Rigorous Diffraction from Two Slits

Give a rigorous explicit solution of the fixed-frequency scalar-wave diffraction problem for two finite slits in the plane, including asymptotics of the solution. This is a broad research direction rather than a fully specified boundary-value problem: the question does not fix the slit geometry or state the incident field and boundary conditions. Deift (2017)

Background

The problem is motivated by Young’s double-slit experiment in optics and its analogue in quantum mechanics. Its classical single-slit precedents are Sommerfeld’s explicit treatment of fixed-frequency scalar-wave diffraction by a semi-infinite slit, described as the half-line $\{x>0\}$ in the plane, and the explicit solution for a single finite slit by Morse and Rubenstein in the 1930s. Deift (2017)

Known results

The established results identified in this account concern the semi-infinite and single finite slit, not the two-slit configuration. Deift states in 2017 that a rigorous solution of diffraction from two slits remains open. He suggests that Fokas’ methods might yield a Riemann–Hilbert formulation, from which relevant asymptotics could follow; this is proposed as a possible approach, not reported as a solution or theorem. Deift (2017)