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A Yukawa tail satisfies the resolvent-tail class while the Coulomb limit does not

Physical scope

The declared short-range scattering model with a Yukawa (screened Coulomb) potential of coupling g>0 and screening parameter mu>0: the explicit split into a compactly supported square-integrable part and a bounded exponentially decaying tail, the verification that the tail satisfies the decaying-tail hypothesis of the resolvent-tail class, hence asymptotic completeness of the model, and the failure of that hypothesis in the unscreened Coulomb limit mu=0, whose treatment belongs to the declared long-range model. Conditional on the declared model and cited mathematics; no measured screening length, coupling or cross section is asserted.

Example

Setup. Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero, action constant $ℏ > 0$ , and declare the Yukawa potential $V_{g, μ} (x) : = - g \frac{e^{- μ ∣ x ∣}}{∣ x ∣} (x \neq 0), g > 0, μ > 0,$ with the origin convention $V_{g, μ} (0) : = 0$ , read in joules, with $g$ in $J m$ and $μ$ in $m^{- 1}$ . Fix the model's reference length $L_{*} > 0$ and write $μ_{*} : = μ L_{*} > 0$ , a dimensionless screening parameter.

Verification

Given: the declared model data and the hypotheses of Asymptotic completeness for the resolvent-tail short-range class, with the decaying-tail example and the explicit exclusion of the Coulomb tail clauses (c), (d).

[F1] Polar integration in dimension three is supplied by Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma; the sphere measure has total mass $4 π$ by Sphere integrals in angle coordinates clause (b). Both suppliers assume Countable Choice, already declared here.

1.1F1givenalgebra

Put $V_{l o c} : = V_{g, μ} 1_{{∣ x ∣ < L_{*}}}$ and $V_{t a i l} : = V_{g, μ} 1_{{∣ x ∣ \geq L_{*}}}$ . The local part is supported in the closed radius- $L_{*}$ ball, and polar integration gives $\int ∣ V_{l o c} ∣^{2} d x = 4 π g^{2} \int_{0}^{L_{*}} e^{- 2 μ r} d r \leq 4 π g^{2} L_{*} < \infty$ . Thus it is compactly supported $L^{2}$ , while the tail is bounded by $g / L_{*}$ , in joules.

2.1step 1.1givenalgebra

For $s = ∣ x ∣ / L_{*} \geq 1$ , $∣ V_{t a i l} (x) ∣ \leq (g / L_{*}) e^{- μ_{*} s}$ . For any dimensionless $δ > 0$ , choose an integer $N > 1 + δ$ ; for $s \geq 1$ , $(1 + s)^{1 + δ} \leq (2 s)^{N}$ , so The exponential dominates every fixed nonnegative integer power at $+ \infty$ at $a = μ_{*}$ and $m = N$ gives $e^{- μ_{*} s} (1 + s)^{1 + δ} \to 0$ . This and continuity on compact intervals give the finite dimensionless constant $C_{δ} : = \sup_{s \geq 0} e^{- μ_{*} s} (1 + s)^{1 + δ}$ . Hence $∣ V_{t a i l} (x) ∣ \leq (g / L_{*}) C_{δ} (1 + ∣ x ∣ / L_{*})^{- 1 - δ}$ on that region; on $∣ x ∣ < L_{*}$ the tail is zero, so the bound holds everywhere.

3.1step 2.1givenalgebra

By 1.1 and 2.1 the declared Yukawa potential has the decaying-tail form of Asymptotic completeness for the resolvent-tail short-range class, with the decaying-tail example and the explicit exclusion of the Coulomb tail clause (c) with any $δ > 0$ ; hence, by that theorem, the model's interacting operator is self-adjoint on $W^{2, 2}$ , its wave operators exist and are asymptotically complete with vanishing singular continuous spectrum, and its scattering operator is unitary.

4.1step 2.1givenalgebra∎

For each $x \neq 0$ , as $μ \to 0^{+}$ the potential $V_{g, μ} (x)$ tends to $- g / ∣ x ∣$ ; assigning any value to the limiting Coulomb potential at the origin gives the same multiplication operator. For that limiting potential, and for every split into a compactly supported part and a remainder, the quotient $(1 + ∣ x ∣ / L_{*})^{1 + δ} ∣ V_{t a i l} (x) ∣ = g (1 + ∣ x ∣ / L_{*})^{1 + δ} / ∣ x ∣$ tends to $+ \infty$ as $∣ x ∣ \to \infty$ for every $δ > 0$ , so the sufficient decay hypothesis of that theorem fails and the Coulomb case is not covered by the short-range class; its treatment is the separate declared long-range model The declared long-range Coulomb model and its Dollard-modified free comparison, for which the theorem supplies no ordinary-comparison statement.

Remarks

What is checked. The example verifies the class hypothesis for the screened potential through explicit integrability and boundedness, and it verifies the failure of the sufficient decay hypothesis in the unscreened limit; it asserts nothing about the actual size of the resolvent tail function $h_{*} (r)$ for the Yukawa potential beyond the class membership.
No measured screening. The screening parameter $μ$ is a declared model parameter, not a measured inverse length, and the limit $μ \to 0^{+}$ is a limit inside the declared family of models.

A Yukawa tail satisfies the resolvent-tail class while the Coulomb limit does not

Example

Verification

Remarks

Depends on

Dependency tree · 2 levels

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