A nonzero function φ in the domain of a linear operator L (on a function space) is an eigenfunction if there exists a scalar λ such that L φ = λ φ; λ is the associated eigenvalue.

Principle

Eigenfunctions are invariant modes under L up to scalar scaling; they reveal the operator's decomposable directions in the function space.

Demonstration

For the Laplace operator on [0,π] with Dirichlet boundary conditions, φ_n(x)=sin(n x) satisfies Δφ_n = -n^2 φ_n, so each φ_n is an eigenfunction with eigenvalue -n^2.

Misapplication

Calling any familiar basis element an eigenfunction without checking it lies in the operator's domain or satisfies boundary conditions; or applying the concept unchanged to nonlinear operators.

Consequence

Eigenfunctions enable modal expansions, solve linear PDEs by separation of variables, and identify resonant frequencies or stable modes of a system.

Reversal

Generalized eigenfunctions or continuous-spectrum solutions replace discrete eigenfunctions in infinite-dimensional or non-compact problems, requiring distributional notions.

Boundary

Applies to linear operators with specified domain and boundary conditions; some operators have no square-integrable eigenfunctions (pure continuous spectrum).

Semantic Tension

Eigenfunction versus arbitrary basis function: eigenfunctions diagonalize the operator action while basis functions may not; eigenfunction differs from an eigenvector only by being a function-valued element.

Synthesis

An eigenfunction is a nontrivial function that the operator scales by a scalar, providing invariant modal components for linear analysis on function spaces.