Rigged Hilbert Spaces in Quantum Mechanics

Rigged Hilbert spaces, often known as Gelfand triplets, extend the standard Hilbert space framework to incorporate both square-integrable states and more singular objects such as generalised eigenvectors. In quantum mechanics this structure reconciles discrete and continuous spectra by embedding a dense space of “test” vectors within a Hilbert space and appending its topological dual. This allows operators with continuous spectrum—such as position and momentum—to act meaningfully on distributional states, yielding Dirac’s bras and kets as bona fide elements of the dual. Practical applications range from scattering theory and resonances to the mathematical formulation of unstable states (Gamow vectors). By endowing the test space with a suitable topology, one achieves well-defined extensions of self-adjoint operators to the dual, ensuring rigorous spectral decompositions. The rigged framework underpins the modern understanding of resonance phenomena, time-asymmetric evolution and the precise role of boundary conditions in quantum dynamics. It also provides a natural setting for special-function expansions and group representations, unifying disparate formulations under a single mathematical umbrella.

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Technical terms

Rigged Hilbert space (Gelfand triplet): A triple Φ ⊂ H ⊂ Φ× where Φ is a space of test vectors, H a Hilbert space, and Φ× its continuous dual of distributions.

Test function space: A densely embedded subspace of smooth, rapidly decaying vectors in which unbounded operators act continuously.

Dual space: The space of continuous linear functionals on Φ, hosting distributional eigenstates and Dirac kets.

Gamow state: A generalised eigenvector with complex energy, representing an unstable or resonant quantum state with exponential decay.

References

1. Groups, Special Functions and Rigged Hilbert Spaces. Axioms (2019).
2. Hermite Functions and Fourier Series. Symmetry (2021).
3. Quantum Mechanics and Its Evolving Formulations. Entropy (2021).
4. A constructive presentation of rigged Hilbert spaces. Journal of Physics Conference Series (2015).
5. Mathematical Models for Unstable Quantum Systems and Gamow States. Entropy (2022).
6. Symmetry Groups, Quantum Mechanics and Generalized Hermite Functions. Mathematics (2022).