This research note proposes an experimental-setting interpretation of self-adjoint realization for boundary-sensitive quantum systems. The starting point is the standard operator-theoretic fact that a formal differential expression does not by itself define quantum dynamics. A Hamiltonian becomes a physical generator of unitary evolution only after a Hilbert space, an operator domain, and a self-adjoint realization have been specified. The note extends the ordinary half-line problem in quantum mechanics to a multidimensional setting in which the coordinate axes act as axial endpoint or boundary sets. A weighted Hilbert space is introduced, and the local radial behavior near an axis is reduced to an effective inverse-square potential problem. The endpoint classification is then described in terms of the usual limit-circle/limit-point distinction. In particular, the phrase “spectral inaccessibility” is used only as an interpretive term; the rigorous statement is endpoint classification, such as limit-point behavior. The main conceptual contribution is the proposal that, in boundary-sensitive effective systems, an experimental setup may participate not only in selecting a state, but also in defining the effective subsystem itself. In this viewpoint, the setup may help select the relevant effective Hilbert space, operator domain, and self-adjoint realization before a subsystem wave function is assigned. Schematically, the proposed viewpoint is \ ₓ₎ₓ (H ₒₘₒ, H ₄₅₅^ (₀), ₒₘₒ). \ Here \ (₀\) is not asserted to be determined by a universal theorem. It is regarded as a model-dependent effective parameter that may arise from the apparatus, boundary region, contact interaction, or environment after reduction to a low-energy subsystem description. The note also clarifies that the statement “no effective wave function has yet been defined” does not deny the possible existence of a total quantum state. Rather, it means that the reduced subsystem wave function is meaningful only after the effective subsystem, its Hilbert space, and its self-adjoint Hamiltonian have been selected. Finally, the note identifies an open problem: the concrete derivation of ₓ₎ₓ ₄₅₅^ (₀) \ from a full self-adjoint total Hamiltonian. This reduction is interpreted as a state-preparation and effective Hamiltonian-engineering problem, rather than merely a measurement-outcome problem. A concrete model of this reduction is left for future work.
hideo umihara (Fri,) studied this question.