On the nature of near-threshold bound and virtual states

Physical states are characterised uniquely by their pole positions and the corresponding residues. Accordingly, in those parameters also the nature of the states should be encoded. For bound states (poles on the real s-axis below the lowest threshold on the physical sheet) there is an established criterion formulated originally by Weinberg in the 1960s, which allows one to estimate the amount of compact and molecular components in a given state. We demonstrate in this paper that this criterion can be straightforwardly extended to shallow virtual states (poles on the real s-axis below the lowest threshold on the unphysical sheet) which should be classified as molecular. We argue that predominantly non-molecular or compact states exist either as bound states or as resonances (poles on the unphysical sheet off the real energy axis) but not as virtual states. We also discuss the limitations of the mentioned classification scheme.

Dynamic Form of Static Dispersion Relations and Projective Spaces

The S-matrix in the static limit of a dispersion relation has a finite order N and is a matrix of meromorfic functions of energy ω in the plane with cuts (-∞, -1],[+1, = ∞). In the elastic case it reduces to N functions Si(ω) conncted by the crossing symmetry matrix A. The problem of analytical continuation of Si(ω) from the physical sheet to unphysical ones can be treated as a nonlinear system of difference equations. It is shown that a global analysis of this system can be carried out effectively in projective spaces PN and PN+1. The connection between spasec PN and PN+1 is discussed.

Complex Singularities in Production Amplitudes

Complex singularities in the ",+N --? "'+ ",+N amplitude corresponding to vertex contractions of the single-loop Feynman diagram exist on the physical sheet of the amplitude irrespective of whether the theory is formulated in terms of timeordered or retarded products_

Some analytic properties of three-pion decay amplitudes

It is shown for all graphs of the ‘strip approximation’ that the K ->3n decay amplitude obeys a Mandelstam double-dispersion relation. The double spectral function turns out to be complex and covers the whole physical domain, so that the Cini-Fubini approximation is no longer valid. Essential singularities are found on the border of the physical sheet. Double-dispersion relations in one energy and in the K mass are also derived.