Degeneracy and crossing of resonance energy surfaces

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Abstract
We investigate the accidental degeneracy of resonances mixed by a Hermitian interaction. We give general expressions for the codimension in parameter space of any degeneracy of resonances. In the case of a degeneracy of two resonances which produces only one simple pole in the S matrix, the codimension is four for time reversal invariant interactions and six when the quantum system is not time reversal invariant. Close to this type of degeneracy the corresponding energy surfaces are two double cones lying in orthogonal subspaces with a common vertex at a double diabolic point. When the degeneracy of two resonances leads to one simple plus one double pole of S the codimension is two irrespective of the time reversal invariant character of the quantum system. Close to this type of degeneracy, the energy surfaces are a hyperbolic cone and a sphere which lie in orthogonal subspaces and touch one another at all points on a diabolic circle.