Asymptotic Analysis - Volume 79, issue 1-2

Authors: BelHadjAli, Hichem | BenAmor, Ali

Article Type: Research Article

Abstract: Let Hβ ± :=−Δ±βδ(|x|=R), be the quantum mechanical Hamiltonian describing a delta interaction living on a sphere of radius R in three dimensions, with strength β>0. Let Dβ ± :=(−Δ+1)−1 −(Hβ ± +1)−1 and D∞ be the strong limit w.r.t. β of the operators Dβ + . For suitable sequence (βn ) we prove uniform convergence of Dβn − towards D∞ with optimal rate, namely βn −1 . Convergence of the above mentioned operators against each other within Schatten–von Neumann ideals Sp , is established as well and the rate of convergence (depending on p) …is determined. At the final step we discuss aspects of differences between negative and positive perturbations at large coupling. Show more

Keywords: rate of convergence, Schatten–von Neumann convergence, uniform convergence

DOI: 10.3233/ASY-2011-1085

Citation: Asymptotic Analysis, vol. 79, no. 1-2, pp. 1-15, 2012

Authors: Hislop, Peter D. | Villegas-Blas, Carlos

Article Type: Research Article

Abstract: We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szegö-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szegö-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non-self-adjoint operators, and second, the Stark perturbation is unbounded so control of the perturbation is …achieved using localization properties of coherent states corresponding to hydrogen atom eigenvalues. Show more

Keywords: hydrogen atom Stark effect, resonance clusters, semiclassical Szegö limit

DOI: 10.3233/ASY-2011-1087

Citation: Asymptotic Analysis, vol. 79, no. 1-2, pp. 17-44, 2012

Authors: Possamaï, Dylan | Soner, H. Mete | Touzi, Nizar

Article Type: Research Article

Abstract: We consider a financial market with liquidity cost as in [Çetin, Jarrow and Protter, Finance and Stochastics 8 (2004), 311–341], where the supply function Sε (s,ν) depends on a parameter ε≥0 with S0 (s,ν)=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, Soner and Touzi [Finance and Stochastics 14(3) (2010), 317–341], of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of ε. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for …the order of the expansion for a European Digital Option. Show more

Keywords: super-replication, liquidity, viscosity solutions, asymptotic expansions

DOI: 10.3233/ASY-2011-1089

Citation: Asymptotic Analysis, vol. 79, no. 1-2, pp. 45-64, 2012

Authors: Hu, Wenqing

Article Type: Research Article

Abstract: We consider a nearly-elastic model system with one degree of freedom. In each collision with the “wall”, the system can either lose or gain a small amount of energy due to stochastic perturbation. The weak limit of the corresponding slow motion, which is a stochastic process on a graph, is calculated. A large deviation type asymptotics and the metastability of the system are also considered.

Keywords: averaging, large deviations, metastability, Markov processes on graphs, random walk

DOI: 10.3233/ASY-2011-1090

Citation: Asymptotic Analysis, vol. 79, no. 1-2, pp. 65-86, 2012

Authors: Du, Zhihua | Du, Jinyuan

Article Type: Research Article

Abstract: In this paper, we study the theory of orthogonal trigonometric polynomials (OTPs). We obtain asymptotics of OTPs with positive and analytic weight functions by Riemann–Hilbert approach and find that they have relations with orthogonal polynomials on the unit circle (OPUC). By the relations and the theory of OPUC, we also get four-terms recurrent formulae, Christoffel–Darboux formula and some algebraic and asymptotic properties of zeros for orthogonal trigonometric polynomials.

Keywords: orthogonal trigonometric polynomials, Riemann–Hilbert approach, orthogonal polynomials on the unit circle, Christoffel–Darboux, recurrence, zeros

DOI: 10.3233/ASY-2012-1096

Citation: Asymptotic Analysis, vol. 79, no. 1-2, pp. 87-132, 2012

Authors: Pauly, Dirk

Article Type: Research Article

Abstract: We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain Ω⊂RN , N≥1, with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a rate r−τ , τ>1, towards the identity. As a canonical application we show that the corresponding eigen-values do not accumulate and that by means of Eidus' limiting absorption principle a Fredholm alternative holds true.

Keywords: Maxwell's equations, exterior boundary value problems, radiating solutions, polynomial and exponential decay of eigen-solutions, variable coefficients, electro-magnetic theory

DOI: 10.3233/ASY-2012-1100

Citation: Asymptotic Analysis, vol. 79, no. 1-2, pp. 133-160, 2012

Authors: Murata, Minoru | Tsuchida, Tetsuo

Article Type: Research Article

Abstract: We obtain, by exploiting Jost functions, asymptotic expansions of Green functions of perturbed Hill operators with respect to the spectral parameter around the minima of the continuous spectra. From the Green function expansions, we derive asymptotic expansions for large time of heat kernels for perturbed Hill operators.

Keywords: long time asymptotics, heat kernel, perturbed Hill operator, Jost function

DOI: 10.3233/ASY-2012-1091

Citation: Asymptotic Analysis, vol. 79, no. 1-2, pp. 161-187, 2012