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 TMF, 1977, Volume 32, Number 1, Pages 88–95 (Mi tmf3138)

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

L. A. Pastur

Abstract: 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 $<\exp\{-\int\limits_0^t q(x(s)) ds\}>$ 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.

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English version:
Theoretical and Mathematical Physics, 1977, 32:1, 615–620

Bibliographic databases:

Citation: L. A. Pastur, “Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schrödinger equations with random potential”, TMF, 32:1 (1977), 88–95; Theoret. and Math. Phys., 32:1 (1977), 615–620

Citation in format AMSBIB
\Bibitem{Pas77}
\by L.~A.~Pastur
\paper Behavior of some Wiener integrals as $t\to\infty$ and the density of states of Schr\"odinger equations with random potential
\jour TMF
\yr 1977
\vol 32
\issue 1
\pages 88--95
\mathnet{http://mi.mathnet.ru/tmf3138}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=449356}
\zmath{https://zbmath.org/?q=an:0353.60053}
\transl
\jour Theoret. and Math. Phys.
\yr 1977
\vol 32
\issue 1
\pages 615--620
\crossref{https://doi.org/10.1007/BF01041435}

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. M. Kozlov, “Averaging of random operators”, Math. USSR-Sb., 37:2 (1980), 167–180
2. S. M. Kozlov, “The method of averaging and walks in inhomogeneous environments”, Russian Math. Surveys, 40:12 (1985), 73–145
3. V. R. Fatalov, “Some asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 157:2 (2008), 1606–1625
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