  Within a well-known decay model describing a particle confined initially
within a spherical $\delta$ potential shell, we consider the situation when the
undecayed state has an unusual energy distribution decaying slowly as
$k\to\infty$; the simplest example corresponds to a wave function constant
within the shell. We show that the non-decay probability as a function of time
behaves then in a highly irregular, most likely fractal way.
