  In the well known Thom's classification, every classical catastrophe is
assigned a Lyapunov function. In the one-dimensional case, due to V. I. Arnold,
these functions have polynomial form $V_{(k)}(x)= x^{k+1} + c_1x^{k-1} +
\ldots$. A natural question is which features of the theory survive when such a
function (say, with an even value of asymptotically dominant exponent $k+1$) is
used as a confining potential in Schr\"{o}dinger equation. A few answers are
formulated. Firstly, it is clarified that due to the tunneling, one of the
possible classes of the measurable quantum catastrophes may be sought in a
phenomenon of relocalization of the dominant part of the quantum particle
density between different minima. For the sake of definiteness we just consider
the spatially even Arnold's potentials and in the limit of the thick barriers
and deep valleys we arrive at a systematic classification of the corresponding
relocalization catastrophes.
