Asymptotic Analysis - Volume 14, issue 1

Authors: Pettersson, Peter

Article Type: Research Article

Abstract: Let M be a compact, connected C∞ manifold with a C∞ Riemannian metric or let M=Rn with Euclidean metric. Let g denote the metric. On M consider the 2×2 system of Schrödinger operators \[P=-h^{2}\Delta +V+h^{2}\mathcal{W},\qquadV=\left(\begin{array}[cc]v_{1}&0\\0&v_{2}\end{array}\right)\] where v1 ,v2 ∈C∞ (M) are non-negative, and W is a C∞ , first order formally selfadjoint differential operator with real coefficients. We study the eigenfunctions of P corresponding to the lowest eigenvalues in the semi-classical limit h→0. They are concentrated near a minimal geodesic γ with respect to the Agmon metric vg, v = min(v1 , v2 ), …connecting two non-degenerate zeros of v (wells). This metric is Lipschitz continuous but not C2 at $\varGamma =\{x\in M;\ v_{1}(x)=v_{2}(x)\}$ . When the derivative of v1 - v2 along γ does not vanish on $\gamma \cap \varGamma $ and the intersection is transversal we obtain WKB expansions of the eigenfunctions. At $\gamma \cap \varGamma $ they are expressed in terms of derivatives Yk,ε =∂k Yε /∂εk of suitable parabolic cylinder functions Yε . As an application of the WKB constructions we compute the splitting due to tunnelling of the lowest eigenvalues of P under a strong symmetry condition. Show more

DOI: 10.3233/ASY-1997-14101

Citation: Asymptotic Analysis, vol. 14, no. 1, pp. 1-48, 1997

Authors: Dall'Aglio, Andrea | Orsina, Luigi

Article Type: Research Article

DOI: 10.3233/ASY-1997-14102

Citation: Asymptotic Analysis, vol. 14, no. 1, pp. 49-71, 1997

Authors: Jentsch, Lothar

Article Type: Research Article

Abstract: The boundary integral method for solving boundary value problems of elastostatics in domains with interface corners leads to a typical asymptotic behaviour of the potential density near the corner. The solution of the boundary value problem is obtained by evaluation of the representing potential. The singular function follows from decompositions of the potential in the sum of a singular and regular part. After reduction to two normal integrals, namely the logarithmic potential and a bipotential on an interval, decompositions were derived for these potentials with densities occurring in the asymptotics. The singular parts are calculated explicitly for real and complex …singular exponents as well as for densities with power logarithmic terms. For integer-valued singular exponents explicit expressions were found for the logarithmic and the bipotential on an interval. Show more

DOI: 10.3233/ASY-1997-14103

Citation: Asymptotic Analysis, vol. 14, no. 1, pp. 73-95, 1997