Diabolical points in the spectra of triangles

`Accidental' degeneracies between energy levels E$_j$ and E$_{j+1}$ of a real Hamiltonian can occur generically in a family of Hamiltonians labelled by at least two parameters X, Y, ... Energy-level surfaces in E, X, Y space have (locally) a double-cone (diabolo) connection and we refer to the degeneracies themselves as `diabolical points'. We studied the family of systems in which a particle moves freely within hard-walled triangles (vibrations of triangular membranes), with X and Y labelling two of the angles. Using an efficient Green-function technique to compute the levels, we found several diabolical points for low-lying levels (as well as some symmetry degeneracies); the lowest diabolical point occurred for levels 5 and 6 of the triangle 130.57$^\circ$, 30.73$^\circ$, 18.70$^\circ$. The conical structure was confirmed by noting that the normal derivative u of the wavefunction $\psi$ at a boundary point changed sign during a small circuit of the diabolical point. The form of the variation of u around a circuit, and the changing pattern of nodal lines of $\psi$, agreed with theoretical expectations. An estimate of the total number of degeneracies $\mathscr{N}_d(j)$, involving levels 1 through $j$, based on the energy-scaling of cone angles and the level spacing distribution, gave $\mathscr{N}_d(j) \sim (j+\frac{1}{2})^{2.5}$, and our limited data support this prediction. Analytical theory confirmed that for thin triangles (where our computational method is slow) there are no degeneracies in the energy range studied.