Diabolical points in the spectra of triangles

Abstract

‘Accidental’ degeneracies between energy levelsEjandEj+1of a real Hamiltonian can occur generically in a family of Hamiltonians labelled by at least two parametersX,Y,... Energy-level surfaces inE,X,Yspace 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), withXandYlabelling 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°, 30.73°, 18.70°. The conical structure was confirmed by noting that the normal derivativeuof the wavefunctionψat a boundary point changed sign during a small circuit of the diabolical point. The form of the variation ofuaround a circuit, and the changing pattern of nodal lines ofψ, agreed with theoretical expectations. An estimate of the total number of degeneraciesNd(j) involving levels 1 throughj, based on the energy-scaling of cone angles and the level spacing distribution, gaveNd(j) ~ (j+ ½)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.