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- Rene Carmona, Leonid Koralov, Stanislav Molchanov
- 2002

We consider the Stochastic Partial Differential Equation u t = κ∆u + ξ(t, x)u, t ≥ 0, x ∈ Z d. The potential is assumed to be Gaussian white noise in time, stationary in space. We obtain the asymptotics of the almost sure Lyapunov exponent γ(κ) for the solution as κ → 0. Namely γ(κ) ∼ c 0 ln(1/κ) , where the constant c 0 is determined by the correlation… (More)

- Abel Klein, Stanislav Molchanov
- 2005

We give a simple, transparent, and intuitive proof that all eigenvalues of the Anderson model in the region of localization are simple. The Anderson tight binding model is given by the random Hamiltonian H ω = −∆ + V ω on 2 (Z d), where ∆(x, y) = 1 if |x − y| = 1 and zero otherwise, and the random potential V ω = {V ω (x), x ∈ Z d } consists of independent… (More)

- Jürgen Gärtner, Wolfgang König, Stanislav Molchanov
- 2007

We consider the parabolic Anderson problem ∂ t u = ∆u + ξ(x)u on R + × Z d with localized initial condition u(0, x) = δ 0 (x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) lies in the vicinity of, or beyond, the double-exponential distribution , we prove the following geometric characterisation of intermittency: with… (More)

- Vo J K An Jak Si C, Stanislav Molchanov
- 1998

We study spectral properties of the discrete Laplacian H on the half space Z d+1 + = Z d Z + with a boundary condition (nn ;1) = tan(n +)(nn 0), where 2 0 1] d. We d e note by H 0 the Dirichlet Laplacian on Z d+1 +. Whenever is independent o ver ratio-nals (H) = R. Khoruzenko and Pastur KP] have s h o wn that for a set of 's of Lebesgue measure 1, the… (More)

We consider an explicitly solvable model (formulated in the Rie-mannian geometry terms) for a stationary wave process in a specific thin domain Ωε with the Dirichlet boundary conditions on ∂Ωε. The transition from the solutions of the scattering problem on Ωε to the solutions of a problem on the limiting quantum graph Γ is studied. We calculate the… (More)

- Vo J K An Jak Si C, Stanislav Molchanov
- 1999

We study scattering properties of the discrete Laplacian H on the half-space Z d+1 + = Z d Z + with the boundary condition (nn ;1) = tan(n +)(nn 0), where 2 0 1] d. We denote by H 0 the Dirichlet Laplacian on Z d+1 +. Khoruzenko and Pastur KP] have shown that if has typical Diophantine properties then the spectrum of H on R n (H 0) is pure point and that… (More)

- Vo J, Jak Si, Stanislav Molchanov
- 2001

We s tudy spectral properties of the discrete Laplacian H ! on the half space Z 2 + = ZZ + with a random boundary condition (nn ;1) = V ! (n)(nn 0). Here, V ! (n) are independent and identically distributed random variables on a probability space ((F P). We show that outside the interval ;4 4] (the spectrum of the Dirichlet Laplacian) the spectrum of H ! is… (More)

- Vo J K An Jak Si C, Stanislav Molchanov
- 1999

We study spectral properties of the discrete Laplacian H on the half-space Z d+1 + = Z d Z + with random boundary condition (nn ;1) = V (n)(nn 0)) the V (n) are independent random variables on a probability space ((F P) and is the coupling constant. It is known that if the V (n) have densities, then on the interval ;2(d + 1) 2(d + 1)] (= (H 0), the spectrum… (More)

- Vo J K An Jak Si C, Stanislav Molchanov
- 1996

We s tudy spectral properties of random Schrr odinger operators h ! = h 0 +v ! (n) o n l 2 (Z) whose free part h 0 is long range. We p r o ve that the spectrum of h ! is pure point f o r t ypical ! whenever the oo-diagonal terms of h 0 decay a s ji ; jj ; for some > 8.

- G Erard, Ben Arous, Stanislav Molchanov, Alejandro F Ram´irez
- 2007

We describe a universal transition mechanism between annealed and quenched regimes in the context of reaction-diffusion in random media. We study the total population size for random walks which branch and annihilate on Z d , with time-independent random rates. The random walks are independent , continuous time, rate 2dκ, simple, symmetric, with κ ≥ 0. A… (More)