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A quantum particle interacting with a thin solenoid and a magnetic flux is described by a five-parameter family of Hamilton operators, obtained via the method of self-adjoint extensions. One of the parameters, the value of the flux, corresponds to the Aharonov-Bohm effect; the other four parameters correspond to the strength of a singular potential barrier. The spectrum and eigenstates are computed and the scattering problem is solved. Ref. SISSA 171/96/FM

A former conjecture of Burr and Rosta [l], extending a conjecture of Erd6s [2], asserted that in any two-colouring of the edges of a large complete graph, the proportion of subgraphs isomorphic to a fixed graph G which are monochromatic is at least the proportion found in a random eolouring. It is now known that the conjecture fails for some graphs G, including G = Kp for p>4.We investigate for which graphs G the conjecture holds. Our main result is that the conjecture fails if G contains K 4 as a subgraph, and in particular it fails for almost all graphs.

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Coherent states are introduced and their properties are discussed for simple quantum compact groups AI,Bt, Ct and D1. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit. The coherent state is interpreted as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compact R-matrix formulation (generalizing this way the q-deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel-Weil construction) is described using the concept of coherent state. The relation between representation theory and non-commutative differential geometry is suggested.B. Jur6o, P. Siovi~ekThe first papers [1,44], which can be viewed as those generalizing coherent states to quantum groups appeared even before the formal birth of quantum groups [11]. A number of papers followed subsequently ([17] and many others). Nevertheless no definition seems to be completely satisfactory. The coherent states are introduced mainly for the simplest quantum groups (q-deformations of the Heisenberg-Weyl, ~u(2) and ~u(1, 1) algebras) in a rather straightforward way which does not suggest a proper generalization to the more general case. Moreover, these states are assumed to be elements of the representation space for an irreducible representation of the corresponding quantized enveloping algebra and they do not reflect the whole underlying Hopf algebra structure.Recently the representation theory for the algebras of quantum functions on compact groups was studied in great detail. It is worth emphasizing too that these results were obtained with the help of the method of orbits [49,45,46]. This lead finally to a quite general definition of the coherent state given by Soibelman [48] related to generalized Pontryagin duals of simple compact groups. So this is in some sense the case dual to the one we wish to consider in the present paper.According to the general philosophy of non-commutative geometry it would be more natural to view the coherent state as a function on an appropriate q-homogeneous space of the corresponding quantum group (dual to the quantized enveloping algebra) with values in the representation space. We hope that such a more sophisticated generalization of the coherent states method to the case of quantum groups could be of interest not only for the representation theory but also for potential applications of quantum groups in physics. Many important ingredients needed for this generalization are already prepared. First of all the representation theory of quantum groups [15,30,40] and the method of induced representations are well developed [33]. The deformations of manifolds playing an import...