Couplings in coupled channels versus wave functions: Application to the X(3872) resonance

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 D⁰D¯^*0 and D⁺D^*- (and c.c. From now on, when we refer to D⁰D¯^*0 , D⁺D^*-, or DD¯^* 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 D⁰D¯^*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 D⁰D¯^*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 Schrödinger 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.