A Non-Reciprocal Elliptic Spectral Solution of the Right-Angle Penetrable Wedge Transmission Problem for $\nu=\sqrt{2}$

Abstract

We consider the two-dimensional time-harmonic transmission problem for an impedance-matched (\rho = 1) right-angle penetrable wedge at refractive index ratio \nu = \sqrt{2}, in the integrable lemniscatic configuration (\theta_w ,\nu,\rho) = (\pi/4,\sqrt{2},1). Starting from Sommerfeld spectral representations, the transmission conditions on the two wedge faces yield a closed spectral functional system for the Sommerfeld transforms Q(\zeta) and S(\zeta). In this special configuration the associated Snell surface is the lemniscatic curve Y^2 = 2(t^4 + 1), uniformized by square-lattice Weierstrass functions with invariants (g_2,g_3) = (4,0). We construct an explicit meromorphic expression for a scattered transform Q_{scat} as a finite Weierstrass--\zeta sum plus an explicitly constructed pole-free elliptic remainder, with all pole coefficients computed algebraically from the forcing pole set. A birational (injective) uniformization is used to avoid label collisions on the torus and to make the scattered-allocation pole exclusion well posed. The resulting closed form solves the derived spectral functional system and satisfies the local regularity constraints imposed at the physical basepoint. However, numerical reciprocity tests on the far-field coefficient extracted from Q_{scat} indicate that the construction is generally non-reciprocal; accordingly we do not claim that the resulting diffraction coefficient coincides with the reciprocal physical transmission scattering solution. The result remains restricted to this integrable lemniscatic case; the general penetrable wedge remains challenging (see [10--12] and, in a related high-frequency penetrable-corner setting, [13]).