,g = \frac{\mathrm d}{\mathrm dx} \left[ p\left( \overline{f'}g-\overline f\,g' \right) \right].$$

After integration,

$$\begin{aligned} \langle f|Lg\rangle_w - \langle Lf|g\rangle_w &= \mathsf B(f,g),\\ \mathsf B(f,g) &= \left[ p\left( \overline{f'}g-\overline f\,g' \right) \right]_a^b. \end{aligned}$$

This is Green’s or Lagrange’s identity. For real functions, the expression inside the brackets is the negative of the modified Wronskian convention used in Linear ODEs, Evolution Operators, and Wronskians. The two statements contain the same conserved boundary flux.

The differential expression is formally symmetric because the bulk terms cancel. The operator on a domain $\mathcal D(L)$ is symmetric only if

$$\mathsf B(f,g)=0 \qquad \text{for every }f,g\in\mathcal D(L).$$

Checking $\mathsf B(f,f)=0$ alone is insufficient. In the regular second-order problem, a self-adjoint domain is obtained when the allowed boundary data form a maximal subspace on which this boundary form vanishes. The maximality clause distinguishes self-adjointness from a symmetric restriction with too many boundary conditions.

Boundary conditions that give a self-adjoint problem

Separated conditions

The most common choice imposes one real condition at each endpoint:

$$\begin{aligned} \alpha_a y(a)+\beta_a (py')(a)&=0,\\ \alpha_b y(b)+\beta_b (py')(b)&=0, \end{aligned}$$

where each real pair $(\alpha_a,\beta_a)$ and $(\alpha_b,\beta_b)$ is nonzero. These conditions include

• Dirichlet: $y=0$;
• Neumann: $py'=0$; and
• Robin: a nontrivial real linear combination of $y$ and $py'$ vanishes.

At a fixed endpoint, any two functions satisfying the same separated condition have proportional boundary vectors $(y,py')$. Their contribution to $\mathsf B(f,g)$ therefore vanishes. Both endpoint contributions vanish separately, and the regular realization is self-adjoint.

Coupled conditions

Self-adjoint boundary conditions need not be separated. Package the boundary data as

$$\Gamma_0y = \begin{pmatrix} y(a)\\ y(b) \end{pmatrix}, \qquad \Gamma_1y = \begin{pmatrix} -(py')(a)\\ (py')(b) \end{pmatrix}.$$

Then

$$\mathsf B(f,g) = \langle\Gamma_1f|\Gamma_0g\rangle_{\mathbb C^2} - \langle\Gamma_0f|\Gamma_1g\rangle_{\mathbb C^2}.$$

In the regular scalar problem, the general self-adjoint boundary domain can be written

$$A\Gamma_0y+B\Gamma_1y=0,$$

where $A,B\in\mathbb C^{2\times2}$ obey

$$\operatorname{rank} \begin{pmatrix} A&B \end{pmatrix} =2, \qquad AB^\dagger=BA^\dagger.$$

The rank condition supplies two independent constraints; the matrix identity makes their boundary-data subspace maximal and isotropic for $\mathsf B$. This boundary-pair parametrization is an equivalent form of the regular self-adjoint classification in Zettl 2005, Chapter 4; Theorem 4.3.1 there also distinguishes the simple separated spectrum from the possible multiplicity-two real coupled spectrum.

Periodic and quasiperiodic conditions are useful coupled examples:

$$\begin{aligned} y(b)&=e^{i\theta}y(a),\\ (py')(b)&=e^{i\theta}(py')(a), \qquad \theta\in\mathbb R. \end{aligned}$$

The phase cancels from the boundary form. At $\theta=0$ the conditions are periodic, while $\theta=\pi$ gives antiperiodic conditions.

Reality, orthogonality, and multiplicity

Let $y_n,y_m$ lie in one self-adjoint domain and satisfy

$$Ly_n=\lambda_n y_n, \qquad Ly_m=\lambda_m y_m.$$

The boundary form vanishes, so

$$0 = \langle y_m|Ly_n\rangle_w - \langle Ly_m|y_n\rangle_w = \left( \lambda_n-\overline{\lambda_m} \right) \langle y_m|y_n\rangle_w.$$

Setting $m=n$ proves that every eigenvalue is real. If $\lambda_m\neq\lambda_n$, the corresponding eigenfunctions are orthogonal in the weighted inner product:

$$\int_a^b \overline{y_m(x)}\,y_n(x)\,w(x)\,\mathrm dx =0.$$

This derivation shows exactly where self-adjoint boundary data and the weight enter.

For a real scalar problem with separated boundary conditions, each eigenvalue is simple. Two eigenfunctions with the same eigenvalue satisfy the same left-endpoint condition, so their modified Wronskian vanishes at $a$; ODE uniqueness then makes them linearly dependent. Coupled conditions do not imply simplicity. For example, periodic boundary conditions for $-\mathrm d^2/\mathrm dx^2$ can pair the two modes $e^{\pm ikx}$ at one eigenvalue.

Completeness is a compact-resolvent result

The central regular theorem is stronger than orthogonality.

Regular Sturm–Liouville spectral theorem. Under the coefficient hypotheses above, any self-adjoint regular boundary domain defines a lower-bounded operator $L$ with compact resolvent. Its eigenvalues, repeated according to finite multiplicity, can be ordered so that $\lambda_n\to+\infty$. There is a weighted-orthonormal basis $\{y_n\}$ of $\mathcal H$ consisting of eigenfunctions of $L$.

Proof status. This is a cited theorem for general regular self-adjoint domains, including coupled conditions; see Zettl 2005, Chapter 4, especially Theorem 4.3.1. Teschl 2012, §5.4, Theorem 5.11, PDF gives the compact-resolvent proof and the stronger simplicity statement for the separated case. The following is a proof sketch of the shared mechanism.

1. Lagrange’s identity and the boundary domain give self-adjointness.
2. Choose a real resolvent point $z_0<\inf\sigma(L)$. Solving $(L-z_0)y=f$ produces a continuous Green kernel on the compact square $[a,b]^2$.
3. The corresponding integral resolvent is compact.
4. The spectral theorem for compact self-adjoint operators supplies a complete orthonormal eigenbasis for the resolvent and therefore for $L$.

The lower-bounded quadratic form of the second-order operator places the only spectral accumulation at $+\infty$. This compactness argument is specific to the regular finite-interval setting; it must not be silently exported to a half-line or a singular radial endpoint.

Normalize the eigenfunctions by

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

Then every $f\in\mathcal H$ has the expansion

$$f = \sum_{n} y_n\langle y_n|f\rangle_w,$$

with convergence in the $L^2_w$ norm, and Parseval’s identity reads

$$\|f\|_w^2 = \sum_n \left| \langle y_n|f\rangle_w \right|^2.$$

Hilbert-space completeness is not automatically pointwise or uniform convergence. Those stronger conclusions require additional regularity and compatibility of $f$ with the endpoint conditions.

The corresponding completeness relation is distributional:

$$\sum_n y_n(x)\overline{y_n(x')} = \frac{\delta(x-x')}{w(x')}.$$

Its meaning is fixed by the measure:

$$\int_a^b w(x')\,\mathrm dx'\, \sum_n y_n(x)\overline{y_n(x')}f(x') =f(x)$$

in the appropriate expansion sense. Omitting the weight from either the coefficient or the delta kernel changes the identity. The orthonormal expansion, Parseval identity, and distributional completeness relation are recorded together in NIST DLMF 2026, §1.18(v), Eqs. (1.18.29)–(1.18.33).

Green kernels and eigenfunction expansions

For $z\notin\sigma(L)$, the resolvent acts diagonally on the eigenbasis:

$$(L-z)^{-1}f = \sum_n \frac{ y_n\langle y_n|f\rangle_w }{ \lambda_n-z }.$$

Writing

$$(L-z)^{-1}f(x) = \int_a^b G_z(x,x')f(x')w(x')\,\mathrm dx',$$

gives the spectral kernel

$$G_z(x,x') = \sum_n \frac{ y_n(x)\overline{y_n(x')} }{ \lambda_n-z }.$$

The series represents the resolvent kernel; it is not a license to exchange pointwise limits and derivatives without a separate convergence argument. The resolvent expansion follows by applying the spectral theorem to $(L-z)^{-1}$; Teschl constructs the corresponding regular Green kernel in Teschl 2012, §5.4, Eqs. (5.60)–(5.69) and Lemma 5.10, PDF.

The source normalization can be written in either operator or differential form:

$$(L_x-z)G_z(x,x') = \frac{\delta(x-x')}{w(x')},$$

or equivalently

$$\left[ -\partial_x\!\left(p(x)\partial_x\right) +q(x)-z w(x) \right] G_z(x,x') = \delta(x-x').$$

Integrating the second equation through $x=x'$ yields the independent flux check

$$\left[ p(x)\partial_xG_z(x,x') \right]_{x=x'^-}^{x=x'^+} =-1.$$

For separated boundary conditions, the same kernel can be constructed from ODE solutions. Let $u_a(z,x)$ satisfy the left boundary condition and $u_b(z,x)$ the right one, and define the constant modified Wronskian

$$W_z = p(x) \left[ u_a(z,x)u_b'(z,x) - u_a'(z,x)u_b(z,x) \right].$$

If $z$ is not an eigenvalue, $W_z\neq0$, and

$$G_z(x,x') = -\frac{ u_a(z,x_<)u_b(z,x_>) }{ W_z }, \qquad x_<:=\min(x,x'), \quad x_>:=\max(x,x').$$

This expression is continuous at $x=x'$, satisfies the two endpoint conditions, and has the required flux jump. It also connects the fundamental solutions and Wronskians developed in Linear ODEs, Evolution Operators, and Wronskians to the spectral expansion above. The construction and its boundary normalization are the separated regular case of the Green-kernel formulas cited from Teschl above.

Self-adjointness gives the kernel check

$$G_z(x,x')^* = G_{\overline z}(x',x).$$

At $z=\lambda_n$, the resolvent does not exist and the spectral denominator must not be used. The equation

$$(L-\lambda_n)y=f$$

is solvable only when $f$ is orthogonal to the full $\lambda_n$-eigenspace, and any solution is nonunique up to addition of an eigenfunction in that eigenspace.

For the general inversion viewpoint, continue to Fundamental Solutions and Green Operators, which develops the construction beyond the regular Sturm–Liouville setting.

QFT-facing example: a scalar field in a cavity

With the site’s $(+---)$ metric convention, consider a free real scalar on $0<x<L$ with $m\geq0$:

$$\left( \partial_t^2-\partial_x^2+m^2 \right)\phi(t,x)=0, \qquad \phi(t,0)=\phi(t,L)=0.$$

The spatial problem is regular Sturm–Liouville with

$$p=w=1, \qquad q=0, \qquad L_x=-\frac{\mathrm d^2}{\mathrm dx^2},$$

and Dirichlet boundary conditions. Its normalized modes and eigenvalues are

$$X_n(x) = \sqrt{\frac{2}{L}} \sin\!\left( \frac{n\pi x}{L} \right), \qquad \lambda_n = \left( \frac{n\pi}{L} \right)^2, \qquad n=1,2,\ldots .$$

They obey

$$\int_0^L X_m(x)X_n(x)\,\mathrm dx = \delta_{mn},$$

and

$$\sum_{n=1}^{\infty} X_n(x)X_n(x') = \delta(x-x')$$

as a distribution on the open interval with the Dirichlet expansion understood. Schwartz’s one-dimensional scalar box in Schwartz 2014, §15.1, especially Eq. (15.5) supplies the corresponding discrete mode sum in a QFT setting.

Expand

$$\phi(t,x) = \sum_{n=1}^{\infty} q_n(t)X_n(x).$$

Projection onto $X_n$ gives an independent oscillator equation for each spatial mode:

$$\ddot q_n(t)+\omega_n^2q_n(t)=0, \qquad \omega_n = \sqrt{m^2+\lambda_n} >0.$$

A commonly used complex mode with oscillator Wronskian normalization is

$$u_n(t,x) = \frac{ X_n(x)e^{-i\omega_n t} }{ \sqrt{2\omega_n} }.$$

The Sturm–Liouville theorem licenses the spatial decomposition; Linear ODEs, Evolution Operators, and Wronskians controls each time-dependent coefficient. Spatial $L^2$ normalization alone is not Klein–Gordon symplectic normalization. Positive frequency, canonical quantization, and the operator interpretation are developed in The Klein–Gordon Field and Its Modes.

For a periodic compact direction of circumference $L$, the coupled boundary conditions instead give

$$X_n(x) = \frac{1}{\sqrt L} e^{2\pi i n x/L}, \qquad \lambda_n = \left( \frac{2\pi n}{L} \right)^2, \qquad n\in\mathbb Z.$$

The $n$ and $-n$ modes are degenerate for $n\neq0$, showing why the separated-problem simplicity statement cannot be exported to coupled conditions. The $n=0$ mode is a zero mode of the spatial operator; for a massless scalar its time equation has $\omega_0=0$ and is not an ordinary positive-frequency oscillator. The Dirichlet, Neumann, and periodic spectra, including periodic multiplicity, are tabulated in NIST DLMF 2026, §1.18(v), Example 1, Eqs. (1.18.36)–(1.18.40).

Radial equations often have a singular endpoint and therefore require a separate endpoint analysis. These variants should not be justified by the cavity theorem without checking their hypotheses.

Where the regular theorem stops

Singular endpoints. An infinite endpoint, or failure of the endpoint integrability conditions for $1/p$, $q$, or $w$, leads to limit-point/limit-circle questions. Extra boundary conditions may be needed, and continuous spectrum or generalized eigenfunctions may appear. Teschl, Teschl 2014, §9.2, especially Theorem 9.6, PDF, gives the limit-point/limit-circle qualification used here.

Indefinite or vanishing weight. The positive Hilbert inner product used above depends on $w>0$. If $w$ changes sign or vanishes in a way that changes the function-space problem, the ordinary self-adjoint spectral theorem does not apply in this form.

Non-self-adjoint boundary data. Complex absorbing or outgoing boundary conditions can be useful, but their eigenvalues need not be real and their right eigenfunctions need not be orthogonal in $\langle\cdot|\cdot\rangle_w$.

Continuous spectrum. A complete discrete sum on a compact regular interval can become a sum plus an integral, or a purely continuous spectral representation, on a noncompact domain.

The theorem-first treatment of unbounded domains and self-adjoint extensions belongs to Self-Adjointness, Extensions, and Unitary Evolution. General spectral measures and continuous-spectrum expansions belong to Spectra, Resolvents, Spectral Measures, and Functional Calculus.

Common pitfalls

Calling a differential expression self-adjoint. The formula for $\ell$ can be formally symmetric while different domains give different operators. State the Hilbert space and boundary domain before invoking the spectral theorem.

Forgetting the weight. Orthogonality uses $w(x)\,\mathrm dx$, and the identity kernel relative to that measure is $\delta(x-x')/w(x')$.

Treating $L^2$ convergence as pointwise convergence. Completeness gives a norm-convergent expansion for every $f\in L^2_w$. Endpoint values and uniform convergence need stronger hypotheses.

Assuming every eigenvalue is simple. Simplicity holds for the real separated scalar problem. Coupled self-adjoint conditions can produce degeneracies.

Inverting at an eigenvalue. When $z\in\sigma(L)$, there is no resolvent. Test the source against the eigenspace and account for the resulting nonuniqueness.

Matching the derivative instead of the flux. At a piecewise-regular interface, the equation controls $py'$. If $p$ jumps, continuity of $y'$ is generally the wrong condition.

Exercises

1. Show directly that two functions satisfying the same real Robin condition at one endpoint make that endpoint’s contribution to the boundary form vanish.

   Check

   Suppose $\alpha y+\beta py'=0$ for both functions, with real $(\alpha,\beta)\neq(0,0)$. If $\beta\neq0$, then $py'=-(\alpha/\beta)y$, so

   $$p\left( \overline{f'}g-\overline f\,g' \right) = -\frac{\alpha}{\beta}\overline f\,g + \frac{\alpha}{\beta}\overline f\,g = 0.$$

   If $\beta=0$, both endpoint values vanish and the same conclusion follows.

2. Derive the reality and orthogonality relation for two eigenfunctions in one self-adjoint domain.

   Check

   Lagrange’s identity and the boundary conditions give

   $$0 = \langle y_m|Ly_n\rangle_w - \langle Ly_m|y_n\rangle_w = \left( \lambda_n-\overline{\lambda_m} \right) \langle y_m|y_n\rangle_w.$$

   Taking $m=n$ makes $\lambda_n$ real. For two distinct real eigenvalues, the inner product must vanish.

3. Verify the cavity spectrum and derive the oscillator frequency.

   Check

   Solving $-X''=\lambda X$ with $X(0)=X(L)=0$ gives $X_n\propto\sin(n\pi x/L)$ and $\lambda_n=(n\pi/L)^2$. The elementary sine integral fixes the normalization $\sqrt{2/L}$. Substituting $\phi=q_n(t)X_n(x)$ into the Klein–Gordon equation yields

   $$\ddot q_n+ \left[ m^2+ \left( \frac{n\pi}{L} \right)^2 \right]q_n = 0.$$

4. Integrate the Green-kernel equation through $x=x'$ and recover the flux jump. What changes when $z$ is an eigenvalue?

   Check

   Integrating

   $$\left[ -\partial_x\!\left(p\partial_x\right)+q-zw \right]G_z = \delta(x-x')$$

   over a shrinking interval leaves only the total derivative:

   $$-\left[ p\partial_xG_z \right]_{x'^-}^{x'^+} = 1.$$

   Hence the jump is $-1$. If $z$ is an eigenvalue, the homogeneous boundary-value problem has a nonzero solution, the Wronskian denominator vanishes, and the inverse exists only after an orthogonality restriction and a choice that removes the eigenspace ambiguity.

References

• NIST Digital Library of Mathematical Functions (accessed August 11, 2026), §1.18, Linear Second Order Differential Operators and Eigenfunction Expansions, especially §§1.18(iv)–(v), for formal versus operator self-adjointness, orthonormality, completeness, Parseval’s identity, and the periodic multiplicity example.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, §2.2 and §15.1, Cambridge University Press, 2014. Book record. This supplies the QFT context for decomposing a scalar field into oscillator modes and for a one-dimensional box mode sum.
• Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Chapter 5, especially §§5.3–5.6 and Theorem 5.11, American Mathematical Society, 2012. Open author edition, PDF; AMS book record. This is the teaching and structural source for regular Sturm–Liouville operators, Lagrange’s identity, compact resolvents, orthonormal eigenfunction bases, oscillation, and periodic boundary conditions.
• Gerald Teschl, Mathematical Methods in Quantum Mechanics, second edition, §§9.1–9.2 and §10.4, American Mathematical Society, 2014. Open author edition, PDF; AMS book record. This supports the maximal regular domain, the boundary-form treatment, limit-point/limit-circle qualifications, and the radial transformation.
• Anton Zettl, Sturm–Liouville Theory, Chapters 3 and 4, Mathematical Surveys and Monographs 121, American Mathematical Society, 2005. AMS book record. This is the structural reference for regular two-point problems and self-adjoint boundary domains; its later chapters distinguish regular from singular endpoint theory.