Rigged Hilbert space (Gel'fand triple) and the nuclear spectral theorem

Intuition Beginner

In quantum mechanics you constantly meet objects like the position ket and the momentum ket — states labelled by a sharp position or a sharp momentum. Physicists write them as if they were vectors in the usual state space and manipulate them with great success. But a state of perfectly sharp position is a spike of infinite height and zero width; a state of perfectly sharp momentum is a plane wave that never decays. Neither has finite length, so neither is an honest vector in the Hilbert space of square-integrable wavefunctions. They live just outside it.

The rigged Hilbert space is the device that gives these outsiders a legal home. The idea is to build a three-layer sandwich. In the middle sits the ordinary Hilbert space of finite-length states. Inside it sits a smaller space of especially well-behaved states: smooth functions that decay fast, the test functions. Outside it sits a larger space of generalized functions: the duals of those test functions. The plane wave and the spike fail to be ordinary states, but they fit comfortably in the outer layer.

The payoff is that the physicist's continuous basis becomes rigorous. You really can expand a state over a continuum of position labels or momentum labels, because the labels index generalized eigenvectors living in the outer layer of the sandwich.

Visual Beginner

Picture three nested rings. The small inner ring is the space of nice test states: smooth, fast-decaying, easy to compute with. The middle ring is the full Hilbert space of finite-length states. The big outer ring is the space of generalized functions, where spikes and plane waves are allowed to live.

A continuous basis is a family of arrows pointing into the outer ring, one arrow for each value of a label such as position or momentum. To read the component of a state along one of these arrows, you pair the state with the corresponding generalized eigenvector. The whole state is reassembled by integrating over the label. The inner ring has to be chosen carefully — small enough that its dual is roomy enough to hold the generalized eigenvectors, and that careful choice is exactly what the word nuclear is doing.

Worked example Beginner

Take the simplest case: a particle on the line, with states described by wavefunctions. The middle ring is the space of wavefunctions with finite total probability. The inner ring is the space of smooth functions that decay faster than any power, together with all their derivatives — the Schwartz test functions.

Now look at the momentum label. The would-be momentum eigenstate with momentum value $p$ is the plane wave whose value at position $x$ is the oscillation of frequency $p$. Its total probability is infinite, so it is not a real state. But pairing it with a test function is perfectly fine: you integrate the test function against the oscillation, and because the test function decays fast, the integral converges to an ordinary number. That number is the value of the Fourier transform of the test function at $p$.

So the plane wave is a legitimate arrow into the outer ring. Reading a state's component at momentum $p$ is computing the Fourier transform at $p$. Reassembling the state by integrating over all $p$ is the Fourier inversion formula. The physicist's momentum-basis expansion is the rigorous Fourier transform, dressed in continuous-basis language.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathcal{H}$ be a separable complex Hilbert space. A rigged Hilbert space, or Gel'fand triple (also equipped Hilbert space), is a chain $$ \Phi ;\subset; \mathcal{H} ;\subset; \Phi' $$ where $\Phi$ is a dense linear subspace of $\mathcal{H}$ carrying a locally convex topology that is finer than the one inherited from $\mathcal{H}$ (so the inclusion $\Phi \hookrightarrow \mathcal{H}$ is continuous), and ${\Phi }^{\prime }$ is the space of continuous antilinear functionals on $\Phi$. Identifying $\mathcal{H}$ with its own dual via the Riesz map gives the second inclusion $\mathcal{H}\hookrightarrow {\Phi }^{\prime }$, where a vector $h\in \mathcal{H}$ acts on $\Phi$ by $\phi \mapsto ⟨h,\phi {⟩}_{\mathcal{H}}$. Both inclusions are continuous and dense. We write the duality pairing $⟨F,\phi ⟩$ for $F\in {\Phi }^{\prime }$, $\phi \in \Phi$, extending the inner product of $\mathcal{H}$.

The triple is called nuclear when $\Phi$ is a nuclear space in the sense of Grothendieck: every continuous linear map from $\Phi$ into a Banach space is nuclear, equivalently the topology of $\Phi$ can be defined by a countable family of Hilbertian seminorms whose successive embeddings are Hilbert-Schmidt. The model example is $\Phi =\mathcal{S}({\mathbb{R}}^n)$, $\mathcal{H}=L^2({\mathbb{R}}^n)$, ${\Phi }^{\prime }={\mathcal{S}}^{\prime }({\mathbb{R}}^n)$, the tempered distributions.

Let $A$ be a self-adjoint operator on $\mathcal{H}$ that maps $\Phi$ continuously into $\Phi$ (so $\Phi$ is a core-carrying invariant test domain). The adjoint action transports $A$ to ${\Phi }^{\prime }$ by $⟨A^{\prime }F,\phi ⟩:=⟨F,A\phi ⟩$. A generalized eigenvector of $A$ with generalized eigenvalue $\lambda \in \mathbb{R}$ is a nonzero $F\in {\Phi }^{\prime }$ with $$ \langle F, A\varphi\rangle = \lambda, \langle F, \varphi\rangle \qquad \text{for all } \varphi \in \Phi, $$ that is, $A^{\prime }F=\lambda F$ in ${\Phi }^{\prime }$. These are the eigendistributions — the rigorous form of Dirac's improper eigenkets. They need not lie in $\mathcal{H}$; the position eigendistribution ${\delta }_{x_0}$ and the momentum eigendistribution $e^{ip\cdot x}$ are the canonical examples on ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$.

Key theorem with proof Intermediate+

Theorem (Gel'fand-Maurin nuclear spectral theorem). Let $\Phi \subset \mathcal{H}\subset {\Phi }^{\prime }$ be a nuclear rigged Hilbert space and let $A$ be a self-adjoint operator on $\mathcal{H}$ with $A\Phi \subseteq \Phi$ continuously. Let $\{E(\cdot )\}$ be the projection-valued spectral measure of $A$ and $\mu$ a basic (scalar spectral) measure on $\mathbb{R}$ equivalent to it. Then for $\mu$-almost every $\lambda \in \mathbb{R}$ there is a nonzero generalized eigenvector $F_{\lambda }\in {\Phi }^{\prime }$ with $A^{\prime }{F}_{\lambda }=\lambda F_{\lambda }$, and the system $\{F_{\lambda }\}$ is complete in the following sense. There is a unitary $$ U : \mathcal{H} ;\xrightarrow{\ \cong\ }; \int_{\mathbb{R}}^{\oplus} \mathcal{H}\lambda , d\mu(\lambda) $$ onto a direct integral of Hilbert spaces, intertwining $A$ with multiplication by $\lambda$, such that for $\phi \in \Phi$ the component $(U\phi )(\lambda )$ is computed by pairing $\phi$ against the generalized eigenvectors at $\lambda$. In particular every $\phi \in \Phi$ admits the generalized eigenfunction expansion $$ \langle \varphi, \psi\rangle{\mathcal{H}} ;=; \int_{\mathbb{R}} \overline{\langle F_\lambda, \psi\rangle},\langle F_\lambda, \varphi\rangle , d\mu(\lambda), \qquad \varphi,\psi \in \Phi. $$

Proof. Reduce first to a cyclic (simple-spectrum) summand. By the multiplicity theory for self-adjoint operators, $\mathcal{H}$ decomposes into a countable orthogonal sum of cyclic subspaces, and it suffices to treat one cyclic piece, where the spectral theorem furnishes a unitary $V:\mathcal{H}\to L^2(\mathbb{R},d\mu )$ carrying $A$ to multiplication by $\lambda$. Fix a Lebesgue point picture: by the Radon-Nikodym theorem the measure $\mu$ disintegrates pointwise, and for $\mu$-a.e.\ $\lambda$ the evaluation-at-$\lambda$ of the $L^2$ representative is defined off a $\mu$-null set.

Consider, for each fixed $\lambda$, the linear functional on $\Phi$ given by $\phi \mapsto (V\phi )(\lambda )$, the value at $\lambda$ of the $L^2$-image of $\phi$. A priori $(V\phi )$ is only an $L^2$ class, so pointwise evaluation is not defined for individual $\lambda$. Nuclearity repairs exactly this. The composite map $\Phi \to L^2(\mathbb{R},d\mu )\to \mathbb{C}$, "transform then evaluate at $\lambda$," is, after restricting to one of the Hilbertian seminorm completions ${\Phi }_n$ of the nuclear space $\Phi$, the composition of a Hilbert-Schmidt embedding ${\Phi }_{n+1}\hookrightarrow {\Phi }_n$ with a bounded map. The kernel theorem (the nuclearity of $\Phi$) produces a measurable field $\lambda \mapsto F_{\lambda }\in {\Phi }^{\prime }$ such that $⟨F_{\lambda },\phi ⟩=(V\phi )(\lambda )$ for $\mu$-a.e.\ $\lambda$, simultaneously for all $\phi$ in the separable space $\Phi$. This is the step that fails without nuclearity: in a bare Hilbert rigging there is no reason a measurable field of evaluation functionals exists $\mu$-a.e.

Each $F_{\lambda }$ is a generalized eigenvector. Indeed $V$ intertwines $A$ with multiplication by $\lambda$, so for $\phi \in \Phi$, $$ \langle F_\lambda, A\varphi\rangle = (V(A\varphi))(\lambda) = \big(\lambda \cdot V\varphi\big)(\lambda) = \lambda,(V\varphi)(\lambda) = \lambda,\langle F_\lambda, \varphi\rangle, $$ valid for $\mu$-a.e.\ $\lambda$; off the null set where this fails we discard $F_{\lambda }$. Thus $A^{\prime }{F}_{\lambda }=\lambda F_{\lambda }$ in ${\Phi }^{\prime }$.

Completeness is the Parseval identity transported through $V$. For $\phi ,\psi \in \Phi$, $$ \langle \varphi, \psi\rangle_{\mathcal{H}} = \langle V\varphi, V\psi\rangle_{L^2(d\mu)} = \int_{\mathbb{R}} (V\varphi)(\lambda),\overline{(V\psi)(\lambda)}, d\mu(\lambda) = \int_{\mathbb{R}} \langle F_\lambda, \varphi\rangle,\overline{\langle F_\lambda, \psi\rangle}, d\mu(\lambda). $$ For the multiplicity-$>1$ case the same argument runs componentwise, yielding for $\mu$-a.e.\ $\lambda$ a finite or countable family of generalized eigenvectors spanning a fibre ${\mathcal{H}}_{\lambda }$, and $U=V$ assembled over the cyclic pieces is the asserted unitary onto ${\int }^{\oplus }{\mathcal{H}}_{\lambda }d\mu (\lambda )$. $\square$

Bridge. This theorem is the foundational reason the Dirac continuous-basis calculus is legitimate: the eigendistributions $⟨x|$ and $⟨p|$ are the $F_{\lambda }$ for the position and momentum operators, and the resolution of the identity $\int |x⟩⟨x|dx=\mathrm{Id}$ is exactly the Parseval identity above. The construction builds toward the spectral-multiplicity and direct-integral picture that appears again in the rigorous treatment of scattering theory and of continuous-spectrum observables, and it is dual to the projection-valued-measure form of the spectral theorem from 02.11.03: where that form integrates projections against $dE(\lambda )$, this form integrates rank-one generalized projections $|F_{\lambda }⟩⟨F_{\lambda }|$ against the scalar measure $d\mu (\lambda )$. Putting these together, the central insight is that nuclearity of the test space is precisely what converts the measure-theoretic "almost everywhere" of the spectral theorem into an everywhere-defined family of eigendistributions; this is exactly the Fourier transform when $A$ is the momentum operator, which generalises to any self-adjoint $A$ with a nuclear invariant core.

Exercises Intermediate+

Advanced results Master

The nuclear spectral theorem extends from a single self-adjoint operator to a finite or countable family of strongly commuting self-adjoint operators $A_1,\dots ,A_k$ that preserve a common nuclear invariant test space $\Phi$. The joint spectral measure on ${\mathbb{R}}^k$ then carries, for almost every joint eigenvalue $\lambda =({\lambda }_1,\dots ,{\lambda }_k)$, a system of joint generalized eigenvectors $F_{\lambda }\in {\Phi }^{\prime }$ satisfying $A_j^{\prime }{F}_{\lambda }={\lambda }_j{F}_{\lambda }$ simultaneously, with a $k$-dimensional direct-integral diagonalization $\mathcal{H}\cong {\int }_{{\mathbb{R}}^k}^{\oplus }{\mathcal{H}}_{\lambda }d\mu (\lambda )$. For the position operators $Q_1,\dots ,Q_n$ on $L^2({\mathbb{R}}^n)$ this recovers the joint eigendistributions ${\delta }_{x_0}$; for the momentum operators it recovers the multidimensional plane waves and the $n$-dimensional Fourier transform.

The choice of rigging is flexible and can always be arranged through a Hilbert-Schmidt chain (Berezansky's construction). Given any self-adjoint $A$ with a cyclic vector, one builds a scale of Hilbert spaces ${\mathcal{H}}_{+}\subset \mathcal{H}\subset {\mathcal{H}}_{-}$ in which ${\mathcal{H}}_{+}\hookrightarrow \mathcal{H}$ is Hilbert-Schmidt — typically ${\mathcal{H}}_{+}=\mathrm{Dom}(T)$ for a positive operator $T$ with $T^{-1}$ Hilbert-Schmidt — and then a nuclear $\Phi ={\bigcap }_m\mathrm{Dom}(T^m)$ with the projective-limit topology. The eigendistributions then live in the negative-index space ${\mathcal{H}}_{-}$ rather than in a bare ${\Phi }^{\prime }$, which sharpens the regularity statement: position and momentum eigendistributions on ${\mathbb{R}}^n$ live in a Sobolev space of negative order $H^{-s}$ for $s>n/2$, precisely the order at which Dirac masses become bounded functionals.

The same machinery diagonalizes self-adjoint operators arising from unitary group representations. For a locally compact group $G$ with a unitary representation on $\mathcal{H}$, the Gel'fand-Maurin theorem provides the direct-integral decomposition into irreducibles when $G$ is type I — the spectral side of Plancherel theory — with the generalized eigenvectors playing the role of distributional matrix coefficients. This is the analytic heart of the Gel'fand programme that runs from the spectral theorem through abstract harmonic analysis on the symmetric spaces $G/K$.

A delicate point is non-uniqueness and regularity of the field $\lambda \mapsto F_{\lambda }$. The eigendistributions are determined only up to a scalar gauge $F_{\lambda }\mapsto c(\lambda )F_{\lambda }$ and up to alteration on $\mu$-null sets; the spectral measure class $[\mu ]$ is the invariant datum, not any one representative measure. On the absolutely continuous spectrum one can normalize the $F_{\lambda }$ to vary continuously, recovering the physicist's smooth continuous basis; on the singular-continuous spectrum no continuous normalization need exist, which is one rigorous diagnostic separating the three spectral types.

Synthesis. The nuclear spectral theorem is the foundational reason the entire Dirac formalism is consistent, and it is dual to the projection-valued-measure spectral theorem of 02.11.03 in a way that putting these together makes precise: the PVM form integrates orthogonal projections against $dE(\lambda )$ and lives entirely inside $\mathcal{H}$, while the rigged form integrates rank-one generalized projections $|F_{\lambda }⟩⟨F_{\lambda }|$ against a scalar measure and reaches outside $\mathcal{H}$ into ${\Phi }^{\prime }$. The central insight is that nuclearity converts the measure-theoretic almost-everywhere of the spectral theorem into an everywhere-defined eigendistribution field; this is exactly the Fourier transform when $A$ is momentum, and it generalises that transform to every self-adjoint operator with a nuclear invariant core. The construction builds toward, and supplies the spectral input to, the direct-integral decomposition of group representations that appears again in Plancherel theory and in the constructive-field-theory measures on ${\mathcal{S}}^{\prime }$, where the same triple $\mathcal{S}\subset L^2\subset {\mathcal{S}}^{\prime }$ carries both the eigendistributions here and the Gaussian path measure there. The bridge is the single structural fact that a small enough test space has a large enough dual to host the continuum.

Full proof set Master

Proposition 1 (the rigging inclusions are continuous and dense). Let $\Phi$ be a dense subspace of a separable Hilbert space $\mathcal{H}$ carrying a locally convex topology finer than the subspace topology from $\mathcal{H}$. Then the canonical map $\mathcal{H}\to {\Phi }^{\prime }$, $h\mapsto ⟨h,\cdot {⟩}_{\mathcal{H}}$, is injective with dense range, and both inclusions $\Phi \hookrightarrow \mathcal{H}\hookrightarrow {\Phi }^{\prime }$ are continuous.

Proof. Continuity of $\Phi \hookrightarrow \mathcal{H}$ is the hypothesis that the $\Phi$-topology is finer. For the map $j:\mathcal{H}\to {\Phi }^{\prime }$, continuity of the pairing $⟨h,\phi {⟩}_{\mathcal{H}}$ in $\phi \in \Phi$ follows because $\phi \mapsto ⟨h,\phi {⟩}_{\mathcal{H}}$ is bounded by $\Vert h\Vert \Vert \phi {\Vert }_{\mathcal{H}}$ and $\Vert \phi {\Vert }_{\mathcal{H}}$ is a continuous seminorm on $\Phi$; so $j(h)\in {\Phi }^{\prime }$. Injectivity of $j$: if $⟨h,\phi {⟩}_{\mathcal{H}}=0$ for all $\phi \in \Phi$, then $h\perp \Phi$, and since $\Phi$ is dense in $\mathcal{H}$ this forces $h=0$. Density of the range $j(\mathcal{H})$ in ${\Phi }^{\prime }$ holds because any $F\in {\Phi }^{\prime }$ annihilating $j(\mathcal{H})$ in the dual pairing would annihilate $\mathcal{H}$, hence the dense $\Phi$, hence vanish. Continuity of $j$ as a map into ${\Phi }^{\prime }$ with its weak-$\ast$ (or strong dual) topology is immediate from the seminorm bound. $\square$

Proposition 2 (eigendistributions of an essentially self-adjoint operator with a nuclear core). Let $A$ be self-adjoint on $\mathcal{H}$ with $A\Phi \subseteq \Phi$ continuously, $\Phi$ nuclear. If $F\in {\Phi }^{\prime }$ satisfies $A^{\prime }F=\lambda F$ with $\lambda \in \mathbb{R}$, then $\lambda$ lies in the spectrum $\sigma (A)$.

Proof. Suppose $\lambda \notin \sigma (A)$. Then $A-\lambda$ has a bounded inverse $R=(A-\lambda {)}^{-1}$ on $\mathcal{H}$. Because $A$ maps $\Phi$ continuously to $\Phi$ and $\Phi$ is a dense invariant core, $R$ maps $\Phi$ into the domain of $A$, and on $\Phi$ the resolvent restricts to a continuous operator $R_{\Phi }:\Phi \to \Phi$ (a standard consequence of $A-\lambda$ being a topological isomorphism of $\Phi$ when $\lambda$ is in the resolvent set and $A$ acts continuously). For $\phi \in \Phi$ write $\phi =(A-\lambda )R_{\Phi }\phi$. Then $$ \langle F, \varphi\rangle = \langle F, (A-\lambda)R_\Phi\varphi\rangle = \langle (A'-\lambda)F, R_\Phi\varphi\rangle = \langle 0, R_\Phi\varphi\rangle = 0 $$ for every $\phi \in \Phi$, so $F=0$, contradicting that a generalized eigenvector is nonzero. Hence $\lambda \in \sigma (A)$. $\square$

Proposition 3 (Parseval transport gives completeness). With the unitary $V:\mathcal{H}\to L^2(\mathbb{R},d\mu )$ diagonalizing a cyclic self-adjoint $A$, and $F_{\lambda }\in {\Phi }^{\prime }$ the eigendistributions with $⟨F_{\lambda },\phi ⟩=(V\phi )(\lambda )$ $\mu$-a.e., the completeness identity $$ \langle \varphi, \psi\rangle_{\mathcal{H}} = \int_{\mathbb{R}} \langle F_\lambda, \varphi\rangle,\overline{\langle F_\lambda, \psi\rangle}, d\mu(\lambda) $$ holds for all $\phi ,\psi \in \Phi$, and conversely any $\phi \in \Phi$ with $⟨F_{\lambda },\phi ⟩=0$ for $\mu$-a.e.\ $\lambda$ is zero.

Proof. Since $V$ is unitary, $⟨\phi ,\psi {⟩}_{\mathcal{H}}=⟨V\phi ,V\psi {⟩}_{L^2(d\mu )}=\int (V\phi )\overline{(V\psi )}d\mu$. Substituting $⟨F_{\lambda },\phi ⟩=(V\phi )(\lambda )$ and $⟨F_{\lambda },\psi ⟩=(V\psi )(\lambda )$ gives the identity. For the converse, if $⟨F_{\lambda },\phi ⟩=(V\phi )(\lambda )=0$ $\mu$-a.e.\ then $V\phi =0$ in $L^2(d\mu )$, and unitarity of $V$ gives $\phi =0$. This is the completeness of the generalized eigenvector system: no nonzero test vector is orthogonal to every eigendistribution. $\square$

Connections Master

• The projection-valued-measure spectral theorem for unbounded self-adjoint operators in 02.11.03 is the dual partner of this unit: that theorem integrates orthogonal projections $dE(\lambda )$ inside $\mathcal{H}$, while the nuclear spectral theorem integrates generalized rank-one projections $|F_{\lambda }⟩⟨F_{\lambda }|$ reaching into ${\Phi }^{\prime }$. The scalar basic measure $\mu$ here is the spectral measure class of 02.11.03, and the multiplicity theory feeding the direct integral is the same.

• The theory of distributions and the Schwartz kernel theorem in 02.14.04 supplies the nuclearity of $\mathcal{S}$ and the kernel theorem that produce the measurable field of eigendistributions $\lambda \mapsto F_{\lambda }$. The triple $\mathcal{S}\subset L^2\subset {\mathcal{S}}^{\prime }$ is the canonical nuclear rigging, and the eigendistributions ${\delta }_{x_0}$ and $e^{ipx}$ are exactly the tempered distributions classified there.

• The Fourier transform and Plancherel theorem in 02.10.04 are the special case of this unit for the momentum operator: the diagonalizing unitary is $\mathcal{F}$, the basic measure is Lebesgue, the eigendistributions are plane waves, and the generalized eigenfunction expansion is Fourier inversion. The nuclear spectral theorem is the operator-theoretic generalisation of Plancherel to an arbitrary self-adjoint observable.

• The Hilbert-space formalism of quantum mechanics in 12.02.01 and the operators-and-observables unit 12.02.02 invoke the rigged Hilbert space as the rigorous home of Dirac's improper kets; this unit discharges that invocation, and the normalization $⟨p|p^{\prime }⟩=\delta (p-p^{\prime })$ used in 12.13.01 is the completeness identity proved here.

• The Radon-Nikodym theorem and direct integrals in 02.07.08 furnish the disintegration of the spectral measure $\mu$ and the measurable field of fibre Hilbert spaces ${\mathcal{H}}_{\lambda }$ that the direct-integral decomposition ${\int }^{\oplus }{\mathcal{H}}_{\lambda }d\mu (\lambda )$ requires.

Historical & philosophical context Master

The rigged Hilbert space was introduced by Israel Gel'fand and his collaborators in the fourth volume of Generalized Functions (Gosizdat 1961; Academic Press translation 1964) ^{[Gel'fand-Vilenkin 1964]}, where the chain $\Phi \subset \mathcal{H}\subset {\Phi }^{\prime }$ — the equipped or rigged Hilbert space — was built precisely to make Dirac's continuous spectra and improper eigenfunctions into rigorous mathematics. The completeness theorem for the generalized eigenfunction system, the nuclear spectral theorem, was proved by Gel'fand together with Arkadii Kostyuchenko and was given its definitive form by Krzysztof Maurin in General Eigenfunction Expansions and Unitary Representations of Topological Groups (PWN 1968) ^{[Maurin 1968]}, using Grothendieck's nuclear-space theory and the kernel theorem; the parallel Hilbert-Schmidt-rigging development is due to Yuri Berezansky ^{[Berezansky 1968]}. Garding and others gave related expansion theorems for elliptic and group-invariant operators in the same period.

Dirac had introduced the bra-ket notation and the improper eigenstates $|x⟩$, $|p⟩$ in The Principles of Quantum Mechanics (Oxford, 1930), with the delta function as a calculational device that von Neumann's 1932 Hilbert-space axiomatization deliberately avoided in favour of projection-valued measures. The rigged Hilbert space reconciled the two: von Neumann's spectral theorem governs the measure-theoretic side inside $\mathcal{H}$, while the Gel'fand triple supplies the eigendistributions outside it, and the two are stitched together by nuclearity. Rafael de la Madrid's survey ^{[de la Madrid 2005]} documents how each manipulation of the Dirac calculus — eigenket expansions, the resolution of the identity, $⟨x|p⟩=(2\pi {)}^{-1/2}{e}^{ipx}$ — corresponds to a theorem about the triple $\mathcal{S}\subset L^2\subset {\mathcal{S}}^{\prime }$, and how resonances and Gamow states require a rigging by Hardy-class test functions rather than Schwartz functions.

Bibliography Master

@book{GelfandVilenkin1964Vol4,
  author    = {Gel'fand, I. M. and Vilenkin, N. Ya.},
  title     = {Generalized Functions, Vol. 4: Applications of Harmonic Analysis},
  publisher = {Academic Press},
  address   = {New York},
  note      = {Translated from the 1961 Russian original by Amiel Feinstein},
  year      = {1964}
}

@book{Maurin1968,
  author    = {Maurin, Krzysztof},
  title     = {General Eigenfunction Expansions and Unitary Representations of Topological Groups},
  publisher = {Polish Scientific Publishers (PWN)},
  address   = {Warsaw},
  series    = {Monografie Matematyczne},
  volume    = {48},
  year      = {1968}
}

@book{Berezansky1968,
  author    = {Berezansky, Yu. M.},
  title     = {Expansions in Eigenfunctions of Selfadjoint Operators},
  publisher = {American Mathematical Society},
  series    = {Translations of Mathematical Monographs},
  volume    = {17},
  year      = {1968}
}

@book{ReedSimonI,
  author    = {Reed, Michael and Simon, Barry},
  title     = {Methods of Modern Mathematical Physics, Vol. I: Functional Analysis},
  publisher = {Academic Press},
  address   = {New York},
  edition   = {Revised and enlarged},
  year      = {1980}
}

@article{deLaMadrid2005,
  author  = {de la Madrid, Rafael},
  title   = {The role of the rigged {H}ilbert space in quantum mechanics},
  journal = {European Journal of Physics},
  volume  = {26},
  number  = {2},
  pages   = {287--312},
  year    = {2005}
}

@book{Dirac1930,
  author    = {Dirac, P. A. M.},
  title     = {The Principles of Quantum Mechanics},
  publisher = {Oxford University Press},
  address   = {Oxford},
  year      = {1930}
}