By Christian Grosche

During this moment variation, a accomplished assessment is given for direction integration in - and three-d (homogeneous) areas of continuing and non-constant curvature, together with an enumeration of all of the corresponding coordinate platforms which enable separation of variables within the Hamiltonian and within the direction essential. The corresponding direction vital ideas are provided as a tabulation. Proposals pertaining to interbasis expansions for spheroidal coordinate structures also are given. particularly, the instances of non-constant curvature Darboux areas are new during this variation.

the amount additionally includes effects at the numerical examine of the homes of a number of integrable billiard platforms in compact domain names (i.e. rectangles, parallelepipeds, circles and spheres) in - and third-dimensional flat and hyperbolic areas. particularly, the dialogue of integrable billiards in circles and spheres (flat and hyperbolic house) and in 3 dimensions are new compared to the 1st variation.

moreover, an outline is gifted on a few fresh achievements within the concept of the Selberg hint formulation on Riemann surfaces, its great generalization, their use in mathematical physics and string concept, and a few extra effects derived from the Selberg (super-) hint formula.

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**Example text**

5 and 6 is to suggest the richness, ﬂexibility and economy of texture available for solutions in a basis function approach. Next, we introduce the total invariant mass-squared M2 for the low-lying physical states in terms of a Hamiltonian H times a dimensionless integer for the total light-front momentum K M2 + P⊥ P⊥ → M2 + const = P+ P− = KH (4) where we absorb the constant into M2 . For simplicity, the transverse functions for both the electron and the photon were taken as eigenmodes of the external trap in our initial application (7) which we discuss here (below, we present results with the external trap removed).

The light-front Hamiltonian quantized within a basis function approach as described here offers a promising avenue that capitalizes on theoretical and computational achievements in quantum many-body theory over the past decade. By way of background, one notes that Hamiltonian light-front ﬁeld theory in a discretized momentum basis (1) and in transverse lattice approaches (2; 3) have shown signiﬁcant promise. I outline here a Hamiltonian basis function approach following Refs. (4–10) that exploits recent advances in solving the non-relativistic strongly interacting nuclear many-body problem (11; 12).

1997). Effective interactions for mesons and baryons in nuclei, Progr. Theor. , 98, 927-941. V. (2004). Clothed particle representation in quantum ﬁeld theory: Mass renormalization, Phys. Rev. D 70, 085011. V. (2007). Relativistic interactions for the meson-two-nucleon system in the clothed-particle unitary representation, Ann. , 322, 736-768. ; Glöckle, W. (1999). Approach towards N-nucleon effective generators of the Poincaré group derived from a ﬁeld theory, Phys. Rev. C 59, 31919-1929. V. (1993).

### Path integrals, hyperbolic spaces, and Selberg trace formulae by Christian Grosche

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