By Isaacs I.M.

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**Example text**

The change of the action when a point is at the distance y from the periodic orbit is 1 ∂ 2 S(y, y) |y=0 δS = y 2 2 ∂y 2 where S(y, y) is the classical action for a classical orbit in a vicinity of the periodic orbit (see Fig. 7). To compute such derivatives it is useful to use the monodromy matrix, mij , which relates initial and ﬁnal coordinates and momenta in a vicinity of periodic orbit in the linear approximation δyf δpf = m11 m12 m21 m22 δyi δpi . As the classical motion preserves the canonical invariant dpdq it follows that det M = 1.

One has δyf = m11 δyi + m12 δpi , δpf = m21 δyi + m22 δpi . But pi = − ∂S , ∂yi pf = ∂S . ∂yf Therefore δpi = − ∂2S ∂2S ∂2S ∂2S δy − δy , δp = δy + δyf . i f f i ∂yi2 ∂yi ∂yf ∂yi ∂yf ∂yf2 From comparison of these two expression one obtains the expressions of the second derivatives of the action through monodromy matrix elements y periodic orbit classical orbit Fig. 7. A periodic orbit and a closed classical orbit in its vicinity Quantum and Arithmetical Chaos 41 1 ∂2S m11 ∂2S m22 ∂2S =− , = , = .

In Fig. 8 the two-point correlation form factors for usual random matrix ensembles are presented. Their explicit formulas can be found in [46], [16]. 2 0 0 1 2 3 t Fig. 8. Two point correlation form factor of classical random matrix ensembles. these classical ensembles small-t behaviour of the form factors is t→0 K(t) −→ with the same β as above. 2 t β (44) Quantum and Arithmetical Chaos 51 The nearest-neighbor distribution, p(s), is deﬁned as the probability density of ﬁnding two levels separated by distance s but, contrary to the two-point correlation function, no levels inside this interval are allowed.