Continuous families of isospectral Heisenberg spin systems and the limits of inference from measurements

We investigate classes of quantum Heisenberg spin systems which have differ ent coupling constants but the same energy spectrum and hence the same ther modynamical properties. To this end we define various types of isospectrali ty and establish conditions for their occurrence. The triangle and the tetr ahedron whose vertices are occupied by spin 1/2 are investigated in some de tail. The problem is also of practical interest since isospectrality presen ts an obstacle to the experimental determination of the coupling constants of small interacting spin systems such as magnetic molecules.