Proposition: Finite positive matrices: large and small radius

Suppose $W={\mathbb{C}}^2$ and $H>0$. Then

with a termwise asymptotic expansion. The leading azimuthal-current terms are obtained by weighting the summands by $-e(q_j+q_k)$. The off-diagonal power is $r^{-2-1/\alpha }$.

For every full finite $H\ge 0$, the small-radius leading rescaled observables are those of $({\mathsf{E}}_0,{\mathsf{E}}_0)$: $r^2\delta F$, $r^2\delta j_{\phi }$, $r^3\delta j_r$, and $r^4\delta T$ in an orthonormal frame have the corresponding limits.

Proof. At small $p$, positive invertibility gives

Set $x=pr$ and use

Gamma reflection and recurrence give the displayed coefficient. A low/high momentum split, the Neumann expansion at small momentum, and exponential decay at high momentum justify the asymptotic integration and the derivative expansions.

At large $p$,

for every full finite $H$. After $p=x/r$, Lemma lem:Cbound supplies an integrable dominating function for each rescaled observable, including its required derivatives. Dominated convergence gives the two-mode ${\mathsf{E}}_0$ limits.