  We investigate classes of quantum Heisenberg spin systems which have
different coupling constants but the same energy spectrum and hence the same
thermodynamical properties. To this end we define various types of
isospectrality and establish conditions for their occurence. The triangle and
the tetrahedron whose vertices are occupied by spins 1/2 are investigated in
some detail. The problem is also of practical interest since isospectrality
presents an obstacle to the experimental determination of the coupling
constants of small interacting spin systems such as magnetic molecules.
