Necessity of the Rollnik-plus-L2 condition

Determine whether, for a potential V∈\mathcal{R}+L_\varepsilon^\infty, membership V∈\mathcal{R}+L^2 is necessary for the finiteness of the double integral ∫_{\mathbb{R}^3}∫_{\mathbb{R}^3}|V(x)|e^{-2m|x-y|}|x-y|^{-2}|V(y)|\,dx\,dy.

Background

Lemma \ref{lemma for T_{m,V} eq if R+L2} establishes that the displayed double integral is finite whenever V belongs to \mathcal{R}+L2. This estimate is then used to prove Hilbert–Schmidt properties needed for equivalence of the Weyl representations associated with two potentials whose difference lies in \mathcal{R}+L2.

The authors explicitly leave unresolved whether the sufficient condition V∈\mathcal{R}+L2 is also necessary under the broader standing assumption V∈\mathcal{R}+L_\varepsilon\infty.

References

Under the assumption $V\in\mathcal{R}+L_\varepsilon\infty,$ the authors do not know whether the condition $V\in\mathcal{R}+L2$ is necessary for int of lemma for T_{m,V eq if R+L2} to hold.

int of lemma for T_{m,V:

$eq if R+L^2} \int_{\mathbb{R}^3}\int_{\mathbb{R}^3}|V(x)|\frac{\mathrm{e}^{-2m|x-y|}}{|x-y|^2}|V(y)|\,\mathrm{d}x\,\mathrm{d}y<\infty. $

— Representation Theory of Canonical Commutation Relations Arising from the Quantization of the Klein-Gordon Equation with an External Potential  (2609.05237 - Matsuzawa et al., 4 Sep 2026) in Remark immediately following Lemma \ref{lemma for T_{m,V} eq if R+L^2}, Section 5, subsection “Quantization of Klein-Gordon equation with an external potential”

It remains unclear to the authors whether Theorem \ref{T_{m,V} ineq}, which holds for potentials in $L2 + L_\varepsilon\infty$, also extends to potentials in $\mathcal{R} + L_\varepsilon\infty$.

— Representation Theory of Canonical Commutation Relations Arising from the Quantization of the Klein-Gordon Equation with an External Potential  (2609.05237 - Matsuzawa et al., 4 Sep 2026) in Remark immediately following the proof of Theorem \ref{T_{m,V} ineq}, Section 5, subsection “Quantization of Klein-Gordon equation with an external potential”