In this article we continue the investigations on invariant metrics on a certain class of weakly pseudoconvex domains which we began in [H 1]. While in that paper the differential metrics of Caratheodory and Kobayashi were estimated precisely, the present paper contains a sharp estimate of the singularity of the Bergman kernel and metric on domains belonging to that class.

The low density limit in the Boson Fock case has been investigated in [1] where also the physical meaning and their motivations have been explained (cf. also [0]). From these papers one knows how the number processes can be obtained from a quantum Hamiltonian model via a certain limit procedure.

Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

The question of whether the existence of a harmonic majorant in a relative neighbourhood of each point of a boundary of a domain D implies the existence of a harmonic majorant in the whole of D has received great attention in recent years and has been dealt with by several authors in different settings. The most general results to date have been achieved in [10] with the Martin boundary. In [9], the author arrives, by independent means, at the conclusions of [10] in the particular case where D is a Lipschitz domain.

In this paper, we answer the question in domains with suitably regular topological frontiers. Our methods rely heavily on the possibility of obtaining an extented-representation for nonnegative superharmonic functions defined near a frontier point. This naturally led to the introduction and the study of new types of regularity for the generalised Dirichlet problem. As well as their suitability in dealing with the question of harmonic majorisation, they present an intrinsic importance as natural extensions of the (classical) regularity. For simplicity reasons, we will treat the finite boundary points and the point at infinity separately.

We shall discuss a differentiable invariant that arises when we consider a class of diffeomorphisms having the pseudo orbit tracing property (abbrev. POTP).

In this article a new contribution to the following question is given: Let Ω ⊂ ⊂ Cn be a bounded pseudoconvex domain with C∞-smooth boundary, q ∈ ∂Ω a fixed point and H a k-dimensional affine complex plane such that q ∈ H and H intersects ∂Ω at q transversally. Let U be a suitably small neighborhood of q, and denote by r a C∞-defining function of Ω on U. Under which conditions on ∂Ω near q is it possible to find an exponent η>0 > 0 such that every holomorphic function f on Ω′ = H ∩Ω∩ U with

where dλ′ denotes the Lebesgue-measure on H, can be extended to a holomorphic function ^f on Ω ∩ U such that even

It was observed in [Su5] that the spectrum of a periodic Schrödinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only finitely many. The purpose of this paper is to show, by a slightly different method, that this is also true for general periodic elliptic self-adjoint operators.