The one-centre Coulomb model: energy observable, flow, stationary states and spectral support

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the kinematics and massive-particle model postulate P5, read through the declared one-centre Coulomb model of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clauses (a)-(c) with parameters $\mathrm{\hslash }>0$, $m>0$ and coupling $\kappa >0$, its assigned energy operator $H_{\kappa }$ on $D(H_{\kappa })=H^2({\mathbb{R}}^3;\mathbb{C})$, the assigned sharp-observable reading Assigned sharp observables and the outcome law of the framework, the flow and preparation reading Assigned energy observable, unitary flow and the two pictures of autonomous evolution, and the spin-zero position law Position and momentum outcome laws of the massive spin-s model (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, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Then:

(a) $(H_{\kappa },\mathrm{J})$ is an assigned sharp observable of the model: $H_{\kappa }$ is densely defined, self-adjoint and bounded below on $D(H_{\kappa })=H^2({\mathbb{R}}^3;\mathbb{C})$, acts by $H_{\kappa }f=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_{w}f-\kappa f\mathrm{/}\mathrm{\mid }\cdot \mathrm{\mid }$, and ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(H_{\kappa })=[0,+\mathrm{\infty });$ every point of $\sigma (H_{\kappa })\cap (-\mathrm{\infty },0)$ is an eigenvalue of finite multiplicity, isolated in $\sigma (H_{\kappa })$, and the set of negative eigenvalues is at most countable;

(b) Flow. The family $U(t):=\exp (-it{H}_{\kappa }\mathrm{/}\mathrm{\hslash })$ is the model's strongly continuous one-parameter unitary group; the Schroedinger equation $i\mathrm{\hslash }\frac{d}{dt}\psi (t)=H_{\kappa }\psi (t)$ holds in the norm-differentiable sense exactly for initial vectors $\psi (0)\in D(H_{\kappa })$, and for other vectors the evolution is continuous but the equation is not asserted; density preparations evolve by $\rho (t)=U(t)\rho (0)U(t{)}^{\ast }$ with $\rho (t)$ a density operator for every $t$;

(c) Stationary states. For $\psi \in H_0$, $\psi \mathrm{\ne }0$, the pure preparation $[\psi ]$ is stationary, $[\psi (t)]=[\psi (0)]$ for all $t$, if and only if $\psi$ is an eigenvector of $H_{\kappa }$; in that case, with eigenvalue $E$, one has $\psi (t)=e^{-itE\mathrm{/}\mathrm{\hslash }}\psi (0)$, the energy outcome law is the point mass ${\delta }_E$ for a normalised $\psi$, and every bounded assigned sharp observable compatible with the energy has a time-independent outcome law in this preparation;

(d) Position reading. In a unit vector $\psi$ the position outcome law is ${\mu }_{\psi }^G(B)={\int }_B\mathrm{\mid }\psi {\mathrm{\mid }}^{2}dx$; if $\psi$ is in addition a unit eigenvector of $H_{\kappa }$ with eigenvalue $E$, then the position density $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ and the energy point mass at $E$ are readings of the same preparation through the declared observables, with no joint-measurement claim;

(e) Spectral support and exact values. For every preparation $\rho$ one has ${\mu }_{\rho }^{H_{\kappa }}(\sigma (H_{\kappa }))=1$, and an exact energy value $\lambda$ has ${\mu }_{\rho }^{H_{\kappa }}(\{\lambda \})=0$ unless $\lambda$ is an eigenvalue, in which case the probability is the total weight of the eigenspace; in particular no exact value in $\mathbb{R}\setminus \sigma (H_{\kappa })$ is admissible, and the item asserts neither that $0$ is an eigenvalue nor any statement about eigenvalues in the positive half-line, which are excluded by the following radial items;

(f) Scope. The statement is conditional on P1, P2, P4, P5 and the declared Coulomb model and the cited mathematics. It asserts no eigenvalue list, no degeneracy, no transition, radiation or line-intensity statement, no identification of $\kappa$, $m$ or $\mathrm{\hslash }$ with measured constants, and no empirical result.

• The continuum is present but unstructured here. The statement records only ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(H_{\kappa })=[0,\mathrm{\infty })$ and the discreteness of the negative part; the exhaustion of the negative spectrum by the Coulomb levels and the absence of positive eigenvalues require the radial theory of the following items, and no scattering interpretation of the continuous spectrum is made on this page.
• Threshold. The value $0$ is called the continuum threshold of the model; the statement does not assert whether $0$ is an eigenvalue, and the following radial items show that it is not.