Schrödinger operators with symmetries

We consider the stationary Schrödinger operator H of a many-body system M with two-body rotation invariant interactions. The operator H is reduced with respect to the symmetries of permutation of identical particles, rotations and reflections, into a direct sum of operators H^τ̃, where τ̃ is an index of the irreducible representations of the symmetry group of the system.The spectra of the operators H^τ̃ were investigated in a series of papers of G.M. Zislin and A.G. Sigalov ([20], [21], [31]-[35]). In a recent paper [3] we have developed the spectral theory of these operators on the basis of the Weinberg equations.In the present work we complete and simplify this theory. In particular we treat in detail the case where the given system can be decomposed into two identical subsystems. For such systems there is a certain coupling between permutation and rotation-reflection symmetries, because a permutation, which interchanges the two subsystems, imposes a reflection on the relative position vector of the two centers of mass. This requires a modification of the theorem on essential spectrum as formulated in [3] in the case where such a division is not possible. The importance of this special case, as exemplified by diatomic molecules, fully justifies such a detailed treatment.This special case was treated by Zislin [34] under the assumption that the interactions are essentially multiplicative, relatively compact two-body interactions. Our method allows for general relatively compact two-body interactions, and can without difficulty be generalized to many-body interactions.Moreover, the method based on the Weinberg equation is suitable for a further analysis of the spectra of these operators.