Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator
Abstract
We introduce a bi-oriented interlaced matrix product on 2 × 2 matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley-Dickson doubling. The opposite orientations ⋄ R and ⋄ L coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators Q, P on L 2 (R), a matrix D and its companion D satisfy, on the common Schwartz domain,
We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations Φ = A + iB and Ψ = A -iB decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space S(R 2 ) ⊂ L 2 (R 2 ) ⊂ S ′ (R 2 ); the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.
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BibTeX
@misc{gazeau2026bioriented,
title = {{Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator}},
author = {Jean-Pierre Gazeau},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28511},
eprint = {2609.28511},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05772840v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator
AU - Jean-Pierre Gazeau
PY - 2026
DA - 2026/09/21
PB - HAL
DO - 10.48550/arxiv.2609.28511
UR - https://hal.science/hal-05772840v1
AB - We introduce a bi-oriented interlaced matrix product on 2 × 2 matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley-Dickson doubling. The opposite orientations ⋄ R and ⋄ L coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators Q, P on L 2 (R), a matrix D and its companion D satisfy, on the common Schwartz domain, We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations Φ = A + iB and Ψ = A -iB decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space S(R 2 ) ⊂ L 2 (R 2 ) ⊂ S ′ (R 2 ); the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.
ER -