Physical interpretation of the Szegő Green function

Ascertain whether the Szegő Green function G_Szegő(z,a), defined via the relation 4π·K_Szegő(z,a)^2 = -(2/π)·∂^2H_Szegő(z,a)/∂z∂\bar{a} with G_Szegő(z,a) = (1/2π)(-log|z−a| + H_Szegő(z,a)) and constant boundary values on each boundary component of a planar domain, admits a concrete physical interpretation analogous to the electrostatic or hydrodynamic Green functions used in vortex dynamics and electrostatics.

Background

In the discussion of kernels on planar domains and their doubles, the square of the Szegő kernel is positioned between the electrostatic and hydrodynamic Bergman kernels. The authors relate this to a Green-type object, denoted G_Szegő(z,a), which shares the same logarithmic singularity as standard Green functions and is constant on boundary components.

While electrostatic and hydrodynamic Green functions have well-established physical meanings (e.g., as stream functions or potentials with prescribed boundary/circulation conditions), the analogous role for the Szegő-based Green function has not been identified. Establishing whether G_Szegő(z,a) corresponds to any physical system would align it with the existing physical interpretations of other Green functions in this framework.