Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schrödinger equations with random potential

The first terms in the asymptotics for $t\to\infty$ of Wiener integrals over the trajectories of $D$-dimensional Brownian motion are derived in the cases when integrated functional has the form $\left<\exp\left\{-\int\limits_0^t q(x(s))\,ds\right\}\right>$ where $q(x)$ is the Gaussian random field or the Poisson field of the form $\sum\limits_j V(x-x_j)$ with showly decreasing positive V(x) or negative $V(x)=(V_0/|x|^\alpha)(1+o(1))$, $|x|\to\infty$, $d<\alpha<d+2$, and $0>\min V(x)=V(0)>-\infty$ respectively. These results are used to obtain asymptotic formulas for density of states on the left end of the spectrum of Schrödinger equation with such random fields as the potentials.