Geodesic
Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schr\"odinger equations with random potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 32 (1977) no. 1, pp. 88-95

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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 $V(x)=(V_0/|x|^\alpha)(1+o(1))$, $|x|\to\infty$, $d\alpha$, 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. 
@article{TMF_1977_32_1_a6,
     author = {L. A. Pastur},
     title = {Behavior of some {Wiener} integrals as $t\to\infty$ and the density of states of {Schr\"odinger} equations with random potential},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {88--95},
     publisher = {mathdoc},
     volume = {32},
     number = {1},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1977_32_1_a6/}
}
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L. A. Pastur. Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schr\"odinger equations with random potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 32 (1977) no. 1, pp. 88-95. http://geodesic.mathdoc.fr/item/TMF_1977_32_1_a6/