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boundary conditions can be either Dirichlet or Neumann. There are 2 3 = 8 quantum mechanical problems, all re- lated to the same classical system. In problems stem- ming from desymmetrization of the regular octagon, the boundary conditions on the triangle sides are chosen to obtain the spectrum for a particular irreducible repre- sentation. The boundary conditions on the hypotenuse N and the shorter leg M must then be the same, i.e., both Dirichlet or both Neumann; four such possibilities exist all of which lead to arithmetical systems with near- Poissonian spectral statistics. The remaining four choices where N and M host different boundary conditions, lead to pseudo-arithmetical systems with the Wigner-Dyson statistics of the orthogonal universality class. 

Form factor and diagonal approximation : The spectral form factor following from Gutzwiller‘s trace formula is a double sum over periodic orbits, 

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where S$_{γ}$, T$_{γ,}$µ$_{γ}$, A$_{γ}$ = A ∗ γ are action, period, Maslov in- dex, and stability coefficient of the orbit γ ; the Heisen- berg time T$_{H}$ = 2 π ℏ Δ , with Δ the mean level spacing, is used as a unit of time such that τ becomes a dimension- less time; the angular brackets 〈 . . . 〉 denote averages over the energy shell and over a small τ interval. 

Of special interest are the pairs of orbits obviously im- mune against destructive interference of their contribu- tions, namely with the same action and Maslov index. For generic chaotic dynamics these are the trivial pairs γ ′ = γ and, if time reversal is allowed, the pairs of mutu- ally time reversed orbits, γ ′ = γ $^{TR}$. Discarding all other pairs one gets Berry’s diagonal approximation [2], 

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here g is the average multiplicity of the action spectrum which is 1 in the absence of time reversal (unitary uni- versality class) or 2 in the presence of time reversal (or- thogonal class); the result yields the first-order term of a power series in τ , in agreement with RMT [4]. 

Turning to arithmetical chaos we can carry over the foregoing reasoning, except that the multiplicity is now exponentially large, g ∝ e l/ $^{2}$. However, orbits with the same length bear no geometric similarity, and it is not at all obvious that their Maslov phases are the same. 

As will be shown, the Maslov phases do indeed coincide for all orbits in a length multiplet, a negligible fraction apart, in the arithmetical case. Consequently, the form factor exhibits almost instant increase at τ ≥ 0, similar to the jump of the integrable case [6], K (0) = 0 and K ( τ ) = 1 for τ > 0. 

In contrast, the Maslov index of pseudo-arithmetical systems will turn out to fluctuate randomly within each fixed-action multiplet. The usual diagonal approxima- tion with g = 2 then holds since in each multiplet only the pairs γ ′ = γ and γ ′ = γ TR escape destructive in- terference. The diagonal approximation thus suggests universal spectral fluctuations. 

The Maslov index of an orbit in our billiard is de- termined only by the number N$_{D}$ of reflections from the sides with the Dirichlet boundary condition: each such reflection changes the Maslov phase by π . There- fore the contribution of an equal-action pair γ, γ ′ is A 2 γ ( − 1) $^{N$_{D}$}$−N ′ D where N$_{D}$, N ′ D respectively refer to γ and γ ′ ; in long orbits both N$_{D}$ and N ′ D are large pseudo- random integers. We shall demonstrate that in arith- metical systems equal-length orbits have, in their over- whelming majority, N$_{D}$ of the same parity (even or odd) such that contributions of all pairs of them are posi- tive and add up. On the contrary, pseudo-arithmetical systems have uncorrelated parities of N$_{D}$, N ′ D and the equal-action contributions mutually cancel, apart from the standard pairs of the orthogonal universality class. 

Numerical observations : The triangular billiard affords symbolic dynamics, and calculating its orbits involves generating all allowed sequences of symbols L, M, N each standing for the visit of the respective side. We denote by λ$_{γ}$, µ$_{γ}$, ν$_{γ}$ the number of symbols L, M, N in an orbit γ ; the total number of symbols in γ is n$_{γ}$ = λ$_{γ}$ + µ$_{γ}$ + ν$_{γ}$ . Studying up to a million orbits we find: 

— Orbits within a given length multiplet Λ almost al- ways have n$_{γ}$ with the same parity. We can therefore speak about Λ$_{g}$- and Λ$_{u}$-multiplets depending on par- ity of n$_{γ}$, γ ∈ Λ. Exceptions are extremely rare and in fact amount to a negligible fraction: E.g., among approx- imately 257000 orbits with the length l$_{γ}$ < 16 grouped into more than 13000 length multiplets, only 4 multiplets are “ gu -degenerate”, i.e., contain orbits with both even and odd number of symbols; these offenders have lengths l = 10 . 6999964 , 12 . 2422622 , 13 . 7571382 , 15 . 2857092. 

— All orbits in a given Λ without gu -degeneracy have λ$_{γ}$ of the same parity which automatically leads to def- inite parity of µ$_{γ}$ + ν$_{γ}$ . On the other hand, µ$_{γ}$ and ν$_{γ}$ separately have no definite parity within Λ. 

These observations, in particular the rarity of multi- plets with gu -degeneracy, suffice to explain the sensitivity 

