Asymptotic Analysis - Volume 74, issue 1-2

Authors: Akkouche, Sofiane

Article Type: Research Article

Abstract: Let H(λ)=−Δ+λb be the combinatorial Schrödinger operator on an infinite connected graph G with a potential b and a non-negative coupling constant λ. When b≡0, it is well known that σ(−Δ)⊂[0,2]. When b≢0, let s(−Δ+λb):=inf σ(−Δ+λb) and M(−Δ+λb):=sup σ(−Δ+λb) be the bounds of the spectrum of the Schrödinger operator. One of the aims of this paper is to study the influence of the potential b on the bounds of the spectrum of −Δ. More precisely, we give a condition on the potential b such that s(−Δ+λb) is strictly positive for λ small enough. We obtain a similar result for the top …of the spectrum. We also prove that for a recurrent bipartite weighted graph, the only Schrödinger operator H=−Δ+b such that σ(H)⊂[0,2] is the Laplacian −Δ. We study independently the asymptotic behaviour of the function λ↦s(−Δ+λb) for non-negative potentials b. We prove that s(∞):=lim λ→+∞ s(−Δ+λb)<+∞ if and only if inf x∈V b(x)=0 and we characterize this limit in the general case. As an application, we give an exact value of this limit for certain Jacobi matrices. Show more

Keywords: Schrödinger operators, spectrum, bounds, spectral functions, weighted graphs

DOI: 10.3233/ASY-2011-1041

Citation: Asymptotic Analysis, vol. 74, no. 1-2, pp. 1-31, 2011

Authors: Dobrokhotov, Sergey | Rouleux, Michel

Article Type: Research Article

Abstract: We extend to the semi-classical setting the Maupertuis–Jacobi correspondence for a pair of Hamiltonians (H(x,hDx ),ℋ(x,hDx )). If ℋ(x,p) is completely integrable, or has merely an invariant Diophantine torus Λ in energy surface {ℋ=ℰ}, then we can construct a family of quasi-modes for H(x,hDx ) at the corresponding energy E. This applies in particular to the linear theory of water waves, and determines trapped modes by an island, from the knowledge of Liouville metrics.

Keywords: Maupertuis principle, quasi-periodic Hamiltonian flows, invariant tori, Birkhoff normal form, Liouville metrics, Maslov theory, linear water waves theory

DOI: 10.3233/ASY-2011-1045

Citation: Asymptotic Analysis, vol. 74, no. 1-2, pp. 33-73, 2011

Authors: Goto, Yukie

Article Type: Research Article

Abstract: In this article, a rigorous mathematical treatment of the dryland vegetation model introduced by Gilad et al. [Phys. Rev. Lett. 98(9) (2004), 098105-1–098105-4, J. Theoret. Biol. 244 (2007), 680–691] is presented. We prove the existence and uniqueness of solutions in (L1 (Ω))3 and the existence of global attractors in L1 (Ω;𝒟), where 𝒟 is an invariant region for the system. A key step is the regularization of the model by adding εΔ to the diffusion term and by approximating the initial data U0 by a sequence {U0,n } of smooth functions in (L1 (Ω))3 . The various a …priori estimates and the maximum principle permit the passage to the limit as ε→0 and n→∞, proving the existence and uniqueness of solutions U in the specified space. Also, we deduce from the a priori estimates that the solution meets the necessary hypotheses (see Theorem 1.1 in Chapter 1 of Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1997) and hence, we obtain the existence of global attractors. Show more

Keywords: desertification, dryland vegetation, parabolic equations, degenerate parabolic equations, porous media equations, regularizations, regularization effect, compactness theorems, maximal attractors

DOI: 10.3233/ASY-2011-1046

Citation: Asymptotic Analysis, vol. 74, no. 1-2, pp. 75-94, 2011

Authors: Bezrodnykh, S.I. | Demidov, A.S.

Article Type: Research Article

Abstract: The inverse Cauchy problem Δu(x)=au(x)+b≥0  for x∈ω,  and  u=0,   ∂u/∂ν=Φ   on γ is shown as having a unique solution within a wide class of simply connected domains ω⋐R2 with smooth boundary γ. Here a and b are real numbers to be determined, and Φ is a given function which is normalized by the condition ∫γ Φ ds=1.

Keywords: inverse Cauchy problem, Grad–Shafranov equation, Vishik–Lyusternik's method, multipole method

DOI: 10.3233/ASY-2011-1047

Citation: Asymptotic Analysis, vol. 74, no. 1-2, pp. 95-121, 2011