An exercise in making connections between classical optics and
quantum mechanics in 2D. Based on a separable, exactly solvable
example composed of square well and P\"{o}schl-Teller potential and
considered at certain critical couplings called, not quite
adequately, ``quantized". The existence of parallels between quantum
bound states at threshold energy (this is the proper meaning of the
puzzling ``zero energy" in the title) and classical optical systems
called ``Maxwell's" (and allowing the rays intersect at two points)
is discussed at length. The use is made of the changes of variables
which are related to conformal mappings in 2D.

MR2563310 Makowski, Adam J. Exact, zero-energy, square-integrable
solutions of a model related to the Maxwell's fish-eye problem. Ann.
Physics 324 (2009), no. 12, 2465--2472. 81Q05 (81Rxx)