The Central Idea
Let \(L\) be a linear operator. An eigenfunction \(\phi\) of \(L\) satisfies
\[ \boxed{ L\phi=\lambda\phi }, \]where \(\lambda\) is the corresponding eigenvalue.
If a collection of eigenfunctions
\[ \phi_1,\phi_2,\phi_3,\ldots \]is sufficiently complete, then a function \(f\) can often be represented as
\[ \boxed{ f(x)=\sum_{n=1}^{\infty}c_n\phi_n(x) }. \]The coefficients \(c_n\) measure how much of each eigenfunction is present in \(f\). This is the same organizing idea that appeared in orthogonal expansions and Fourier series, now attached to a general operator.
From Eigenvectors to Eigenfunctions
In linear algebra, an eigenvector \(v\) of a matrix \(A\) satisfies
\[ Av=\lambda v. \]The matrix changes the magnitude of the eigenvector but not its direction.
An eigenfunction behaves similarly. A linear operator acts on a function and returns the same function multiplied by a scalar:
\[ L\phi=\lambda\phi. \]The function space may be infinite-dimensional, but the structural idea is the same. Eigenfunctions are directions in function space that the operator treats especially simply.
A Familiar Eigenfunction
Consider the differentiation operator
\[ L=\frac{d}{dx}. \]For
\[ \phi(x)=e^{ax}, \]differentiation gives
\[ L\phi =\frac{d}{dx}e^{ax} =ae^{ax}. \]Therefore,
\[ \boxed{ \frac{d}{dx}e^{ax}=a e^{ax} }, \]so \(e^{ax}\) is an eigenfunction of differentiation with eigenvalue \(a\).
Complex Exponentials and Frequency
For the derivative operator,
\[ \frac{d}{dx}e^{i\omega x} =i\omega e^{i\omega x}. \]Thus each complex exponential is an eigenfunction of differentiation.
For the second derivative,
\[ \frac{d^2}{dx^2}e^{i\omega x} =-\omega^2e^{i\omega x}. \]This helps explain why Fourier methods are so effective for constant-coefficient differential equations: differentiation becomes multiplication by a scalar in the eigenfunction coordinates.
Boundary Conditions Change the Problem
An eigenvalue problem is usually not determined by the differential operator alone. Boundary conditions are part of the mathematical system.
Consider
\[ -\phi''(x)=\lambda\phi(x), \qquad 0<x<L, \]with boundary conditions
\[ \phi(0)=0, \qquad \phi(L)=0. \]Only certain values of \(\lambda\) admit nonzero solutions satisfying both conditions. Those permitted values form the spectrum of this boundary-value problem.
Solving the Basic Dirichlet Eigenvalue Problem
Begin with
\[ -\phi''=\lambda\phi, \qquad \phi(0)=\phi(L)=0. \]For positive \(\lambda\), write
\[ \lambda=\mu^2. \]Then
\[ \phi''+\mu^2\phi=0, \]with general solution
\[ \phi(x)=A\cos(\mu x)+B\sin(\mu x). \]The condition \(\phi(0)=0\) gives \(A=0\), so
\[ \phi(x)=B\sin(\mu x). \]The condition \(\phi(L)=0\) requires
\[ \sin(\mu L)=0. \]Hence
\[ \mu L=n\pi, \qquad n=1,2,3,\ldots \]and therefore
\[ \boxed{ \lambda_n=\left(\frac{n\pi}{L}\right)^2 }. \]The Corresponding Eigenfunctions
For each eigenvalue
\[ \lambda_n=\left(\frac{n\pi}{L}\right)^2, \]an associated eigenfunction is
\[ \boxed{ \phi_n(x)=\sin\left(\frac{n\pi x}{L}\right) }. \]Multiplying an eigenfunction by any nonzero constant gives another eigenfunction for the same eigenvalue. We therefore often choose a normalization that makes calculations convenient.
Why the Spectrum Is Discrete
Without boundary conditions, the differential equation
\[ -\phi''=\lambda\phi \]permits many values of \(\lambda\). The boundary conditions restrict the permitted modes.
For the fixed-end conditions \(\phi(0)=\phi(L)=0\), only wavelengths that fit exactly into the interval survive. This produces the discrete sequence
\[ \lambda_1,\lambda_2,\lambda_3,\ldots \]rather than a continuous range of eigenvalues.
Physical Interpretation: A Vibrating String
A string fixed at both ends has normal modes
\[ \sin\left(\frac{n\pi x}{L}\right). \]These are exactly the eigenfunctions of the spatial operator associated with the wave equation under fixed-end boundary conditions.
The first mode has one half-wave across the string, the second has two, and so on. A complicated initial displacement can be decomposed into these modes, each of which then evolves independently in time.
Orthogonality of Eigenfunctions
For the fixed-end problem, the sine eigenfunctions satisfy
\[ \int_0^L \sin\left(\frac{m\pi x}{L}\right) \sin\left(\frac{n\pi x}{L}\right) \,dx =0, \qquad m\ne n. \]Thus eigenfunctions associated with distinct eigenvalues are orthogonal under the usual inner product
\[ \langle f,g\rangle =\int_0^L f(x)\overline{g(x)}\,dx. \]Orthogonality is what allows coefficients to be isolated one at a time.
Normalization
For
\[ \phi_n(x)=\sin\left(\frac{n\pi x}{L}\right), \]the squared norm is
\[ \|\phi_n\|^2 =\int_0^L \sin^2\left(\frac{n\pi x}{L}\right) \,dx =\frac{L}{2}. \]Therefore the normalized eigenfunctions may be written as
\[ \boxed{ e_n(x)=\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) }. \]They satisfy
\[ \langle e_m,e_n\rangle=\delta_{mn}, \]where \(\delta_{mn}\) is the Kronecker delta.
Expansion Coefficients
Suppose
\[ f(x)=\sum_{n=1}^{\infty}c_n\phi_n(x) \]and the eigenfunctions are orthogonal. Take the inner product with \(\phi_m\):
\[ \langle f,\phi_m\rangle =\sum_{n=1}^{\infty} c_n\langle\phi_n,\phi_m\rangle. \]Orthogonality eliminates every term except \(n=m\), giving
\[ \langle f,\phi_m\rangle =c_m\langle\phi_m,\phi_m\rangle. \]Therefore,
\[ \boxed{ c_m =\frac{\langle f,\phi_m\rangle} {\langle\phi_m,\phi_m\rangle} }. \]Normalized Coefficients
If the eigenfunctions are orthonormal, then
\[ \langle e_m,e_n\rangle=\delta_{mn}. \]The expansion
\[ f=\sum_{n=1}^{\infty}a_ne_n \]has the especially simple coefficients
\[ \boxed{ a_n=\langle f,e_n\rangle }. \]This is the function-space analogue of resolving a vector into components along orthonormal coordinate directions.
Example: Sine Expansion on \((0,L)\)
Expand a suitable function \(f\) in the fixed-end eigenfunctions:
\[ f(x) =\sum_{n=1}^{\infty} b_n\sin\left(\frac{n\pi x}{L}\right). \]Since the squared norm is \(L/2\),
\[ b_n =\frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) \,dx. \]Thus
\[ \boxed{ b_n =\frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) \,dx }. \]This familiar Fourier sine coefficient is therefore an eigenfunction projection coefficient.
Fourier Series Are Eigenfunction Expansions
Fourier series are not a separate idea from eigenfunction methods. They are a particularly important example.
Sines and cosines arise as eigenfunctions of second-derivative operators under common boundary conditions, while complex exponentials arise naturally under periodic conditions.
The broader principle is:
\[ \boxed{ \text{Choose basis functions adapted to the operator and boundary conditions.} } \]Self-Adjoint Operators
Orthogonality of eigenfunctions is closely tied to self-adjointness. Informally, an operator \(L\) is self-adjoint with respect to an inner product when
\[ \langle Lf,g\rangle =\langle f,Lg\rangle \]for functions satisfying the relevant domain and boundary conditions.
This is the infinite-dimensional analogue of a real symmetric matrix. Just as symmetric matrices have especially well-behaved eigenvectors, self-adjoint operators have especially well-behaved eigenfunctions and spectra.
Why Distinct Eigenvalues Give Orthogonal Eigenfunctions
Suppose \(L\) is self-adjoint and
\[ L\phi_m=\lambda_m\phi_m, \qquad L\phi_n=\lambda_n\phi_n. \]Then
\[ \langle L\phi_m,\phi_n\rangle =\lambda_m\langle\phi_m,\phi_n\rangle. \]Self-adjointness also gives
\[ \langle L\phi_m,\phi_n\rangle =\langle\phi_m,L\phi_n\rangle =\lambda_n\langle\phi_m,\phi_n\rangle \]when the eigenvalues are real. Therefore,
\[ (\lambda_m-\lambda_n) \langle\phi_m,\phi_n\rangle=0. \]If \(\lambda_m\ne\lambda_n\), then
\[ \boxed{ \langle\phi_m,\phi_n\rangle=0 }. \]Sturm–Liouville Problems
A broad class of eigenfunction problems has the form
\[ -\frac{d}{dx} \left(p(x)\frac{dy}{dx}\right) +q(x)y =\lambda w(x)y, \]together with suitable boundary conditions.
This is a Sturm–Liouville problem. The functions \(p(x)\), \(q(x)\), and \(w(x)\) determine the operator, and \(w(x)\) serves as a weight in the natural inner product.
Weighted Inner Products
For a Sturm–Liouville problem, the natural inner product is often
\[ \boxed{ \langle f,g\rangle_w =\int_a^b f(x)\overline{g(x)}w(x)\,dx }. \]Orthogonality then means
\[ \int_a^b \phi_m(x)\overline{\phi_n(x)}w(x)\,dx =0, \qquad m\ne n. \]The weight is not an optional decoration. It is part of the geometry of the function space generated by the differential problem.
Weighted Expansion Coefficients
If \(\{\phi_n\}\) is orthogonal with respect to weight \(w\), then
\[ f(x)=\sum_{n=1}^{\infty}c_n\phi_n(x) \]has coefficients
\[ \boxed{ c_n =\frac{ \int_a^b f(x)\overline{\phi_n(x)}w(x)\,dx }{ \int_a^b |\phi_n(x)|^2w(x)\,dx } }. \]This is the same projection formula as before, expressed in the appropriate weighted inner product.
Common Boundary Conditions
Boundary conditions help determine both eigenvalues and eigenfunctions. Common types include:
- Dirichlet: the function value is prescribed;
- Neumann: the derivative is prescribed;
- Robin: a linear combination of value and derivative is prescribed;
- periodic: values and often derivatives match across the endpoints.
Different boundary conditions on the same differential expression can produce different spectra and different eigenfunction families.
Neumann Eigenfunctions
Consider
\[ -\phi''=\lambda\phi, \qquad \phi'(0)=\phi'(L)=0. \]The eigenfunctions are cosine modes:
\[ \phi_n(x)=\cos\left(\frac{n\pi x}{L}\right), \qquad n=0,1,2,\ldots \]with eigenvalues
\[ \boxed{ \lambda_n=\left(\frac{n\pi}{L}\right)^2 }. \]The \(n=0\) mode is the constant function. Its presence reflects the fact that a constant has zero derivative at both endpoints.
Periodic Eigenfunctions
On a periodic interval, complex exponentials are natural eigenfunctions:
\[ \phi_n(x)=e^{i2\pi nx/L}, \qquad n\in\mathbb Z. \]For the second-derivative operator,
\[ -\phi_n'' =\left(\frac{2\pi n}{L}\right)^2\phi_n. \]Fourier series therefore arise directly as eigenfunction expansions of a periodic differential operator.
Legendre Polynomials as Eigenfunctions
Not every eigenfunction family is trigonometric. Legendre's differential equation can be written as
\[ -\frac{d}{dx} \left[(1-x^2)\frac{dy}{dx}\right] =\lambda y, \qquad -1<x<1. \]Under the appropriate regularity conditions, the eigenvalues are
\[ \lambda_n=n(n+1), \]and the eigenfunctions are the Legendre polynomials
\[ P_0(x),P_1(x),P_2(x),\ldots \]These functions are orthogonal on \([-1,1]\) with weight \(1\).
Other Important Eigenfunction Families
Different geometries and operators generate different natural bases. Examples include:
- Legendre polynomials for problems with spherical structure;
- Bessel functions for radial problems in cylindrical geometry;
- spherical harmonics for angular structure on spheres;
- Hermite functions in oscillator and Gaussian-weighted problems;
- Chebyshev polynomials in approximation and spectral computation.
The useful basis is dictated by the operator, interval or geometry, weight, and boundary conditions.
Completeness
Orthogonality alone is not enough to guarantee that every target function can be reconstructed. The eigenfunctions must also be complete in the function space under consideration.
Informally, completeness means that no nonzero function in the space is orthogonal to every basis function. Equivalently, finite linear combinations of the eigenfunctions can approximate every function in the space as closely as the chosen norm allows.
When completeness holds, the eigenfunctions play the role of a genuine basis.
Pointwise Equality versus Norm Convergence
The statement
\[ f=\sum_{n=1}^{\infty}c_n\phi_n \]must be interpreted carefully. Depending on the problem, convergence may mean:
- pointwise convergence at individual values of \(x\);
- uniform convergence across an interval;
- mean-square or \(L^2\) convergence;
- convergence in a stronger derivative-based norm.
Spectral expansions are often naturally justified first in an \(L^2\) sense. Additional smoothness may yield stronger forms of convergence.
Partial Sums
In computation we cannot usually use infinitely many terms. Instead we form the partial sum
\[ S_N(x)=\sum_{n=1}^{N}c_n\phi_n(x). \]If the expansion converges, then increasing \(N\) improves the approximation in the relevant sense.
The coefficients often decay faster when \(f\) is smoother. This is one reason spectral methods can achieve high accuracy with relatively few modes for smooth problems.
Best Approximation in an Orthogonal Subspace
Let
\[ V_N=\operatorname{span}\{\phi_1,\ldots,\phi_N\}. \]The orthogonal projection
\[ S_N=\sum_{n=1}^{N}c_n\phi_n \]is the best approximation to \(f\) from \(V_N\) in the norm induced by the inner product:
\[ \boxed{ \|f-S_N\| \le \|f-v\| \quad \text{for every }v\in V_N }. \]The coefficient formula is therefore not merely convenient. It is the least-squares-optimal choice within the selected modal subspace.
Parseval-Type Identities
For a complete orthonormal eigenfunction basis,
\[ f=\sum_{n=1}^{\infty}a_ne_n, \]one has the energy identity
\[ \boxed{ \|f\|^2 =\sum_{n=1}^{\infty}|a_n|^2 }. \]This is the generalized form of Parseval's identity. It says that the total squared size of the function equals the total squared size of its modal coefficients.
Why Eigenfunction Coordinates Simplify Operators
Suppose
\[ f=\sum_n c_n\phi_n \]and
\[ L\phi_n=\lambda_n\phi_n. \]By linearity,
\[ Lf =\sum_n c_nL\phi_n =\sum_n \lambda_n c_n\phi_n. \]Thus
\[ \boxed{ c_n \quad\longmapsto\quad \lambda_n c_n }. \]The operator becomes diagonal in the eigenfunction basis. This is the central computational advantage of spectral coordinates.
Solving an Operator Equation Mode by Mode
Suppose we want to solve
\[ Lu=f. \]Expand
\[ u=\sum_n a_n\phi_n, \qquad f=\sum_n b_n\phi_n, \]where \(L\phi_n=\lambda_n\phi_n\). Then
\[ Lu =\sum_n \lambda_n a_n\phi_n. \]Matching coefficients gives
\[ \lambda_n a_n=b_n. \]Whenever \(\lambda_n\ne0\),
\[ \boxed{ a_n=\frac{b_n}{\lambda_n} }. \]The differential equation has been converted into scalar algebra for each mode.
Zero Eigenvalues Require Special Attention
If \(\lambda_n=0\), the equation
\[ \lambda_n a_n=b_n \]cannot be solved by division. A solution may require the corresponding forcing coefficient \(b_n\) to vanish.
This is an example of a compatibility condition. The presence of a null space changes both existence and uniqueness and must be checked before an inverse operator is claimed.
Separation of Variables
Eigenfunction expansions arise naturally when a partial differential equation is separated into spatial and temporal parts.
Suppose
\[ u(x,t)=X(x)T(t). \]Substitution into a PDE often produces a spatial eigenvalue problem
\[ LX=\lambda X \]together with a simpler time equation depending on the same \(\lambda\).
The full solution is then assembled by summing separated eigenmodes.
Heat Equation Example
Consider the heat equation
\[ u_t=\kappa u_{xx}, \qquad 0<x<L, \]with fixed zero-temperature boundary conditions
\[ u(0,t)=u(L,t)=0. \]Using the spatial eigenfunctions
\[ \phi_n(x)=\sin\left(\frac{n\pi x}{L}\right), \]write
\[ u(x,t)=\sum_{n=1}^{\infty}a_n(t) \sin\left(\frac{n\pi x}{L}\right). \]Each Heat Mode Evolves Independently
Since
\[ \phi_n'' =-\left(\frac{n\pi}{L}\right)^2\phi_n, \]substitution into the heat equation gives, mode by mode,
\[ a_n'(t) =-\kappa \left(\frac{n\pi}{L}\right)^2a_n(t). \]Therefore,
\[ a_n(t) =a_n(0) e^{-\kappa(n\pi/L)^2t}. \]The solution is
\[ \boxed{ u(x,t) =\sum_{n=1}^{\infty} a_n(0) e^{-\kappa(n\pi/L)^2t} \sin\left(\frac{n\pi x}{L}\right) }. \]Why High Heat Modes Decay Faster
The \(n\)-th heat mode contains the factor
\[ e^{-\kappa(n\pi/L)^2t}. \]Since the exponent grows like \(n^2\), high-frequency spatial modes decay much more rapidly than low-frequency modes.
This gives a spectral explanation for diffusion: fine-scale spatial variation is smoothed out faster than broad-scale variation.
Wave Equation Example
For the wave equation
\[ u_{tt}=c^2u_{xx}, \]with fixed ends, the same spatial eigenfunctions appear. If
\[ u(x,t) =\sum_{n=1}^{\infty}a_n(t) \sin\left(\frac{n\pi x}{L}\right), \]then each coefficient satisfies
\[ a_n''(t) +c^2\left(\frac{n\pi}{L}\right)^2a_n(t)=0. \]Each spatial eigenmode therefore behaves like an independent harmonic oscillator.
Modal Frequencies of a Fixed String
The angular frequency of the \(n\)-th string mode is
\[ \boxed{ \omega_n=\frac{cn\pi}{L} }. \]Thus the spatial eigenvalue
\[ \lambda_n=\left(\frac{n\pi}{L}\right)^2 \]determines the temporal oscillation rate through
\[ \omega_n=c\sqrt{\lambda_n}. \]This is a direct connection between operator spectra and physical resonance.
Initial Conditions Determine the Modal Coefficients
In the heat equation, the initial condition
\[ u(x,0)=f(x) \]requires
\[ f(x) =\sum_{n=1}^{\infty}a_n(0) \sin\left(\frac{n\pi x}{L}\right). \]Therefore,
\[ \boxed{ a_n(0) =\frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) \,dx }. \]The initial state is projected onto the spatial eigenmodes, after which each mode evolves according to its own scalar time equation.
Example: One Pure Eigenmode in the Heat Equation
Suppose
\[ u(x,0)=\sin\left(\frac{3\pi x}{L}\right). \]The initial condition already equals a single eigenfunction. Therefore no projection calculation is needed. Only the third coefficient is nonzero.
The solution is
\[ \boxed{ u(x,t) =e^{-9\kappa\pi^2t/L^2} \sin\left(\frac{3\pi x}{L}\right) }. \]An eigenmode keeps its spatial shape while its amplitude changes with time.
Example: Expanding \(f(x)=x\) on \((0,L)\)
Write
\[ x =\sum_{n=1}^{\infty} b_n\sin\left(\frac{n\pi x}{L}\right), \qquad 0<x<L. \]The coefficients are
\[ b_n =\frac{2}{L} \int_0^L x\sin\left(\frac{n\pi x}{L}\right) \,dx. \]Integration by parts gives
\[ \int_0^L x\sin\left(\frac{n\pi x}{L}\right) \,dx =-\frac{L^2}{n\pi}(-1)^n. \]Therefore,
\[ \boxed{ b_n =\frac{2L}{n\pi}(-1)^{n+1} }. \]The Resulting Sine Expansion
The function \(f(x)=x\) on \((0,L)\) has the sine expansion
\[ \boxed{ x =\frac{2L}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin\left(\frac{n\pi x}{L}\right) }. \]This is simultaneously a Fourier sine series and an eigenfunction expansion for the fixed-end second-derivative operator.
Operator-Adapted Coordinates
Ordinary coordinates are chosen for convenience. Eigenfunction coordinates are chosen because they simplify the operator itself.
In the original function description, applying \(L\) may require derivatives, integrals, or boundary calculations. In eigenfunction coordinates, the same action may reduce to
\[ c_n\mapsto\lambda_n c_n. \]This is why eigenfunction methods are central to differential equations, quantum mechanics, vibration analysis, diffusion, acoustics, electromagnetism, and numerical spectral methods.
Discrete and Continuous Spectra
Some operators have discrete eigenvalues
\[ \lambda_1,\lambda_2,\lambda_3,\ldots \]and lead naturally to sums of eigenfunctions.
Other problems involve a continuous spectral parameter and lead to integrals rather than sums. The Fourier transform is the most familiar example of this continuous-spectrum viewpoint.
Some operators possess both discrete and continuous spectral components.
Expansion Sums and Transform Integrals
A discrete eigenfunction expansion has the form
\[ f(x)=\sum_n c_n\phi_n(x). \]A continuous spectral representation has the schematic form
\[ f(x)=\int c(\lambda)\phi_{\lambda}(x)\,d\lambda. \]The conceptual role is the same: decompose the function into modes adapted to an operator. The mathematical machinery changes according to the type of spectrum.
Eigenfunction Expansions and Green's Functions
Eigenfunctions can also be used to construct inverse operators and Green's functions. If
\[ L\phi_n=\lambda_n\phi_n \]with a complete orthonormal basis and nonzero eigenvalues, the inverse acts formally as
\[ L^{-1}f =\sum_n \frac{\langle f,\phi_n\rangle}{\lambda_n} \phi_n. \]The factor \(1/\lambda_n\) tells us how strongly the inverse operator responds to each mode.
Spectral Filtering
Because an operator acts mode by mode, we can intentionally modify modal coefficients. For example, a filtered approximation may be written
\[ f_N^{\mathrm{filtered}} =\sum_{n=1}^{N} \sigma_n c_n\phi_n, \]where \(\sigma_n\) attenuates selected modes.
This generalizes ordinary frequency filtering. The spectral variable need not be physical frequency; it may be any eigenvalue index associated with the operator.
Numerical Spectral Methods
In a spectral numerical method, an unknown solution is approximated by a finite expansion such as
\[ u_N(x)=\sum_{n=0}^{N}a_n\phi_n(x). \]The basis may consist of Fourier modes, Chebyshev polynomials, Legendre polynomials, or other functions suited to the geometry and boundary conditions.
The differential equation is then converted into equations for the coefficients \(a_n\).
Why Spectral Methods Can Converge Rapidly
For sufficiently smooth functions, expansion coefficients in a well-chosen basis often decrease rapidly as the mode number increases.
Consequently, relatively few modes may capture a smooth solution with high accuracy. This behavior is sometimes called spectral convergence when the error decreases faster than any fixed algebraic power of \(1/N\) under suitable assumptions.
Discontinuities and nonsmooth features reduce this advantage because their high-mode coefficients decay more slowly.
Choosing the Right Eigenfunction Basis
A useful basis should reflect the problem rather than merely being familiar. Ask:
- What operator governs the problem?
- What boundary conditions must be satisfied?
- What interval or geometry is involved?
- Is there a natural weight function?
- Are the basis functions orthogonal or orthonormal?
- Is completeness known in the function space of interest?
- Will the operator become simple or diagonal in this basis?
The best basis is usually the one that turns the governing structure into the simplest coefficient relations.
Example: Projection onto Two Modes
Suppose \(\phi_1\) and \(\phi_2\) are orthonormal. The best two-mode approximation to \(f\) is
\[ S_2 =\langle f,\phi_1\rangle\phi_1 +\langle f,\phi_2\rangle\phi_2. \]The residual
\[ r=f-S_2 \]satisfies
\[ \langle r,\phi_1\rangle =\langle r,\phi_2\rangle=0. \]No other linear combination of these two modes can produce a smaller inner-product norm error.
Example: Applying an Operator to an Expansion
Suppose
\[ f=4\phi_1-2\phi_2+3\phi_3 \]and
\[ L\phi_1=2\phi_1, \qquad L\phi_2=5\phi_2, \qquad L\phi_3=-\phi_3. \]Then
\[ Lf =4(2\phi_1)-2(5\phi_2)+3(-\phi_3). \]Therefore,
\[ \boxed{ Lf=8\phi_1-10\phi_2-3\phi_3 }. \]No differentiation or integration is required once the modal representation is known.
Example: Solving a Diagonal Operator Equation
Suppose
\[ Lu=f, \]where
\[ f=6\phi_1+10\phi_2 \]and
\[ L\phi_1=3\phi_1, \qquad L\phi_2=5\phi_2. \]Write
\[ u=a_1\phi_1+a_2\phi_2. \]Matching coefficients gives
\[ 3a_1=6, \qquad 5a_2=10. \]Hence
\[ \boxed{ u=2\phi_1+2\phi_2 }. \]Common Error: Treating Every Function as an Eigenfunction
A function \(f\) is an eigenfunction only if applying the operator returns a scalar multiple of the same function:
\[ Lf=\lambda f. \]For example, under differentiation,
\[ \frac{d}{dx}x^2=2x, \]which is not a constant multiple of \(x^2\). Therefore \(x^2\) is not an eigenfunction of the first-derivative operator.
Common Error: Ignoring Boundary Conditions
Solving the differential equation alone does not determine the spectrum.
For
\[ -\phi''=\lambda\phi, \]sine, cosine, or exponential families may appear depending on the domain and boundary conditions. Always carry the boundary conditions through the eigenvalue calculation.
Common Error: Using the Wrong Inner Product
In a weighted Sturm–Liouville problem, the correct orthogonality relation uses the weight \(w(x)\):
\[ \int_a^b \phi_m\overline{\phi_n}w\,dx=0. \]Omitting the weight can produce incorrect coefficients even when the differential equation itself has been solved correctly.
Common Error: Assuming Orthogonality Implies Completeness
A set of functions can be mutually orthogonal without spanning the entire function space.
Orthogonality makes coefficient extraction simple. Completeness is the additional property that makes reconstruction possible for every function in the intended space.
Common Error: Dividing by a Zero Eigenvalue
In an equation such as
\[ a_n=\frac{b_n}{\lambda_n}, \]the formula is valid only when \(\lambda_n\ne0\).
A zero eigenvalue signals a null-space direction. Existence or uniqueness may require a separate compatibility condition or an additional normalization.
Common Error: Confusing Eigenvalues with Expansion Coefficients
The eigenvalue \(\lambda_n\) belongs to the operator-mode pair
\[ L\phi_n=\lambda_n\phi_n. \]The coefficient \(c_n\) belongs to the particular function being expanded:
\[ f=\sum_n c_n\phi_n. \]Changing \(f\) changes the coefficients \(c_n\), but it does not change the eigenvalues of the fixed operator problem.
Common Error: Expecting Pointwise Convergence Everywhere
A valid eigenfunction expansion may converge in mean square without converging uniformly or pointwise at every location.
Always identify the convergence concept that the theorem or application guarantees. Stronger convergence usually requires stronger smoothness and compatibility assumptions.
A Reliable Procedure for Eigenfunction Expansions
- Identify the linear operator.
- Include the boundary conditions and domain.
- Solve the eigenvalue problem \(L\phi=\lambda\phi\).
- Determine the permitted eigenvalues and eigenfunctions.
- Identify the correct inner product and weight.
- Normalize the eigenfunctions when useful.
- Project the target function onto the eigenfunctions.
- Write the expansion or finite partial sum.
- Use the eigenvalue relation to simplify operator actions.
- Check zero modes, convergence assumptions, and boundary compatibility.
A Reliable Procedure for PDE Mode Expansions
- Separate the spatial and temporal variables when appropriate.
- Derive the spatial eigenvalue problem from the PDE and boundary conditions.
- Find the spatial eigenfunctions and eigenvalues.
- Expand the solution in the spatial eigenfunctions.
- Derive the scalar time equation for each modal coefficient.
- Use the initial condition to determine the starting coefficients.
- Solve each scalar time equation.
- Reassemble the full solution as a modal sum.
- Check the PDE, boundary conditions, and initial conditions.
Practice 1: Identify an Eigenfunction
Let
\[ L=\frac{d}{dx}. \]Show that \(e^{4x}\) is an eigenfunction and identify its eigenvalue.
Practice 2: Second-Derivative Eigenvalue
Let
\[ L=-\frac{d^2}{dx^2}. \]Find the eigenvalue corresponding to
\[ \phi(x)=\sin(5x). \]Practice 3: Fixed-End Spectrum
For
\[ -\phi''=\lambda\phi, \qquad 0<x<2, \]with \(\phi(0)=\phi(2)=0\), find \(\lambda_n\) and one corresponding eigenfunction \(\phi_n\).
Practice 4: Projection Coefficient
Suppose \(\phi_n\) are orthogonal but not normalized. Write the general formula for the coefficient \(c_n\) in
\[ f=\sum_n c_n\phi_n. \]Practice 5: Normalized Projection
Suppose \(e_1,e_2,e_3,\ldots\) are orthonormal and
\[ \langle f,e_1\rangle=3, \qquad \langle f,e_2\rangle=-1. \]Write the best two-mode approximation to \(f\).
Practice 6: Operator Action in Modal Coordinates
Suppose
\[ f=2\phi_1-3\phi_2 \]and
\[ L\phi_1=4\phi_1, \qquad L\phi_2=7\phi_2. \]Find \(Lf\).
Practice 7: Solve Mode by Mode
Suppose
\[ Lu=8\phi_1-9\phi_2, \]where
\[ L\phi_1=2\phi_1, \qquad L\phi_2=3\phi_2. \]Find \(u\) in the form \(a_1\phi_1+a_2\phi_2\).
Practice 8: Heat-Mode Decay
For the heat equation on \((0,L)\), compare the decay rates of the \(n=1\) and \(n=4\) modes. By what factor is the magnitude of the exponent coefficient larger for the fourth mode?
Practice 9: Weighted Coefficient Formula
Write the coefficient formula for an orthogonal Sturm–Liouville family \(\{\phi_n\}\) with weight \(w(x)\) on \([a,b]\).
Practice 10: Identify the Structural Advantage
Explain why expanding a function in eigenfunctions of \(L\) can simplify the application of \(L\).
Practice Answers
Practice 1
Differentiate:
\[ \frac{d}{dx}e^{4x}=4e^{4x}. \]Therefore,
\[ \boxed{ \lambda=4 }. \]Practice 2
Since
\[ \phi''(x)=-25\sin(5x), \]we have
\[ -\phi''(x)=25\phi(x). \]Hence,
\[ \boxed{ \lambda=25 }. \]Practice 3
With \(L=2\),
\[ \lambda_n =\left(\frac{n\pi}{2}\right)^2. \]A corresponding eigenfunction is
\[ \boxed{ \phi_n(x) =\sin\left(\frac{n\pi x}{2}\right), \qquad \lambda_n =\left(\frac{n\pi}{2}\right)^2 }. \]Practice 4
For an orthogonal family,
\[ \boxed{ c_n =\frac{\langle f,\phi_n\rangle} {\langle\phi_n,\phi_n\rangle} }. \]Practice 5
For an orthonormal basis, the projection coefficients are the inner products. Thus
\[ \boxed{ S_2=3e_1-e_2 }. \]Practice 6
Apply \(L\) term by term:
\[ Lf =2(4\phi_1)-3(7\phi_2). \]Therefore,
\[ \boxed{ Lf=8\phi_1-21\phi_2 }. \]Practice 7
Write
\[ u=a_1\phi_1+a_2\phi_2. \]Then
\[ Lu=2a_1\phi_1+3a_2\phi_2. \]Matching coefficients gives
\[ 2a_1=8, \qquad 3a_2=-9. \]Hence
\[ \boxed{ u=4\phi_1-3\phi_2 }. \]Practice 8
Heat-mode decay contains
\[ e^{-\kappa(n\pi/L)^2t}. \]The exponent coefficient grows as \(n^2\). Therefore,
\[ \frac{4^2}{1^2}=16. \]The fourth mode has an exponent coefficient
\[ \boxed{16} \]times as large as the first mode.
Practice 9
The weighted projection formula is
\[ \boxed{ c_n =\frac{ \int_a^b f(x)\overline{\phi_n(x)}w(x)\,dx }{ \int_a^b |\phi_n(x)|^2w(x)\,dx } }. \]Practice 10
If \(L\phi_n=\lambda_n\phi_n\) and
\[ f=\sum_n c_n\phi_n, \]then
\[ Lf=\sum_n \lambda_n c_n\phi_n. \]Thus the operator acts by scalar multiplication on each coefficient. A potentially complicated operator calculation becomes a diagonal, mode-by-mode calculation.
What to Remember
The defining relation is
\[ \boxed{ L\phi_n=\lambda_n\phi_n }. \]When the eigenfunctions form a suitable complete orthogonal family,
\[ \boxed{ f=\sum_n c_n\phi_n }. \]- eigenfunctions are the function-space analogues of eigenvectors;
- boundary conditions help determine the spectrum;
- self-adjoint problems naturally produce orthogonal eigenfunctions;
- Sturm–Liouville problems may require weighted inner products;
- projection determines expansion coefficients;
- normalization simplifies those coefficients;
- completeness is required for full reconstruction;
- partial sums give finite-dimensional approximations;
- the operator becomes diagonal in its eigenfunction basis;
- zero eigenvalues require null-space and compatibility checks;
- separation of variables turns many PDEs into eigenfunction expansions;
- heat modes decay independently while wave modes oscillate independently;
- Fourier series are one important special case of the broader method;
- spectral numerical methods approximate solutions using finite modal expansions.
The most important habit is to ask which functions make the governing operator simple. Those functions often provide the natural coordinates of the problem.
Where We Go Next
Eigenfunction expansions show that a linear operator can become simple when we describe functions in coordinates adapted to its eigenmodes. The next step is to view this not just as an expansion technique, but as a structural decomposition of the operator itself.
The next lesson, Spectral Decomposition, develops that viewpoint. We will connect matrix diagonalization with self-adjoint operators, spectral projectors, discrete and continuous spectra, functional calculus, and the idea that an operator can be reconstructed from its spectral components.