A mathematical structure used to represent quantum states and the operations performed on them. It specifies how states are expressed, related, combined, and decomposed into components that support calculation. It does not by itself determine physical predictions without a mapping to observables and measurement rules. It enables precise computation of probabilities and expectation values from state and operator inputs. The concept is generally stable, though notation choices and computational methods evolve over time.

Principle

From the spectral theorem and rigged Hilbert space formalism: self-adjoint operators admit spectral decompositions that include continuous parts represented by generalized eigenfunctions which are normalized in a delta-function sense and used to expand physical (normalizable) states as integrals over the spectrum.

Demonstration

Free-particle momentum eigenfunctions ψk(x)=e^{ikx}/√(2π) are generalized eigenfunctions of the momentum operator with eigenvalue k; scattering solutions of the Schrödinger equation at positive energy are generalized eigenfunctions of the Hamiltonian, used to compute S-matrix elements.

Misapplication

Treating generalized eigenfunctions as L2-normalizable basis vectors, using their pointwise values to assign probabilities without forming wavepackets, or ignoring boundary conditions and self-adjointness that determine whether such generalized solutions are physically admissible.

Consequence

They permit spectral expansions of physical states as integrals (continuous superpositions), the practical calculation of scattering amplitudes and transition probabilities, and an operational bridge between formal eigenvalue equations and measurable continuous-spectrum observables.

Reversal

Discrete, square-integrable eigenfunctions belonging to the Hilbert space (normalizable bound states) that are orthonormal in the usual sense and correspond to pure-point spectrum rather than continuous spectrum.

Boundary

Valid for self-adjoint (or properly extended) operators exhibiting continuous spectrum; generalized eigenfunctions are not in L2 and require distributional interpretation and rigged Hilbert space machinery; they may fail to exist for non-self-adjoint operators or be nonphysical without correct boundary conditions.

Semantic Tension

Tension between calling these objects 'eigenfunctions'—which suggests ordinary function-space membership—and their true status as distributional eigenvectors normalized by delta functions; also tension between the physicist's heuristic use and the mathematician's rigour about domains and extensions of operators.

Synthesis

A generalized eigenfunction is a distributional solution of an operator eigenvalue equation associated with continuous spectrum: not square-integrable but delta-normalized, it enables integral spectral decompositions and provides the basis for describing scattering and continuum observables when used with wavepackets and proper operator domain specifications.