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The Lamb shift is an unique signature of time-independent quantum vacuum fluctuations
and self-interaction in Hydrogen. A novel generalized nonlinear equation of motion is con-
structed with a convolution kernel that can accommodate self-interaction in an essential non—
local manner. The kernel can be constructed out of the time-independent Maxwell vacuum
solutions. The solutions vary with dimension and thus so does the convolution kernel. The
kernel can also be chosen to form a linear equation, the Korteweg—De Vries equation, or the
Gross-Pitaevskii equation. Solutions are sought on the space L2(R), but are also considered
on the space of distributions not necessarily only by trivial set containment. The Maxwell
kernel for d = 1 is linear and thus approximate solutions can be constructed for the nonlinear
equation of motion on compact subsets of R . The equation itself exhibits superlinearity, which
is a critical property of such nonlinear equations of motion. In particular, it will be proved
that there is an unique rotationally invariant distributional solution in d = 3, and for d = 1 it
will be proved that there is in fact an elliptic modular particular solution that admits a precise
algebraic and geometric interpretation of the quantum vacuum.
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