Asymptotic Analysis - Volume 13, issue 2

Authors: Sordoni, Vania

Article Type: Research Article

Abstract: In this paper we study the bottom of the spectrum of a semiclassical Schrödinger operator Pm (h)=−(h2 /2)Δm +Vm in high dimension m. We assume that Vm is convex and satisfies some conditions uniformly with respect to m. We get a complete asymptotic expansion in powers of h with an explicit control of the coefficients and of the remainder terms with respect to m of its lowest eigenvalue and we show that its first eigenfunctions decays exponentially outside a ℓ2 -ball of radius $\sqrt{m}$ centered at the point where Vm reaches its minimum, as h→0.

DOI: 10.3233/ASY-1996-13201

Citation: Asymptotic Analysis, vol. 13, no. 2, pp. 109-129, 1996

Authors: Sordoni, Vania

Article Type: Research Article

Abstract: In this paper we study the bottom of the spectrum of the semiclassical anharmonic oscillator Pm (h)=−h2 Δm +Vm (x) where Vm (x)=μΣj=1 m x2 j +((g)/(mn−1 ))(Σj=1 m xj 2 )n , μ∈R, g∈R+ and n∈N, n>1, when the number m of interacting particles is large. Denoting by λ(m,h) its lowest eigenvalue, we prove that lim m→+∞ λ(m,h)/m exists and has a complete asymptotic expansion in powers of h, when the Planck's constant h tends to 0. For h fixed, we also obtain an expansion in powers of m−1 for the first eigenvalues of Pm . Moreover, we …consider integrals of the form I(β,m)=∫Rm =e−βVm (x)  dx where β is a large parameter and we prove the existence of the limit, as m→+∞, of the quantity (1/m)ln I(β,m) and that this limit has an asymptotic expansion in power of β−1 for large values of β. Show more

DOI: 10.3233/ASY-1996-13202

Citation: Asymptotic Analysis, vol. 13, no. 2, pp. 131-166, 1996

Authors: Dauge, Monique | Gruais, Isabelle

Article Type: Research Article

Abstract: This paper is the first of a series of two, where we study the asymptotics of the displacement in a thin clamped plate made of a rigid “monoclinic” material, as the thickness of the plate tends to 0. The combination of a polynomial Ansatz (outer expansion) and of a boundary layer Ansatz (inner expansion) yields a complete multi-scale asymptotics of the displacement and leads to optimal error estimates in energy norm. We investigate the polynomial Ansatz in Part I, and the boundary layer Ansatz in Part II. If ε denotes the small parameter in the geometry, we first construct …the algorithm for an infinite “even” Ansatz involving only even powers of ε, which is a natural extension of the usual Kirchhoff-Love Ansatz. The boundary conditions of the clamped plate being only satisfied at the order 0, we try to compensate for them by boundary layer terms: we rely on a result proved in Part II giving necessary and sufficient conditions for the exponential decay of such terms. In order to fulfill these conditions, the constructive algorithm for the boundary layer terms has to be combined with an “odd” polynomial Ansatz. The outcome is a two-scale asymptotics involving all nonnegative powers of ε, the in-plane space variables xα , the transverse scaled variable x3 and the quickly varying variable r/ε where r is the distance to the clamped part of the boundary. Show more

DOI: 10.3233/ASY-1996-13203

Citation: Asymptotic Analysis, vol. 13, no. 2, pp. 167-197, 1996

Authors: Comte, Myriam | Mironescu, Petru

Article Type: Research Article

DOI: 10.3233/ASY-1996-13204

Citation: Asymptotic Analysis, vol. 13, no. 2, pp. 199-215, 1996