  The research explores a high irregularity, commonly referred to as
intermittency, of the solution to the non-stationary parabolic Anderson
problem: \begin{equation*}
  \frac{\partial u}{\partial t} = \varkappa \mathcal{L}u(t,x) +
\xi_{t}(x)u(t,x) \end{equation*} with the initial condition \(u(0,x) \equiv
1\), where \((t,x) \in [0,\infty)\times \mathbb{Z}^d\). Here, \(\varkappa
\mathcal{L}\) denotes a non-local Laplacian, and \(\xi_{t}(x)\) is a correlated
white noise potential. The observed irregularity is intricately linked to the
upper part of the spectrum of the multiparticle Schr\"{o}dinger equations for
the moment functions \(m_p(t,x_1,x_2,\cdots,x_p) = \langle
u(t,x_1)u(t,x_2)\cdots u(t,x_p)\rangle\).
  In the first half of the paper, a weak form of intermittency is expressed
through moment functions of order $p\geq 3$ and established for a wide class of
operators $\varkappa \mathcal{L}$ with a positive-definite correlator $B=B(x))$
of the white noise. In the second half of the paper, the strong intermittency
is studied. It relates to the existence of a positive eigenvalue for the
lattice Schr\"odinger type operator with the potential $B$. This operator is
associated with the second moment $m_2$. Now $B$ is not necessarily
positive-definite, but $\sum B(x)\geq 0$.
