Uncertainty principle for operators commuting with translations. II

Continuation of the authors' paper (RZhMat., 1980, 4B820). Convolution operators with semirational symbols (s.s.) are studied. Uniqueness theorems are proved for logarithmic potentials, as well as compatibility theorems for pairs of equations where K is a kernel with s.s., E is a sufficiently “sparse” subset of the line, f is an “unknown” function. Versions are considered of the “two constants theorem” of Hadamard, relating to uniqueness properties of operators with s.s.     

Deformation of CR-manifolds,minimal points and CR-manifolds with the microlocal analytic extension property

LetM be a smooth CR-manifold embedded into ? ⁿ . Letp be a point inM and letC be a small truncated cone inM (in suitable Euclidean coordinates onM) with vertexp which “symmetry axis” is a real vector in the complex tangent space. Then one can deformM into a smooth CR-manifoldM _d letting fixed all points outsideC in such a way thatp is a minimal point ofM _d . This result is used to give a new proof of the fact that wedge extendability of continuous CR-functions propagates along the CR-orbits of a CR-manifold. It allows also to prove the following natural result which was conjectured by Trepreau. LetM be a smooth generic CR-manifold in ? ⁿ . SupposeM consists of one single CR-orbit. Then each continuous CR-function onM is wedge extendable at any point ofM. Uniqueness theorems for continuous CR-functions are derived.     

Experimental study of high spin states in the ground state bands of158,160Er,164,166Yb,and168Hf

Ground state rotational bands in the deformed doubly-even rare earth nuclei^158,160Er,^164,166Yb, and¹⁶⁸Hf have been observed in (α, 8n γ) reactions. The γ-spectra associated with these reactions were studied in-beam using conventional spectroscopic methods. In all five nuclei the nuclear moment of inertia of the ground state rotational states was found to increase abruptly as higher spin states were attained. In a plot of the moment of inertia as a function of the angular velocity all these five nuclei display “backbending” curves. Moreover, in¹⁵⁸Er and¹⁶⁶Yb, the curves after passing through a maximum bend subsequently downwards.     

Observation of back- and forwardbending in the182Os ground-state band

From a study of theγ-decay of high-spin states populated using the (α, 8n) reaction, the ground-state rotational band in¹⁸²Os has been established up to spin 20⁺. A plot of the moment of inertia as a function of the square of the rotational frequency for this ground-state band results in a back-and-forwardbending curve, indicating that an explanation in terms of pure Coriolis decoupling in thei _13/2 neutron orbital requires revision. The moment of inertia reaches a maximum of only 74% of the rigid rotor value.     

DWBA and CCBA analysis of selected transitions in the C(d,τ) B reaction at Ed = 80 MeV

Angular distribution of the reaction ¹²C(d,τ) ¹¹B leading to the 3/2⁻ ground state and the 4.44 MeV excited (5/2⁻ state ¹¹B have measured at E_d = 80 MeV up to about 70° and 40°, respectively. The inclusion of second-order processes in the framework of CCBA calculations gives a good reproduction of the 3/2⁻ as well as of the DWBA forbidden 3/2⁻ angular distribution. The strong dependence of the cross section on the sign of the deformation supports a positive β₂ value for ¹¹B.     

10+ isomers observed in even-even 78-neutron nuclei

Isomeric states with spinI ^π=10⁺ were found in the nuclei¹³⁸Nd and¹⁴⁰Sm. These states are interpreted as having a [vh _11/2]^?2 configuration. The systematical behaviour of these isomers in the isotones¹³⁶Ce,¹³⁸Nd and¹⁴⁰Sm is discussed.     

Observation of anomalous behaviour of the nuclear moment of inertia at large rotational frequencies in Yb and Hf

The (α, 8n) reaction has been used to produce high-spin states in ¹⁶⁴Yb and ¹⁶⁸Hf. In both cases a sharp increase of the moment of inertia as a function of rotational frequency was observed at I = 12.     

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the Continuity Principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.     

The pluripolar hull of a graph and fine analytic continuation

We show that if the graph of an analytic function in the unit disk D is not complete pluripolar in C² then the projection of its pluripolar hull contains a fine neighborhood of a point . Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function f in D extends to a function ℱ which is defined on a fine neighborhood of a point and is finely analytic at p then the pluripolar hull of the graph of f contains the graph of ℱ over a smaller fine neighborhood of p. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them C^∞ functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.