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We consider the Laplacian in a strip R(0,d) with the boundary condition which is Dirichlet except at the segment of a length 2a of one of the boundaries where it is switched to Neumann. This operator is known to have a non-empty and simple discrete spectrum for any a>0. There is a sequence 0<a1<a2< of critical values at which new eigenvalues emerge from the continuum when the Neumann window expands. We find the asymptotic behavior of these eigenvalues around the thresholds showing that the gap is in the leading order proportional to (aan)2 with an explicit coefficient expressed in terms of the corresponding threshold-energy resonance eigenfunction.

The present paper which represents the fourth part of the series devoted to analysis of a simple Lee-type model of two-particle decay deals with three problems. The first one concerns relation of the model to the scattering theory. We prove asymptotic completeness for the elastic scattering of the two light particles and show that for a sufficiently weak coupling this system has just one resonance whose position is the same as that of the pole which yield the main contribution to the decay law. The second problem concerns spectral concentration; we prove its occurrence for families of intervals around<i>E</i> that shrink slower than quadratically in<i>g</i>. Finally, necessary and sufficient conditions for the existence of bound states are discussed.

It has been a successful practice to define a canonical pre-ordering on a normed space using the inclusion of faces of its closed dual unit ball. This pre-ordering reflects some geometric property in a natural way. In this article, we will give an algebraic description of this pre-ordering in the case of complex C*-algebras as well as that of their self-adjoint parts. In developing our theory we introduce the essential support of an element, which is closely related to the notion of peak projections studied recently by Blecher and Hay. As applications, we give some interesting facts about weak*-closed faces, and will identify the quasi-maximal elements and the quasi-minimal elements with respects to this pre-ordering. They are closely related to the extreme points and the smooth points of the unit sphere of the C*-algebra.