Abstract
The eigenvalues of a quantum system depending on two parameters become degenerate at isolated points in the parameter space, which are called diabolical points because of their double-cone structure. Varying one parameter produces near-degeneracies termed avoided crossings. Some results on the density of these objects in parameter space can be obtained by requiring consistency with the level repulsion exhibited by the distribution of energy-level spacings. The density of diabolical points and the distribution of their ellipticities are obtained exactly for the Gaussian orthogonal ensemble random-matrix model. The distribution of their separations in parameter space is also investigated.
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