Geodesic
Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schrödinger equations with random potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 32 (1977) no. 1, pp. 88-95 
L. A. Pastur. Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schrödinger 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/
@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},
     year = {1977},
     volume = {32},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1977_32_1_a6/}
}
TY  - JOUR
AU  - L. A. Pastur
TI  - Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schrödinger equations with random potential
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1977
SP  - 88
EP  - 95
VL  - 32
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1977_32_1_a6/
LA  - ru
ID  - TMF_1977_32_1_a6
ER  - 
%0 Journal Article
%A L. A. Pastur
%T Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schrödinger equations with random potential
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1977
%P 88-95
%V 32
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1977_32_1_a6/
%G ru
%F TMF_1977_32_1_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] M. Donsker, S. Varadhan, Commun. Pure and Appl. Mathem., 28 (1975), 525 | DOI | MR | Zbl

[2] M. Kats, Veroyatnost i smezhnye voprosy v fizike, «Mir», 1965 | Zbl

[3] D. Ryuel, Statisticheskaya mekhanika, «Mir», 1971

[4] S. M. Rytov, Vvedenie v statisticheskuyu radiofiziku, «Nauka», 1965

[5] L. A. Pastur, Funkts. analiz i ego prim., 6 (1972), 93 | MR | Zbl

[6] L. A. Pastur, TMF, 6 (1971), 415 | MR | Zbl

[7] V. S. Videnskii, Matem. sb., 62 (1963), 121 | MR | Zbl

[8] I. M. Lifshits, UFN, 83 (1964), 617 | DOI | Zbl