Analytic continuation of group representations

Some general methods for analytically continuing group representations are presented. In favorable cases, this enables one to recognize the generalization of the principal, discrete and supplementary series.Work performed under the auspices of the United States Atomic Energy Commission.     

Analytic continuation of group representations II

The paper deals with the general background connecting the ideas of analytic continuation and contraction of Lie algebras and their representations. A connection is also established between the Kodaira-Spencer deformation theory and the theory of cohomology of Lie algebras.Supported by the Office of Naval Research, NONR 3656(09).     

Analytic continuation of group representations. VI

The Gell-Mann formula for analytically continuing group representations is worked out explicitly for more cases than in previous work, and extended to certain pseudo-Riemannian symmetric spaces. The method of finding the asymptotic behavior of matrix elements of group representations introduced in Part V is developed in more detail and it is shown how it leads to new mathematical problems in the theory of dynamical systems and Hilbert space theory.Work supported by the U.S. Atomic Energy Commission.     

Analytic continuation of group representations. III

The connection between the ideas of contraction and analytic continuation of Lie algebras and their representations is discussed, with particular emphasis on the contraction of the Poincaré to the Galilean group.This research was supported in part by the Office of Air Force Scientific Research AF 49 (638)-1440.     

Analytic continuation of group representations. IV

The main problem, deforming a subalgebra of a Lie algebra, is treated algebraically, requiring an extensive development of methods of defining multiplications on Lie algebra cohomology cochains. Some applications to differential geometry are also presented.Work supported by the U.S. Atomic Energy Commission.     

Analytic continuation of harmonic sums

We present a method for calculating any (nested) harmonic sum to arbitrary accuracy for all complex values of the argument. The method utilizes the relation between harmonic sums and (derivatives of) Hurwitz zeta functions, which allows a harmonic sum to be calculated as an expansion valid for large values of its argument. A program for implementing this method is also provided.     

Analytic continuation at first-order phase transitions

We study the analytic structure of thermodynamic functions at first-order phase transitions in systems with short-range interactions and in particular in the two-dimensional Ising model. We analyze the nature of the approximation of the d=2 system by anN × strip. Investigation of the structure of the eigenvalues of the transfer matrix in the vicinity of H=0 in the complexH plane allows us to define a new function which provides rapidly convergent approximations to the stable free energyf and its derivatives for allH 0. This new function is used for numerical calculation of the coefficients C_n in the power series expansions of the magnetizationm in the form m(H)=1 + C_n(H-H ₀ )ⁿ for various H₀ 0. The resulting series are studied by conventional methods. We confirm recent series analysis results on the existence of the droplet model type essential singularity at H=0. Evidence is found for a spinodal at H=H_sp(Ti < 0.     

Analytic continuation of Moyal and operator algebras

We consider the analytic continuation of the commutator[q ^m, pⁿ for values of m,n positive integers. This is carried out both in the Moyal algebra and the operator algebra.     

Analytic continuation of sturmian expansions for two-photon ionization

We show that the sturmian expansions of transition amplitudes for two-photon ionization may be handled as ordinary (single variable) power series, and we describe a simple method for performing their analytic continuation above the photoelectric threshold. Numerical results are presented for the hydrogenic states 1s, 2s, and 2p.     

Analytic continuation of the Regge pole residue function

We show that the Regge residue functiony(s) is particularly well suited for performing unitarity bootstrap calculations. The reason is that firstlyy(s) has only one, viz. the right hand, cut along which its value can be evaluated from direct channel unitarity using a parameterfree representation for the partial wave amplitudeS(l, s). Secondly its values for negative real argument follow directly from large-energy scattering with the exchange of one Regge pole in the crossed channel. These values can be evaluated from the sameS(l, s) representation by partial wave sums. Then all one needs for a bootstrap system is an analytic connection of these 2 different pieces of information. We show that this can be achieved by logarithmic dispersion relations. This bootstrap system is supposed to compete favorably with the old unitaryN/D calculations. We finally also propose a new parameter free representation ofS(l, s) which applies equally well as that of Cheng. One main result is that Im α(s) has to decrease exponentially for larges.     

Analytic continuation and reciprocity relation for collinear splitting in QCD

It is well-known that direct analytic continuation of the DGLAP evolution kernel(splitting functions)from space-like to time-like kinematics breaks down at three loops.We identify the origin of this breakdown as due to splitting functions not being analytic functions of external momenta.However,splitting functions can be constructed from the squares of(generalized)splitting amplitudes.We establish the rules of analytic continuation for splitting amplitudes,and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions.In this way we derive the time-like splitting functions at three loops without ambiguity.We also propose a reciprocity relation for singlet splitting functions,and provide non-trivial evidence that it holds in QCD at least through three loops.     

Analytic continuation of the multiple Daehee q-l-functions associated with Daehee numbers

In this paper, we present a new generating function which is related to Daehee numbers. By using the Mellin transformation formula and the Cauchy theorem for this generating function, we define multiple Daehee q-l-functions and q-zeta functions We also give the values of this function at negative integers.     

Analytic continuation in two-color QCD: new results on the critical line

We test the method of analytic continuation from imaginary to real chemical potential in two-color QCD, which is free from the sign problem. In particular, we consider the analytic continuation of the critical line to real values of the chemical potential.     

Electronic properties of disordered systems as derived by analytic continuation from renormalization group theory

The averaged single particle Green function for electrons moving in a gaussian random potential is calculated by analytic continuation from the order parameter susceptibility of a thermodynamic system with order parameter dimension O. New results are obtained and discussed.     

Analytic continuation of scattering data to the region of negative energies for systems that have one and two bound states

An exactly solvable potential model is used to study the possibility of deducing information about the features of bound states for the system under consideration (binding energies and asymptotic normalization coefficients) on the basis of data on continuum states. The present analysis is based on an analytic approximation and on the subsequent continuation of a partial-wave scattering function from the region of positive energies to the region of negative energies. Cases where the system has one or two bound states are studied. The α+d and α+¹²C systems are taken as physical examples. In the case of one bound state, the scattering function is a smooth function of energy, and the procedure of its analytic continuation for different polynomial approximations leads to close results, which are nearly coincident with exact values. In the case of two bound states, the scattering function has two poles—one in the region of positive energies and the other in the region of negative energies between the energies corresponding to the two bound states in question. Padéapproximants are used to reproduce these poles. The inclusion of these poles proves to be necessary for correctly describing the properties of the bound states.