Asymptotic Analysis - Volume 63, issue 1-2

Authors: Helffer, Bernard | Pankrashkin, Konstantin

Article Type: Research Article

Abstract: The two-dimensional Schrödinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields and a weak coupling we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like operators. This shows the existence of parts of Cantor structure in the spectrum for special values of the magnetic flux.

Keywords: Schrödinger operator, periodic perturbation, Cantor spectrum, magnetic field, zero-range potential

DOI: 10.3233/ASY-2008-0923

Citation: Asymptotic Analysis, vol. 63, no. 1-2, pp. 1-27, 2009

Authors: Baraket, Amel Atallah | Mechergui, Chokri

Article Type: Research Article

Abstract: We study the microlocal analytic smoothing effect for the solutions of the Schrödinger equation associated with the harmonic oscillator using the method of Martinez, Nakamura and Sordoni (Comm. Pure Appl. Math. 59 (2006), 1330–1351) of microlocal weights estimates.

Keywords: FBI transform, propagation of singularities, Schrödinger equation, wave front set

DOI: 10.3233/ASY-2008-0924

Citation: Asymptotic Analysis, vol. 63, no. 1-2, pp. 29-48, 2009

Authors: Zakharov, Sergei V.

Article Type: Research Article

Abstract: The Cauchy problem for a quasi-linear parabolic equation with a small parameter ε at the higher derivative is considered. The derivative of the initial function is of order O(1/ρ), where ρ is another small parameter. We study the asymptotics of the solution of the problem in parameters ε and ρ tending to zero.

Keywords: Cauchy problem, two-parameter asymptotics

DOI: 10.3233/ASY-2008-0927

Citation: Asymptotic Analysis, vol. 63, no. 1-2, pp. 49-54, 2009

Authors: Grasselli, Maurizio | Muñoz Rivera, Jaime E. | Squassina, Marco

Article Type: Research Article

Abstract: We consider a coupled linear system describing a thermoviscoelastic plate with hereditary effects. The system consists of a hyperbolic integrodifferential equation, governing the temperature, which is linearly coupled with the partial differential equation ruling the evolution of the vertical deflection, presenting a convolution term accounting for memory effects. It is also assumed that the thermal power contains a memory term characterized by a relaxation kernel. We prove that the system is exponentially stable and we obtain a closeness estimate between the system with memory effects and the corresponding memory-free limiting system, as the kernels fade in a suitable sense.

Keywords: thermoviscoelastic plate, memory effects, singular limit, exponential decay

DOI: 10.3233/ASY-2008-0928

Citation: Asymptotic Analysis, vol. 63, no. 1-2, pp. 55-84, 2009

Authors: Paoletti, Roberto

Article Type: Research Article

Abstract: A Weyl law for Toeplitz operators was proved by Boutet de Monvel and Guillemin for general Toeplitz structures. In the setting of positive line bundles, we revisit this theme in light of local asymptotic techniques based on the microlocal theory of the Szego kernel. By pairing this approach with classical arguments used to estimate the spectral function of a pseudodifferential operator, we first establish a local Weyl law (that is, a pointwise estimate on the spectral function of the Toeplitz operator). The global Weyl law follows by integration.

Keywords: Toeplitz operators, Weyl law, positive line bundle, Szegö kernel, Fourier integral operator

DOI: 10.3233/ASY-2008-0929

Citation: Asymptotic Analysis, vol. 63, no. 1-2, pp. 85-99, 2009

Authors: Shi, Wei | Wong, R.

Article Type: Research Article

Abstract: In this paper, we use a new method to derive hyperasymptotic expansions for the modified Bessel function of the third kind of purely imaginary order. Our approach is based on the method of Hadamard-series expansion developed by Paris (Proc. Roy. Soc. Lond. A 457 (2001), 2855–2869). Modifications are made to deal with the presence of an infinite number of saddle points, and numerical computation is used to illustrate the accuracy of our results.

Keywords: modified Bessel function of the third kind of purely imaginary order, hyperasymptotic expansion, Hadamard series

DOI: 10.3233/ASY-2008-0930

Citation: Asymptotic Analysis, vol. 63, no. 1-2, pp. 101-123, 2009