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By Christian Grosche
During this moment version, a finished evaluation is given for direction integration in - and 3-dimensional (homogeneous) areas of continuing and non-constant curvature, together with an enumeration of the entire corresponding coordinate platforms which enable separation of variables within the Hamiltonian and within the direction indispensable. The corresponding course critical ideas are offered as a tabulation. Proposals pertaining to interbasis expansions for spheroidal coordinate platforms also are given. specifically, the situations of non-constant curvature Darboux areas are new during this version.
the amount additionally includes effects at the numerical research of the houses of a number of integrable billiard structures in compact domain names (i.e. rectangles, parallelepipeds, circles and spheres) in - and 3-dimensional flat and hyperbolic areas. specifically, the dialogue of integrable billiards in circles and spheres (flat and hyperbolic area) and in 3 dimensions are new compared to the 1st version.
moreover, an outline is gifted on a few fresh achievements within the thought of the Selberg hint formulation on Riemann surfaces, its tremendous generalization, their use in mathematical physics and string idea, and a few extra effects derived from the Selberg (super-) hint formulation.
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Extra info for Path Integrals, Hyperbolic Spaces and Selberg Trace Formulae: 2nd Edition
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50) ˙ denotes any classical Lagrangian at most quadratic in x and x, ˙ and Here L(x, x) S[xCl ] = tt L(xCl , x˙ Cl )dt the corresponding classical action evaluated along the classical solution xCl satisfying the boundary conditions xCl (t ) = x , xCl (t ) = x (we assume that the classical dynamics allows only a single classical path). g. DeWitt [123], Morette [414], van Vleck [504] and references therein). The explicit evaluation of S[xCl ] may have any degree of complexity due to complicated classical solutions of the Euler-Lagrange equations as the classical equations of motion.
Perturbation expansions (Feynman and Hibbs [164], Devreese et al. [184]–[186]), and point interactions (Bauch [49], Goovaerts et al. [184, 185] and Refs. [205, 216, 222, 217, 225]). Also 1/r- [186, 370] and 1/r2 -potentials [370] can be taken into account by such a formalism. 2. , by Feynman and Hibbs [164], Giachetti et al. [180], and Feynman and Kleinert [165]. 3. , V (x) → V (x/ζ(t))/ζ 2 (t) (x ∈ IRD ) [140, 218]. 4. Point interactions in two and three dimensions [4, 89, 217]. 5. Path integral representation of the one-dimensional Dirac particle according to Feynman and Hibbs [164], Ichinose and Tamura [282], Jacobson [293], and Jacobson and Schulman [294].
Then ΔVP F simplifies further into ΔVP F = −2 8m 2D a (D − 4)f,a2 + 2f · f,aa . 12) This implies, that if the dimension of the space is D = 2, the quantum correction ΔVP F vanishes. 13) with h = h(a, b), u = u(a, b). Then the quantum potential is of the form 2 ΔVP F = h2 8m u2 2u(u,aa + u,bb ) − (u2,a + u2,b ) . 15) with h = h(a, b), u = u(a, b), the quantum potential reads 2 ΔVP F = 2m h2 ab u,aa + u,bb . u These specific examples of ΔVP F will be useful in the sequel. 16) CHAPTER 2. 1 Transformation Techniques Point Canonical Transformations Indispensable tools in path integral techniques are transformation rules.