content_image/434233.jpg content_image/434234.jpg
4 , q$_{0}$ = 2 has the property x $^{2}$+ y 2 = 1 and since that “nor- malization” is preserved we indeed conclude the preser- vation of the inequalities under study. 

Finally, let us demonstrate that there are no lengths of the type ac outside the equidistant-length ladder just established. Momentarily assuming the existence of such a freak length we can apply the length reducing transfor- mation (14) until arriving at a reduced length l ′′ ∈ [0 , l I $_{0}$] which must obey 

<img src='content_image/434233.jpg'>

On the other hand, the conservation of x + y sub- jects the freak to ( x ′′ ) 2 + ( y ′′ ) 2 = ( m ′′ − n $^{$_{′′}$√$_{2}$}$) 2 / 4 + ( p ′′ − q $^{$_{′′}$√$_{2}$}$) 2 / 4 α 2 = x 2 + y 2 ≤ 2 , and thus to ∣ m ′′ n $^{$_{′′}$√$_{2}$}$∣ ∣ < $_{2}$√$_{2}$$_{,}$ ∣ p ′′ − q $^{$_{′′}$√$_{2}$}$∣ ∣ < $_{2}$√$_{2}$$_{α}$$_{.}$ Upon checking the remaining finite number of quadruplets of integers m ′′ , n ′′ , p ′′ , q ′′ with lengths inside the interval 0 < l ′′ < l I 0 we conclude that no freak can exist. 

The discussion of case ad proceeds in close analogy with the previously treated ac . The only differences are the replacements of (i) the quartic irrationality α by β and (ii) the minimal length l I 0 by l II 0 = 2 s II = 4 . 8969, the latter corresponding to u$_{0}$ $_{= 6+4}$√$_{2}$$_{,}$ v$_{0}$ $_{= 4+4}$√$_{2.}$ All other lengths are given by kl II 0 , k = 1 , 2 , . . . . The proof repeats the previous case word for word. 

To treat case bd we employ γ = αβ to write the equa- tions for the lengths as 

The smallest length corresponds to u$_{m}$ = 2 + √$_{2}$$_{, v}$$_{m}$ = √$_{2 and is exactly}$ s $^{I}$, i.e., half of the smallest length of case ac . Solutions of (16) do not have in general the group property of the case ac ; in particular, if l is an exceptional length of the type bd then 2 l is not. Indeed, squaring the rhs of (16) we get √$_{2 (}$$_{αu}$ ± v ) 2 / 4 which cannot be an algebraic number of the type β ( αu + v ) / 2. On the other hand, all odd multiples of l do belong to the admissible 

<img src='content_image/434234.jpg'>

type. The same reasoning as in the cases ac, ad shows that all solutions can be represented as ( k + 1 / 2) l I $_{0}$, k = 0 , 1 , . . . . 

The remaining case bc is related to bd by the replace- ments α ↔ β and l I 0 → l II 0 . All solutions can be written as ( k + 1 / 2) l II 0 , k = 0 , 1 , . . . . 

We thank Eugene Bogomolny for useful discussions. Support by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft is gratefully ac- knowledged. 

[1] O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52 , 1 (1984); G. Casati, F. Valz-Gris, and I. Guarneri, Lett. Nuovo Cim. 28 , 279 (1980) 

[2] M. V. Berry, Proc. R. Soc. Lond. A 400 , 229 (1985) 

[3] M. Sieber and K. Richter, Physica Scripta T 90 , 128 (2001); M. Sieber, J. Phys. A 35 , L613 (2002) 

[4] S. M¨ uller, S. Heusler, P. Braun, F. Haake, and A. Alt- land, Phys. Rev. Lett. 93 , 014103 (2004); Phys. Rev. E 72 , 046207 (2005). 

[5] S. M¨ uller, S. Heusler, A. Altland, P. Braun, and F. Haake, New J. Phys. 11 , 103025 (2009) 

[6] F. Haake, Quantum Signatures of Chaos , 3rd ed. (Springer, Berlin, 2010). 

[7] J. Bolte, G. Steil and F. Steiner, Phys. Rev. Lett. 69 , 2188 (1992) 

[8] R. Aurich, E.B. Bogomolny, and F. Steiner, Physica D 48 , 91 (1991) 

[9] J. Bolte, Int. J. Mod. Phys. 7 , 4451 (1993) 

[10] E.B. Bogomolny, B. Georgeot, M.-J. Giannoni, and C. Schmit, Phys. Rep. 291 , 219–324 (1997) 

[11] E. Bogomolny, in Frontiers in Number Theory, Physics and Geometry , Proceedings of Les Houches winter school 2003, Springer-Verlag (2006) 

[12] J. Marklof, in Encyclopedia of mathematical physics , Vol. 1, edited by J.-P. Francoise, G. L. Naber, and T. S. Tsun, Academic Press/Elsevier Science, Oxford, 2006 

[13] R. Aurich and F. Steiner, Physica D 39 , 169 (1989) 

[14] R. Aurich, F. Scheffler, and F. Steiner, Phys. Rev. E 51 , 4173 (1995) 

[15] H. Ninnemann, Int. J. Mod. Phys. B 9 , 1647 (1995) 

[16] O. Bohigas, M.-J. Giannoni, and C. Schmit, in Spectral Fluctuations, Random Matrix Theories and Chaotic Mo- tion , Lecture Notes in Physics, Vol. 262 (Springer, Hei- delberg, 1986) 

