The declared central-potential model: state space, assigned energy operator, spherical reading and bound-state vocabulary

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5, the identical-particle and spin postulate P6, and the preparation and apparatus discipline P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Fix a mass parameter $m>0$ and a spin label $s=0$, so that the framework's one-particle model The massive spin-s particle model of the framework: state space, operators, units and frame readings reads its state space as $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ with the pairing linear in the first argument, its componentwise position family $Q_1,Q_2,Q_3$, its componentwise momentum family $P_1,P_2,P_3$ and the action constant $\mathrm{\hslash }>0$; the spin-zero internal factor carries the trivial representation of P6. Units are those of Dimensions, SI unit conventions and quantity types of the framework: position is read in metres, momentum in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, energy and potential energy in joules, and $\mathrm{\hslash }$ in $\mathrm{J}\mathrm{s}$. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) The declared central-potential model. As an additional model assumption of the pair, adopt the central-potential model. Its declared state space is $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ of clause (a), with the position and momentum families $Q_j$, $P_k$ of that clause (The massive spin-s particle model of the framework: state space, operators, units and frame readings). A declared central potential of the model is a function $V:{\mathbb{R}}^3\to \mathbb{R}$ of the form $V(x):=v(\mathrm{\mid }x\mathrm{\mid })$ for $x\mathrm{\ne }0$, where $v:(0,\mathrm{\infty })\to \mathbb{R}$ is real, bounded and measurable and $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}:=\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}={\sup }_{r>0}\mathrm{\mid }v(r)\mathrm{\mid }$ is finite; the assignment $V(0):=0$ is a declared convention that does not affect the $L^2$ theory. The potential energy is read in joules. The model declares the Born position law of the framework for a unit class $\psi$, ${\mu }_{\psi }^G(B):={\int }_B\mathrm{\mid }\psi (x){\mathrm{\mid }}^{2}dx(B\subseteq {\mathbb{R}}^3\text{ Borel}),$ the spin-zero reading of clause (b) of Position and momentum outcome laws of the massive spin-s model. The pair asserts no derivation of a central force law from $H_0$ or from P5: the restriction to a declared central potential is an assumption of the model, and an assignment of a concrete material system to the model is an identification in the sense of clause (iii) of P7 and carries the corresponding uncertainty.

(c) The assigned energy operator. Let ${\mathcal{F}}_2$ be the unitary Plancherel transform of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b), $c:=2{\pi }^2{\mathrm{\hslash }}^2\mathrm{/}m$, and let $H_0^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}:={\mathcal{F}}_2^{\ast }{M}_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_2,D(H_0^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}):=\{f\in H_0:\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f\in H_0\}$ be the free operator of that clause with $d=3$ and $b=0$. With $M_V$ the bounded multiplication operator of clause (a) of the same theorem, the assigned energy operator of the model with central potential $V$ is $H_V:=H_0^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}+M_V,D(H_V):=D(H_0^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}})\mathrm{.}$ The declared energy unit is the joule. By clause (c) of the cited theorem $H_V$ is self-adjoint on $D(H_V)$, bounded below by $-\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$; by The three-dimensional free Schrödinger operator as the weak Laplacian on its Sobolev domain the domain is the Sobolev space $W^{2,2}({\mathbb{R}}^3;\mathbb{C})$ and $H_{V}f=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_{w}f+Vf(f\in D(H_V)),$ ${\mathrm{\Delta }}_w=D_1^2+D_2^2+D_3^2$ the weak Laplacian of that lemma. If $\psi \in D(H_V)\setminus \{0\}$ and $E\in \mathbb{R}$ satisfy $H_V\psi =E\psi$, the time-independent equation $-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_w\psi +V\psi =E\psi$ is an equality in $L^2({\mathbb{R}}^3;\mathbb{C})$ and hence in distributions; it is a partial differential equation in the three spatial variables. If $V\in C^2(\mathrm{\Omega })$ on an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$, then the class $\psi$ has a representative in $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,\alpha }(\mathrm{\Omega })$ for every $0<\alpha <1\mathrm{/}2$, and the equation holds pointwise there. To justify this, write $f:=(2m\mathrm{/}{\mathrm{\hslash }}^2)(E-V)\psi$, so $-\mathrm{\Delta }\psi =f$ in distributions. On every relatively compact open subset of $\mathrm{\Omega }$, the weak product rule (obtained by local smooth mollification of the $C^2$ factor and passage to the $L^2$ limit) gives, for every multi-index $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }\le 2$, $D^{\beta }f=\frac{2m}{{\mathrm{\hslash }}^2}{\sum }_{\eta \le \beta }\left(\genfrac{}{}{0px}{}{\beta }{\eta }\right)D^{\eta }(E-V)D^{\beta -\eta }\psi \in L^2;$ the factors $D^{\eta }(E-V)$ are bounded on compact subsets and $\psi \in W^{2,2}({\mathbb{R}}^3)$, so $f\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$. Integration by parts against compactly supported smooth test functions makes $\psi \in H^1(\mathrm{\Omega })$ a local weak solution in the sense of Local weak solutions of a divergence-form operator for $L=-D_i({\delta }_{ij}{D}_j)=-\mathrm{\Delta }$, whose divergence coefficients are $a^{ij}={\delta }_{ij}$ and $b^i=c=0$. They satisfy the uniform ellipticity condition of Uniformly elliptic divergence-form operators and their sesquilinear forms with $\theta =1$ and have the required local coefficient derivatives of every order. The interior regularity theorem Interior $H^{k+2}$ elliptic regularity with $k=2$ therefore gives $\psi \in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^4(\mathrm{\Omega })$. Every ball compactly contained in $\mathrm{\Omega }$ is a bounded smooth domain and hence a $W^{4,2}$ extension domain by Bounded C^k domains admit integer-order Sobolev extension. The higher-order Sobolev embedding Higher-order Sobolev embedding with $n=3$, $k=4$, $p=2$ then gives a representative with continuous derivatives through order two and locally Hölder-continuous second derivatives of every exponent $0<\alpha <1\mathrm{/}2$; this is the meaning of $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,\alpha }(\mathrm{\Omega })$ here. These representatives agree on overlaps because they are continuous and represent the same almost-everywhere class. Both sides of the equation are then continuous and agree almost everywhere, so they agree pointwise. The model adopts $(H_V,\mathrm{J})$ as its declared energy observable; its self-adjointness is the cited theorem's, and no further spectral property is presupposed here.

(d) The declared spherical reading. Let $U$ be the unitary of Angular radial unitary reduction from $H_0$ onto $K={⨁}_{(l,\bar{m})}{L}^2((0,\mathrm{\infty }),dr;\mathbb{C})$, written $U\psi =(u_{l\bar{m}}{)}_{l\ge 0,-l\le \bar{m}\le l}$, and let $(Y_{l\bar{m}})$ be the complete orthonormal family of that theorem; $l,\bar{m}$ are integers, while $m>0$ continues to denote the mass. For a fixed $(l,\bar{m})$, let ${\iota }_{l\bar{m}}:L^2((0,\mathrm{\infty }),dr;\mathbb{C})\to K$ insert its argument in that coordinate and zero in every other coordinate, and define the isometric spherical lift $J_{l\bar{m}}:=U^{-1}{\iota }_{l\bar{m}}$, so that $J_{l\bar{m}}u(x)=\frac{u(r)}{r}{Y}_{l\bar{m}}(\mathrm{\Omega })(x=r\mathrm{\Omega },r>0)\mathrm{.}$ The spherical sector $(l,\bar{m})$ is the range of this lift. Its admissible reduced domain is fixed by the assigned energy domain: ${\mathcal{D}}_{l\bar{m}}:=\{u\in L^2((0,\mathrm{\infty }),dr;\mathbb{C}):J_{l\bar{m}}u\in W^{2,2}({\mathbb{R}}^3;\mathbb{C})\}\mathrm{.}$ Thus every weak Cartesian derivative through order two of the lift must belong to $L^2$ on all of ${\mathbb{R}}^3$, including neighbourhoods of the origin. This condition and the global square integrability specify admissibility at both endpoints; no independent half-line boundary parameter is part of the model. In particular $C_c^{\mathrm{\infty }}((0,\mathrm{\infty });\mathbb{C})\subseteq {\mathcal{D}}_{l\bar{m}}$ by clause (d) of the cited theorem. The effective potential is $v_l^{\mathrm{e}\mathrm{f}\mathrm{f}}(r):=v(r)+\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2m{r}^2}(r>0),$ whose second summand is called the angular-momentum barrier; both summands, and $v_l^{\mathrm{e}\mathrm{f}\mathrm{f}}$, are read in joules. For $E\in \mathbb{R}$, the model's declared radial equation in the sector $(l,\bar{m})$ is $u\in {\mathcal{D}}_{l\bar{m}},H_V{J}_{l\bar{m}}u=E{J}_{l\bar{m}}u\text{in }{L}^2({\mathbb{R}}^3;\mathbb{C})\mathrm{.}$ A radial eigenpair additionally requires $u\mathrm{\ne }0$. The associated one-dimensional differential expression is $u⟼-\frac{{\mathrm{\hslash }}^2}{2m}{u}^{\mathrm{\prime }\mathrm{\prime }}+(v(r)+\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2m{r}^2})u,u:(0,\mathrm{\infty })\to \mathbb{C},$ where $u^{\mathrm{\prime }\mathrm{\prime }}$ means the distributional second derivative on $(0,\mathrm{\infty })$ (Weak derivative of a locally integrable function). The expression alone, without the specified lifted domain, does not define a radial eigenvalue problem. On $C_c^{\mathrm{\infty }}((0,\mathrm{\infty });\mathbb{C})$, its equation with right-hand side $Eu$ is equivalent almost everywhere to the declared radial equation above by the spherical-reduction theorem of this page; pointwise equivalence additionally requires continuity of $v$. No full-domain differential reduction or endpoint classification is presupposed here. The angular-momentum quantum numbers $l$ and $\bar{m}$ appearing in the reading are labels of the orthonormal family $(Y_{l\bar{m}})$; the identification of $l$ with a total angular momentum of a material system is a separate assignment of the framework's angular-momentum items and is not made here.

(e) Stationary and bound-state vocabulary. A stationary vector of the model with potential $V$ and energy $E\in \mathbb{R}$ is a nonzero class $\psi \in D(H_V)$ with $H_V\psi =E\psi$; the pair $(\psi ,E)$ is an eigenpair of the model. A bound state of the model is a stationary vector with energy $E<0$ whose representative is normalised, $\mathrm{\parallel }\psi \mathrm{\parallel }=1$; its eigenvalue is its binding energy, read in joules, relative to the chosen energy reference $0$. Here $E<0$ is a vocabulary convention; no assertion that $0$ is a spectral continuum threshold follows for an arbitrary bounded central potential. A continuum threshold requires a separate spectral determination for the particular operator. In particular, adding a constant potential shifts the free operator and its spectrum by that constant without changing its eigenvectors. The model reads the energy outcome law of the pure preparation of a unit eigenvector with eigenvalue $E$ as the point mass ${\delta }_E$, the reading of clause (c) of Assigned sharp observables and the outcome law of the framework, and the position density of that preparation as $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ by Position and momentum outcome laws of the massive spin-s model clause (b), with $s=0$ and the state, position and choice hypotheses of (a). These are declarations of the model under P2; they construct no sample, detector or measured value, and they assert no stability, decay or formation claim.

(f) Status and scope. Clauses (a)-(e) fix the data of the pair: a declared state space, position and momentum assignments, an assigned energy operator for a real bounded measurable central potential, a declared spherical reading with its effective potential and radial equation, and a stationary-state and bound-state vocabulary. They adopt P1, P2, P4, P5, P6, P7, the choice hypotheses of the cited mathematics, and the additional central-potential model assumption and spherical reading of clauses (b) and (d); in particular the central form of the potential, the model's use of the full three-dimensional space as its state space and the identification of the spherical sectors with angular-momentum labels are declarations of the pair. The model excludes from its scope, until their own items introduce them: the Coulomb potential $\propto -1\mathrm{/}r$ and other unbounded or singular potentials; spin and magnetic couplings; relativistic corrections; finite nuclear size; the radiation extension, spontaneous emission and radiation reaction; external electromagnetic fields; and any many-body or shielding modification of a central potential. No numerical value of $m$, $\mathrm{\hslash }$ or of any potential is fixed here, and no empirical result is asserted.