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A Multidimensional Extension of the Half-Line Problem in Quantum Mechanics and an Experimental-Setting Interpretation of Self-Adjoint Realization

hideo umihara

Published May 15, 2026• 0 citations
Publisher

Abstract

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 \[\Psi_{\rm tot}\quad\longrightarrow\quad\left(\mathcal H_{\rm sys},H_{\rm eff}^{(\theta_{\rm app})},\psi_{\rm sys}\right).\] Here \(\theta_{\rm app}\) 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 \[H_{\rm tot}\quad\longrightarrow\quadH_{\rm eff}^{(\theta_{\rm app})}\] 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.

DOI
10.5281/zenodo.20191984

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This paper proposes an apparatus-dependent interpretation of self-adjoint realization in boundary-sensitive quantum systems. The main claim is not a new classification theorem for self-adjoint extensions, but a conceptual and operational separation of apparatus dependence into two levels. The first level is structural effective-system selection. In the weighted-endpoint model studied here, this is represented by a model parameter gamma, which specifies the effective Hilbert-space measure or, after unitary transformation, the corresponding inverse-square coupling. The second level is self-adjoint-domain selection after the effective Hilbert space and formal Hamiltonian have already been fixed. In Lebesgue-measure models, this domain choice is represented concretely by a Robin parameter. The paper illustrates this two-level structure through several boundary-sensitive models. A Robin--Dirichlet finite-box model shows that the textbook infinite square well is recovered as the Dirichlet realization, while perfect reflection alone does not uniquely select Dirichlet boundary conditions. A weighted endpoint model distinguishes structural measure selection from residual self-adjoint extension freedom. A thin-wall model gives a concrete asymptotic mechanism by which a resonantly scaled microscopic boundary layer produces an effective Robin condition. Finally, the momentum representation of the Dirichlet box is interpreted as the full-line spectral representation selected by a momentum-measurement setting, rather than as an intrinsic momentum representation of the closed box. The resulting hierarchy is: experimental setting, effective quantum system, state preparation, and measurement statistics. This hierarchy does not modify ordinary quantum measurement theory. It identifies a prior layer that becomes especially relevant in boundary-sensitive systems: before state preparation and measurement are described, the effective quantum system to which the state belongs must be specified.
For a particle moving on a half-line or in an interval the operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}=\ensuremath{-}i{\ensuremath{\partial}}_{x}$ is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ fails. Based upon an alternative concept for a self-adjoint momentum operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}$, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ and the momentum operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}$ are endowed with self-adjoint extension parameters that characterize the corresponding domains $D(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H})$ and $D({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R})$ in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The alternative concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.
For a particle moving on a half-line or in an interval the operator pˆ = −i∂x is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on pˆ fails. Based upon an alternative concept for a self-adjoint momentum operator pˆR, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian Hˆ and the momentum operator pˆR are endowed with self-adjoint extension parameters that characterize the corresponding domains D(Hˆ ) and D(pˆR ) in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The alternative concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.
Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.
Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.
This work addresses the structural problem of defining physical observables in regimes where the spectral decomposition of self-adjoint operators is not purely discrete. In particular, it is demonstrated that, outside the ultraviolet regime, the identification between mass and eigenvalues ceases to be consistent when the associated spectral measure has a continuous component. Starting exclusively from minimal spectral hypotheses -- self - adjoint operator, resolvent, spectral measure, and its density -- it is established, without resorting to ad hoc hypotheses, that the observable physical structure is determined by the spectral measure associated with the state. As a consequence, it is proven that the physical dimension of a system is not given by the dimension of the Hilbert space nor by the cardinality of the spectrum, but by the structure of the dominant spectral regions of the density, defining the notion of effective spectral dimension. In this framework, physical observables emerge as functionals of the spectral measure. In particular, mass is characterized as the first-order spectral moment, while the spectral width quantifies the dispersion associated with the energy distribution. It is further shown that dominant spectral regions act as effective degrees of freedom, providing a distributed description of the dynamics. The developed formalism recovers the discrete limit of quantum mechanics, is compatible with the spectral representation of quantum field theory, and admits a natural interpretation in terms of scale flow. Explicit falsifiability criteria are also established, and the structural limitations of the proposed framework are delineated. The central result is the formulation of a principle: physical dimension and observables emerge from the effective spectral structure, implying the necessary substitution of pointlike descriptions with distributed descriptions in non-discrete regimes.