The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units

(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).

Choose a reference length $L_{\ast }>0$ in metres and put $y=x\mathrm{/}{L}_{\ast },E_{\ast }:=\frac{{\mathrm{\hslash }}^2}{2m{L}_{\ast }^2},T_{\ast }:=\mathrm{\hslash }\mathrm{/}{E}_{\ast }\mathrm{.}$ Here $y$ is dimensionless, $E_{\ast }$ is in joules and $T_{\ast }$ in seconds. The unitary coordinate dilation from physical-coordinate $L_x^2$ to $L_y^2$ is $(D_{\ast }f)(y)=L_{\ast }^{3\mathrm{/}2}f(L_{\ast }y)$, with inverse $(D_{\ast }^{\ast }g)(x)=L_{\ast }^{-3\mathrm{/}2}g(x\mathrm{/}{L}_{\ast })$. Lebesgue dilation (For a nonzero real $c$, dilation by $c$ multiplies Lebesgue outer measure by $\mathrm{\mid }c{\mathrm{\mid }}^n$, and reflection in the origin preserves it) gives its norm identity, first for simple functions and then by nonnegative approximation. This convention makes all coordinates and spectral shifts in the tail criterion below dimensionless; physical stationary equations retain $x$.

(b) The declared scattering model and its potential classes. As an additional model assumption of the pair, adopt the short-range scattering model. Its declared state space is $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ of clause (a), with the position and momentum families of that clause. A declared short-range potential of the model is a real-valued measurable function $V:{\mathbb{R}}^3\to \mathbb{R}$ satisfying either

1. (compact-support class) $V$ is bounded, $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$, and $V=0$ almost everywhere outside some ball $B_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}:=\{x:\mathrm{\mid }x\mathrm{\mid }\le R_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}\}$ with $R_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}<\mathrm{\infty }$; or
2. (resolvent-tail class) $V=V_2+V_{\mathrm{b}}$ with real $V_2\in L^2({\mathbb{R}}^3)$ and real bounded $V_{\mathrm{b}}$, and, with the dimensionless reduced potential $W_{\ast }(y):=V(L_{\ast }y)\mathrm{/}{E}_{\ast }=W_{\ast ,2}(y)+W_{\ast ,\mathrm{b}}(y)$ and $A_{0,\ast }:=-{\mathrm{\Delta }}_y^{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}$ on $W^{2,2}({\mathbb{R}}_y^3)$, the function $h_{\ast }(r):=\parallel M_{W_{\ast }}(A_{0,\ast }-iI{)}^{-1}{\mathbf{1}}_{\{\mathrm{\mid }y\mathrm{\mid }\ge r\}}\parallel ,r\ge 0,$ is finite for every dimensionless $r$ and belongs to $L^1([0,\mathrm{\infty }),dr)$. Equivalently the physical-coordinate operator is $M_V(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}-i{E}_{\ast }I{)}^{-1}{\mathbf{1}}_{\{\mathrm{\mid }x\mathrm{\mid }\ge L_{\ast }r\}}$: conjugation by $D_{\ast }$ gives exactly the displayed operator. Thus physical cutoffs have radius $L_{\ast }r$, and integration in physical radius $R$ gives ${\int }_0^{\mathrm{\infty }}{h}_{\ast }(R\mathrm{/}{L}_{\ast })dR=L_{\ast }{\int }_0^{\mathrm{\infty }}{h}_{\ast }(r)dr$. The reference length is fixed as part of this unit convention.

The function $V$ is read as a potential energy in joules; the model asserts no derivation of $V$ from P5 or from any force law, and an assignment of a concrete target to $V$ is a tested identification in the sense of clause (iii) of P7 carrying the corresponding uncertainty. The pure Coulomb potential $V(x)=-\kappa \mathrm{/}\mathrm{\mid }x\mathrm{\mid }$, $\kappa >0$, is not in either class: it is a declared long-range model with its own definition and comparison construction on this page.

(c) Assigned energy observables. 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 $H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}:={\mathcal{F}}_2^{\ast }{M}_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_2,D(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}):=\{f\in H_0:\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f\in H_0\}$ be the free operator of that clause with $d=3$ and $b=0$; by The three-dimensional free Schrödinger operator as the weak Laplacian on its Sobolev domain its domain is the Sobolev space $W^{2,2}({\mathbb{R}}^3;\mathbb{C})$ (Integer-order Sobolev spaces and their norms) and $H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}f=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_{w}f$ with ${\mathrm{\Delta }}_w$ the weak Laplacian (Weak derivative of a locally integrable function). For equations in physical coordinates write $A_{0,x}:=-{\mathrm{\Delta }}_x^{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}$ and $W_x(x):=2mV(x)\mathrm{/}{\mathrm{\hslash }}^2$, so $H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}=({\mathrm{\hslash }}^2\mathrm{/}2m)A_{0,x}$. Both $A_{0,x}$ and $W_x$ have inverse-square-length units; this energy division alone is not a spatial nondimensionalization. Stationary items may abbreviate $W_x$ as $W$ (or its central radial profile as $w$), keeping physical $k$ and $r$. The dimensionless operators instead satisfy $D_{\ast }{H}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}{D}_{\ast }^{\ast }=E_{\ast }{A}_{0,\ast },D_{\ast }H{D}_{\ast }^{\ast }=E_{\ast }(A_{0,\ast }+M_{W_{\ast }}),W_{\ast }(y)=L_{\ast }^2{W}_x(L_{\ast }y)\mathrm{.}$ Indeed testing the weak derivative definition after $x=L_{\ast }y$ gives ${\mathrm{\partial }}_y^{\alpha }{D}_{\ast }f=L_{\ast }^{\mathrm{\mid }\alpha \mathrm{\mid }}{D}_{\ast }({\mathrm{\partial }}_x^{\alpha }f)$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le 2$; applying the inverse dilation gives the converse, so $D_{\ast }$ maps the physical Sobolev domain onto the dimensionless one. Multiplication transforms by direct substitution, proving both operator identities on their exact domains. The subscript $w$ in ${\mathrm{\Delta }}_w$ above denotes weak derivatives, not a coordinate or a unit convention. With $M_V$ the maximal multiplication operator, $D(M_V)=\{f\in L^2:Vf\in L^2\}$ (bounded and everywhere defined in the compact-support class), the assigned energy observables of the model with declared potential $V$ are the free operator $H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ and the interacting operator $H:=H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}+M_V,D(H):=D(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}})\mathrm{.}$ The declared energy unit is the joule. For the compact-support class, clause (c) of the cited theorem makes $H$ self-adjoint on $D(H)$ and bounded below by $-\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$. For the resolvent-tail class, Resolvent-tail short-range completeness (Statement and proof step 1.1) applies to $W_{\ast }=W_{\ast ,2}+W_{\ast ,\mathrm{b}}$: $\mathrm{\parallel }{W}_{\ast ,2}{\mathrm{\parallel }}_2=E_{\ast }^{-1}{L}_{\ast }^{-3\mathrm{/}2}\mathrm{\parallel }{V}_2{\mathrm{\parallel }}_2<\mathrm{\infty }$, $\mathrm{\parallel }{W}_{\ast ,\mathrm{b}}{\mathrm{\parallel }}_{\mathrm{\infty }}=E_{\ast }^{-1}\mathrm{\parallel }{V}_{\mathrm{b}}{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$, and its tail hypothesis is exactly clause (b)2. That supplier proves $H_y^2\subseteq D(M_{W_{\ast }})$ and self-adjointness of $A_{0,\ast }+M_{W_{\ast }}$ on $H_y^2=W_y^{2,2}$. The displayed unitary dilation and positive scaling therefore give $D(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}})\subseteq D(M_V)$ and self-adjointness of $H$ on precisely $W_x^{2,2}$, without assuming that $V_2$ is bounded. The model adopts $(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}},\mathrm{J})$ and $(H,\mathrm{J})$ as assigned sharp observables in the sense of Assigned sharp observables and the outcome law of the framework clause (a), and reads their flows by Assigned energy observable, unitary flow and the two pictures of autonomous evolution with $U_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(t)=e^{-it{H}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\mathrm{/}\mathrm{\hslash }}$ and $U(t)=e^{-itH\mathrm{/}\mathrm{\hslash }}$.

(d) Short-range scattering vocabulary. For a declared short-range potential $V$ the model declares the free asymptotic comparison to be the pair of strong limits ${\mathrm{\Omega }}_{\pm }:={\mathrm{s-lim}}_{t\to \pm \mathrm{\infty }}{e}^{itH\mathrm{/}\mathrm{\hslash }}{e}^{-it{H}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\mathrm{/}\mathrm{\hslash }},$ which are declared wave operators of the model when both limits exist, and declares the scattering operator $S:={\mathrm{\Omega }}_{+}^{\ast }{\mathrm{\Omega }}_{-}$ when both exist and ${\mathrm{\Omega }}_{\pm }$ are isometries. The model says that the pair is asymptotically complete for this $V$ when the two ranges agree with the subspace $H_{\mathrm{a}\mathrm{c}}(H):=\{\psi \in H_0:{\mu }_{\psi }^H\ll \lambda \},$ where ${\mu }_{\psi }^H(B):=⟨E_H(B)\psi ,\psi ⟩$ is the scalar spectral measure of the projection valued measure $E_H$ of the self-adjoint operator $H$, and with $H_{\mathrm{c}}(H):=H_{\mathrm{p}\mathrm{p}}(H{)}^{\perp },H_{\mathrm{p}\mathrm{p}}(H):=\stackrel{}{\mathrm{span}}\{\psi \mathrm{\ne }0:H\psi =\lambda \psi \text{ for some }\lambda \in \mathbb{R}\};$ here $H_{\mathrm{p}\mathrm{p}}(H)$ is the closed linear span of the eigenvectors of $H$. An asymptotic free state of a vector $\psi \in H_0$ is a vector ${\psi }_{\mathrm{i}\mathrm{n}}$ such that $\parallel e^{-itH\mathrm{/}\mathrm{\hslash }}\psi -e^{-it{H}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\mathrm{/}\mathrm{\hslash }}{\psi }_{\mathrm{i}\mathrm{n}}\parallel \to 0$ as $t\to -\mathrm{\infty }$; the model reads this limit relation through the wave-operator identity $\psi ={\mathrm{\Omega }}_{-}{\psi }_{\mathrm{i}\mathrm{n}}$ only when the wave operator exists. These are declarations of the model under P4; they construct no preparation, detector or measured count, and no definition of a cross section appears here.

(e) Status and scope. Clause (a) adopts P1, P2, P4, P5, P6 and P7 and fixes the pair's spin-zero one-particle data, mass, action constant and choice hypotheses. Clause (b) further adopts the compact-support and resolvent-tail potential classes as additional pair-level model assumptions; neither class follows from those postulates or from the cited mathematical results. Clause (c) declares the assigned energy operators and their common domain; the cited mathematics establishes self-adjointness where stated, including the resolvent-tail case supplied by the cited mathematical theorem. Clause (d) declares the comparison and spectral vocabulary without asserting that the limits exist or are complete. No identification of $m$, $\mathrm{\hslash }$ or $V$ with measured data is asserted. Existence of the limits, completeness of the ranges and every cross section are separate results of the pair.

• Why two classes. The compact-support class is the minimal physically transparent case in which the pair's completeness theorem is proved; the resolvent-tail class is the wider class in which the same conclusion is proved from an integrable decay hypothesis on the shifted free resolvent tail. The Coulomb potential fails both classes and is treated separately by the long-range definition of this page.
• Names. The symbols ${\mathrm{\Omega }}_{\pm }$, $S$, $H_{\mathrm{a}\mathrm{c}}$, $H_{\mathrm{p}\mathrm{p}}$ and $H_{\mathrm{c}}$ are declared here once and used with exactly this meaning on this page and its companion.