Abstract: We perform an analytical study of the scattering matrix and bound states in problems with many physical coupled channels. We establish the relationship of the couplings of the states to the different channels, obtained from the residues of the scattering matrix at the poles, with the wave functions for the different channels. The couplings basically reflect the value of the wave functions around the origin in coordinate space. In the concrete case of the X(3872) resonance, understood as a bound state of D-0(D) over bar*(0) and D+D*(-) (and c.c. From now on, when we refer to D-0(D) over bar*(0), D+D*(-), or D (D) over bar* we are actually referring to the combination of these states with their complex conjugate in order to form a state with positive C-parity), with the D-0(D) over bar*(0) loosely bound, we find that the couplings to the two channels are essentially equal leading to a state of good isospin I = 0 character. This is in spite of having a probability for finding the D-0(D) over bar*(0) state much larger than for D+D*(-) since the loosely bound channel extends further in space. The analytical results, obtained with exact solutions of the Schrodinger equation for the wave functions, can be useful in general to interpret results found numerically in the study of problems with unitary coupled channels methods.

Abstract: Charged particle therapy, or so-called hadrontherapy, is developing very rapidly. There is large pressure on the scientific community to deliver dedicated accelerators, providing the best possible treatment modalities at the lowest cost. In this context, the Italian research Foundation TERA is developing fast-cycling accelerators, dubbed 'cyclinacs'. These are a combination of a cyclotron (accelerating ions to a fixed initial energy) followed by a high gradient linac boosting the ions energy up to the maximum needed for medical therapy. The linac is powered by many independently controlled klystrons to vary the beam energy from one pulse to the next. This accelerator is best suited to treat moving organs with a 4D multipainting spot scanning technique. A dual proton/carbon ion cyclinac is here presented. It consists of an Electron Beam Ion Source, a superconducting isochronous cyclotron and a high-gradient linac. All these machines are pulsed at high repetition rate (100-400 Hz). The source should deliver both C6+ and H-2(+) ions in short pulses (1.5 μs flat-top) and with sufficient intensity (at least 10(8) fully stripped carbon ions per pulse at 300 Hz). The cyclotron accelerates the ions to 120 MeV/u. It features a compact design (with superconducting coils) and a low power consumption. The linac has a novel C-band high-gradient structure and accelerates the ions to variable energies up to 400 MeV/u. High RF frequencies lead to power consumptions which are much lower than the ones of synchrotrons for the same ion extraction energy. This work is part of a collaboration with the CLIC group, which is working at CERN on high-gradient electron-positron colliders.