By Pierre Emile Cartier, Pierre E. Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove

This ebook provides pedagogical contributions on chosen themes touching on quantity thought, Theoretical Physics and Geometry. The elements are composed of lengthy self-contained pedagogical lectures by means of shorter contributions on particular topics geared up through topic. so much classes and brief contributions cross as much as the new advancements within the fields; a few of them keep on with their author?s unique viewpoints. There are contributions on Random Matrix idea, Quantum Chaos, Non-commutative Geometry, Zeta features, and Dynamical platforms. The chapters of this publication are prolonged models of lectures given at a gathering entitled quantity thought, Physics and Geometry, held at Les Houches in March 2003, which collected mathematicians and physicists.

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**Extra resources for Frontiers in number theory, physics, and geometry I**

**Example text**

These relations give D f (d(y, SgS −1 (y)))dµ = S −1 (D) f (z, g(z))dµ and the last integral is taken over the image of the fundamental domain under the transformation S −1 . Therefore S D f (d(y, SgS −1 (y)))dµ = S S −1 (D) f (d(z, g(z)))dµ . For diﬀerent S images S −1 (D) are diﬀerent and do not overlap. The integrand does not depend on S and S D f (d(y, SgS −1 (y)))dµ = where Dg f (d(z, g(z)))dµ S −1 (D) . Dg = S −1 The sum of all images S (D) will cover the whole upper half plane but we have to sum not over all S but only over S factorized by the action the group of matrices commuting with a ﬁxed matrix g.

Now dg (E) = λ20 ∞ dx −∞ 1 (λ2 − 1)2 (x2 + y 2 ) λ2 y 2 F dy . y2 Introducing a new variable ξ = xy one gets dg (E) = λ20 1 = ln λ20 After the substitution dy y ∞ ∞ −∞ −∞ F F (1 + ξ 2 ) (1 + ξ 2 ) (λ2 − 1)2 λ2 (λ2 − 1)2 λ2 dξ dξ . Quantum and Arithmetical Chaos 27 y 2 i λ0 i x Fig. 4. Fundamental domain of multiplication group u = (1 + ξ 2 ) one obtains (λ2 − 1)2 λ2 ∞ ln λ2 dg (E) = √ 0 u0 F (u) √ du u − u0 u0 where (λ2 − 1)2 1 = λ2 + 2 − 2 . λ2 λ The variable u is connected with the distance by cosh d = 1 + u/2 and the function F (2(cosh d − 1)) has the form u0 = ∞ 1 F (2(cosh d − 1)) = √ 2 2π 2 d sin kr √ dr .

The free motion equations x x = kt + y with a ﬁxed vector k. One has k= x−y t and the conservation of energy |k|2 = E determines the time of motion t= |x − y| √ . E Therefore S(x, y) = √ E|x − y| which, evidently, is the solution of the free Hamilton–Jacobi equation. The next order equation 2∇S∇A + ∆SA = 0 38 Eugene Bogomolny is equivalent to the conservation of current. Indeed, for the semiclassical wave function (35) 1 J = (Ψ ∗ ∇Ψ − Ψ ∇Ψ ∗ ) = A2 ∇S 2i and ∇J = A(2∇A∇S + A∆S) = 0 . The solution of the above transport equation has the form A(x, y) = ∂2S π 1 det − ∂ti⊥ ∂tf ⊥ (2π )(f +1)/2 ki kf 1/2 where ti⊥ and tf ⊥ are coordinates perpendicular to the trajectory in the initial, y, and ﬁnal, x, points respectively and ki , kf are the initial and ﬁnal momenta.