Asymptotic Analysis - Volume 5, issue 3

Authors: Gohberg, I. | Kaashoek, M.A.

Article Type: Research Article

Abstract: A generalization of Szegö's first limit theorem is deduced for a general class of block Toeplitz operators with Hilbert–Schmidt entries. Analogous formulas for Wiener–Hopf integral operators are derived, and by discretization a direct connection between the two types of asymptotics is established. For the continuous case the results are specified further for operators with a rational matrix symbol.

DOI: 10.3233/ASY-1992-5301

Citation: Asymptotic Analysis, vol. 5, no. 3, pp. 187-220, 1992

Authors: März, Christoph

Article Type: Research Article

Abstract: We study the spectrum of a one-dimensional semiclassical periodic Schrödinger operator near the potential maximum. The potential is supposed to be analytic and to have only one point of maximum over the period, which in addition is assumed to be non-degenerate. The spectrum is the union of bands, and some energy value is belonging to the spectrum, if and only if the trace of the operator of translation by the period acting on the two-dimensional space of solutions of the corresponding stationary Schrödinger equation lies in [−2,2]. By application of a result by Helffer and Sjöstrand we can take …the dilation generator as a model near the potential maximum, and by application of a Fourier integral operator we find local solutions that have a standard WKB-form away from the point of maximum. So we may extend them over the whole period in order to compute the asymptotics of the trace of the translation matrix such that we finally obtain the size of the bands and the gaps separating them. Show more

DOI: 10.3233/ASY-1992-5302

Citation: Asymptotic Analysis, vol. 5, no. 3, pp. 221-267, 1992

Authors: Finzi Vita, Stefano | Tchou, Nicoletta Anna

Article Type: Research Article

Abstract: We consider the asymptotic behaviour of the solutions of a sequence of elliptic problems with homogeneous Dirichlet boundary conditions, involving singular potentials which can be Borel measures with infinite values. Problems of this kind, introduced in (Dal Maso and Mosco, 1986) and (Dal Maso and Mosco, 1987) are called ‘relaxed Dirichlet problems’ since they form the smallest family of equations, stable under the convergence of solutions in L2 , which includes Dirichlet problems with zero boundary conditions on many small holes (see also (Cionarescu and Murat, 1982) and (Dal Maso, 1987)). After extraction of a subsequence, the solutions converge …weakly in H1 0 (Ω) to the solution of a limit problem of the same type. We determine a corrector, i.e. an explicit expression constructed from the limit of the solutions and from some generalized capacitary potentials, the difference of which with the solution tends strongly to zero in H1 0 (Ω). Under suitable regularity assumptions on the limit problem, we are able to prove a strong convergence results and an abstract error estimate. Moreover, we prove in the general case that there is no need of correctors in the singular set of the measure of the limit problem. Show more

DOI: 10.3233/ASY-1992-5303

Citation: Asymptotic Analysis, vol. 5, no. 3, pp. 269-281, 1992