By Pierre Emile Cartier, Pierre E. Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove
This e-book offers pedagogical contributions on chosen subject matters bearing on quantity thought, Theoretical Physics and Geometry. The components are composed of lengthy self-contained pedagogical lectures through shorter contributions on particular topics equipped through topic. so much classes and brief contributions move as much as the new advancements within the fields; a few of them stick with their author?s unique viewpoints. There are contributions on Random Matrix concept, Quantum Chaos, Non-commutative Geometry, Zeta services, and Dynamical platforms. The chapters of this e-book are prolonged models of lectures given at a gathering entitled quantity conception, Physics and Geometry, held at Les Houches in March 2003, which accrued mathematicians and physicists.
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Extra info for Frontiers in number theory, physics, and geometry I
This is the famous Selberg trace formula. It connects eigenvalues of the Laplace–Beltrami operator for functions automorphic with respect to a discrete group having only hyperbolic elements with classical periodic orbits. 9 Density of Periodic Orbits To ﬁnd the density of periodic orbits for a discrete group let us choose the test function h(r) in (29) as h(r) = e−(r 2 +1/4)T ≡ e−ET with a parameter T > 0. Its Fourier transforms is g(u) = ∞ 2 e−T /4 h(k)e−iku dk = √ e−u /4T . 2 πT −∞ 1 2π In the left hand side of the Selberg trace formula one obtains e−En T = 1 + n e−En T En >0 where we take into account that for any discrete group there is one zero eigenvalue corresponding to a constant eigenfunction.
Assume that g is in the diagonal form. Then g(z) = λ2 z and cosh d(z, g(z)) = 1 + (λ2 − 1)2 (x2 + y 2 ) . 2λ2 y 2 Because λ0 is real the transformation z = λ20 z gives y = λ20 y and the funda2 mental domain of Sg = λ2m 0 z has the form of a horizontal strip 1 < y < λ0 indicated in Fig. 4. 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 .
G. ) x . N (p < x) = ln x As ln p ≡ Tp this expression has the form similar to number of periodic orbits of chaotic systems with h = 1 N (Tp < T ) = eT . T Due to these similarities number-theoretical zeta functions play the role of simple (but by far non-trivial) models of quantum chaos. Notice that the overall signs of the oscillating part of trace formulas for the Riemann zeta function and dynamical systems are diﬀerent. According to Connes  it may be interpreted as Riemann zeros belong not to a spectrum of a certain self-adjoint operator but to an ’absorption’ spectrum.