Diabolical points in the spectra of triangles

TLDR

Accidental degeneracies between adjacent energy levels in real Hamiltonians with at least two parameters form generic “diabolical points” characterized by a local double‑cone structure in energy‑parameter space. The authors examined a family of hard‑walled triangular systems, labeling two angles as parameters, and employed an efficient Green‑function method to compute the energy spectra. They identified multiple diabolical points among low‑lying levels, with the lowest occurring for levels 5 and 6 in a 130.57°–30.73°–18.70° triangle; the conical structure was confirmed by sign changes of the normal derivative and nodal‑line patterns, and their estimate N_d(j) ≈ (j+½)^{2.5} agrees with data while thin triangles show no degeneracies in the studied range.

Abstract

‘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°, 30.73°, 18.70°. The conical structure was confirmed by noting that the normal derivative u of the wavefunction ψ 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 ψ , agreed with theoretical expectations. An estimate of the total number of degeneracies N d ( j ) involving levels 1 through j , based on the energy-scaling of cone angles and the level spacing distribution, gave N d ( 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.