Positivity of the renormalized two-dimensional strip kernel

Determine whether the renormalized Neumann strip kernel \(G_t\) defined for \(n=2\) is positive for all distinct points in the strip.

Background

For dimensions n≥3n\geq3, the strip kernel is defined by a convergent positive reflection series. In the two-dimensional case, the analogous reflection series diverges and must be renormalized by subtracting a leading asymptotic term.

The authors establish that the renormalized kernel is locally bounded below, which is sufficient for their energy arguments, but explicitly leave its global positivity unresolved.

References

Thus in particular, although we do not know that the kernel is positive when n=2, it is certainly bounded below provided x and y lie in some bounded subset of the strip.

— Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity  (2609.35438 - Clark et al., 28 Sep 2026) in Section 2, Definition “Strip kernel with Neumann boundary conditions”