The unbounded-operator domain-restriction problem in quantum mechanics was first raised in the foundations-of-physics literature by Heathcote (1990), restated by Carcassi, Calderón & Aidala (2025)

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The technical trigger for the whole Hilbert-space-physicality debate (claim-lu-2026-argues-full-hilbert-space-is-physical-against-carcassi-and-heathcote) is a basic fact of functional analysis. In the textbook axioms of quantum mechanics, observables — position, momentum, most Hamiltonians — are represented by self-adjoint operators. But the Hellinger–Toeplitz theorem forces a problem: an unbounded self-adjoint operator cannot be defined on the whole Hilbert space, only on a dense proper subset, its domain. Zhonghao Lu's paper states the result directly: "According to the Hellinger-Toeplitz theorem, an unbounded self-adjoint operator A can only be defined on a subset of the Hilbert space, namely its domain Dom(A)." Since the standard expectation-value and dynamical-evolution axioms are written in terms of such operators, they fail to apply to any state outside that domain — which is what motivates restricting "physical" states to some proper subset in the first place.

Lu credits this observation, within the foundations-of-physics literature specifically, to A. Heathcote's 1990 paper, with Gabriele Carcassi, Francisco Calderón & Christine A. Aidala's 2025 paper independently restating it 35 years later: "In the foundations of physics literature, this problem is first noticed by Heathcote in 1990, and then by Carcassi, Calderón, and Aidala in 2025." Carcassi et al.'s own proposed fix — restricting physical states to Schwartz spaces — is the specific proper-subset candidate Lu's paper argues against.