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...I think there
is a real physical object with the zero-width boundary. This is a
leading edge (wave-front) of any real perturbation. According to the
Sommerfeld-Brillouin theorem, it always propagates with the invariant
speed. If we have a laser pulse that can be used as a signal,
then it has sharply-defined front with no precursor. The corresponding
wave-function is exactly zero ahead of the pulse and non-zero in the
region of the pulse. The corresponding shape must be described by a
function that is non-analytic - there has to be a discontinuity
starting from derivative of a certain order....
Moses Fayngold,
NJIT