  Recently, the possibility of so-called twinlike field theories has been
demonstrated, that is, of different field theories which share the same
topological defect solution with the same energy density. Further, purely
algebraic conditions have been derived which the corresponding Lagrangians have
to obey in order that the field theories be twins of each other. A further
diagnostical tool which, in general, allows to distinguish the topological
defects of a given theory from the corresponding defects of its twins is the
spectrum of linear fluctuations about these defects. Very recently, however,
explicit examples of twin theories have been constructed such that not only
their shapes and energy densities coincide, but also their linear fluctuation
spectra are the same. Here we show that, again, there exist purely algebraic
conditions for the Lagrangian densities which imply that the corresponding
field theories are twins and that the fluctuation spectra about their defects
coincide. These algebraic conditions allow to construct an infinite number of
twins with coinciding fluctuation spectra for a given theory, and we provide
some explicit examples. The importance of this result is related to the fact
that coinciding defects with coinciding energy densities and identical
fluctuation spectra are almost indistinguishable physically, that is,
indistinguishable in a linear or semiclassical approximation. This implies that
the measurable physical properties of a kink, in general, do not allow to
determine the theory which provides the kink uniquely. Instead, in principle an
infinite number of possible theories has to be considered.
