Generalized Eigenfunction Expansions

A generalized eigenfunction expansion expresses a spectral transform through continuous functionals on a test space. Its validity rests on a self-adjoint operator, an appropriate rigging, a spectral measure, and any multiplicity labels. A formal list of solutions to a differential equation does not establish completeness or identify the correct integration measure.

Required background. The Unbounded Spectral Theorem supplies the Hilbert-space spectral representation. Gelfand Triples defines generalized functionals. Test Function Spaces explains nuclear riggings and operator continuity.

A nuclear spectral theorem supplies continuous coefficients

One useful scalar version has these hypotheses: $\mathcal H$ is separable; $A$ is self-adjoint with simple spectral multiplicity; $\Phi$ is a nuclear Fréchet space continuously and densely included in $\mathcal H$; $\Phi$ is a core for $A$; and $A:\Phi\to\Phi$ is continuous. The core condition identifies the closure of the test-space operator with the chosen self-adjoint realization.

The nuclear spectral theorem gives a scalar spectral representation with measure $\mu$ and continuous linear coefficient functionals $\ell_\lambda$ on $\Phi$, for $\mu$-almost every $\lambda$, such that

$$\ell_\lambda(A\phi)=\lambda\ell_\lambda(\phi),$$

and

$$\langle\phi,\psi\rangle =\int\overline{\ell_\lambda(\phi)} \ell_\lambda(\psi)\,d\mu(\lambda).$$

The corresponding ket is $F_\lambda(\phi)=\overline{\ell_\lambda(\phi)}\in\Phi^\times$. It satisfies the antidual eigen-equation. The measure and coefficient normalization depend on the chosen spectral representation; there is no universal Lebesgue measure for every self-adjoint operator.

The theorem is imported from nuclear spectral theory. Its proof combines the ordinary spectral transform with nuclearity to obtain continuous evaluation functionals almost everywhere; intertwining identifies their eigenvalue equation, and the transform’s isometry gives the norm identity. The crucial step is continuity of the spectral coefficients on $\Phi$. Ordinary $L^2$ functions alone have no pointwise evaluation maps that would prove it.

This statement asserts almost-everywhere completeness, not that every formal generalized eigenfunctional belongs to the physical spectral representation or that a chosen formula is valid at every threshold.

Weak reconstruction and Hilbert reconstruction are distinct

The symbolic formula

$$|\psi\rangle=\int|\lambda\rangle \ell_\lambda(\psi)\,d\mu(\lambda)$$

means the preceding scalar identity when paired with a test vector. Its left-hand side is embedded in the antidual. The individual integrand need not be a Hilbert-space vector.

For general $\psi\in\mathcal H$, the spectral transform extends by isometry, and its coefficients are equivalence classes in $L^2(\mu)$. Spectral cutoffs reconstruct $\psi$ in Hilbert norm. This does not automatically give pointwise convergence of a coordinate-space integral or continuity of an individual generalized bra on all of $\mathcal H$.

Atoms of $\mu$ describe point spectrum. Absolutely continuous and singular continuous parts require their actual measures. A notation containing only a discrete sum and a Lebesgue integral needs hypotheses excluding an omitted singular continuous sector.

Multiplicity requires more than one coefficient at an energy

For nonsimple spectrum, the scalar representation is replaced by a direct integral with multiplicity spaces. Schematically,

$$\|\psi\|^2 =\int\sum_\alpha|\psi_\alpha(\lambda)|^2\,d\mu(\lambda),$$

where the available labels can depend on $\lambda$. Suppressing them can lose part of the state even if every displayed eigen-equation is correct. Complete Sets of Commuting Observables discusses the algebraic meaning of spectral multiplicity.

For a free particle on the line, $H=P^2/(2m)$ has two momentum branches at every energy $E>0$. Set $p(E)=\sqrt{2mE}$ and write

$$a_\pm(E)=\sqrt{\frac{m}{p(E)}}\, \widetilde\psi(\pm p(E)).$$

Changing variables separately on the two momentum half-lines yields

$$\int_{\mathbb R}|\widetilde\psi(p)|^2dp =\int_0^\infty \bigl(|a_+(E)|^2+|a_-(E)|^2\bigr)dE.$$

The square-root Jacobian makes the energy transform unitary. Both branches are required. The factor is singular at $E=0$, but the transform is an almost-everywhere statement; it does not assert a normalized threshold ket or justify a pointwise threshold limit.

Differential eigenfunctions need a completeness argument

Solving $(H-E)u=0$ produces candidate functions. To turn them into a spectral transform, one still needs the self-adjoint boundary conditions, normalization measure, multiplicity, and an isometry or inversion theorem. For scattering, existence of asymptotic wave operators and completeness add further claims beyond solving the stationary equation.

The free Fourier example verifies these ingredients explicitly. More general Sturm–Liouville, distorted Fourier, or scattering constructions use their corresponding spectral theorems. A rigging organizes the generalized vectors but does not replace those proofs.

Exercise: equal energy laws with different momenta

Let $g(E)$ be normalized and supported in a compact interval strictly above zero. Compare states with energy amplitudes $(a_+,a_-)=(g,0)$ and $(0,g)$. Do they have the same energy law? Can an energy-only coefficient recover both states?

Solution

Both have density $|g(E)|^2$ for energy. Their momentum supports lie on opposite half-lines, and their means are $\pm\int p(E)|g(E)|^2dE$, which are finite and nonzero. A scalar energy probability law cannot distinguish them. The multiplicity label retains the information that the energy eigenvalue alone omits; relative phases between the branches carry further state information.

References

• Gel’fand, I. M., and N. Ya. Vilenkin, Generalized Functions, Vol. 4: Applications of Harmonic Analysis, translated by Amiel Feinstein, Academic Press, 1964, Chapter I, publisher edition — the nuclear spectral theorem and generalized eigenfunction expansions.
• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — Fourier spectral transforms and continuous-spectrum models.
• Teschl, Gerald, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, author’s edition — spectral representations, spectral types, and Sturm–Liouville theory.