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.
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