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Previous efforts to extract resonance widths by analytic continuation of stabilization graphs were limited by non-analytic behavior of the eigenvalues as functions of the stabilization parameter. We describe a procedure which avoids this difficulty by numerical analytic continuation of the characteristic polynomial, truncated to low order, of the hamiltonian matrix.

A method is developed for the numerical evaluation of any term in the Born series for the off-shell partial-wave T matrix. At negative energies, the integrals may be evaluated by using any appropriate quadrature formula. Numerical analytic continuation of the negative-energy results to positive energies is carried out by means of Pade approximants. The method enables the rate of convergence of eigenfunction expansions for the two-body off-shell amplitude to be accelerated. Numerical results are presented for a Yukawa potential and for a neutron-proton /sup 3/S/sub 1/ interaction. 9 tables.

The usual higher Hilbert-Riesz transforms are principal value convolution transforms with kernels Yj(x)¦x¦−j−n, where Yj, is a homogeneous harmonic polynomial of degree j. With the help of the algebra attached to the Laplace operator, we write the kernels in a vectorial form, and consider more generally the kernel xj¦x¦−j+λ−n, where for Re λ < 0, the expression is defined as a distribution by analytic continuation. We find explicit formulas for the analytic continuation, conditions under which two such kernels can be convolved, and formulas for the convolution.

The structure of the non-compact gaugings of N = 8 supergravity is analyzed. The SO(p,q) gaugings are shown to be related to the SO(8) model by a form of analytic continuation. This gives a method for calculating the scalar dependence of the potential and Yukawa interactions in terms of the corresponding quantities in the de Wit-Nicolai model. The truncations to N = 4 supergravity coupled to N = 4 Yang-Mills multiplets are given.

of differential equations in the complex plane. Here 3’ is an n-dimensional vector, B(Z) is an n x n matrix of functions analytic at z = 0, and D = diag(d, ,..., d,), with (I-; = 0, 1, or 2, i = l,..., n. A classical result for this system was given in 1926 by F. Lettenmeyer [6], who proved that (1.1) has at least (12 tr D) linearly independent solutions analytic at z = 0. For Lettenmeyer’s theorem to be meaningful, of course we must have tr D < n. Some results for the case in which & = 2, i = l,..., n, have been obtained by L. J. Grimm and L. M. Hall [l]. Th is result, like Lettenmeyer’s, gives at best a lower bound for the number of analytic solutions. In this paper we develop a procedure which yields the exact number of linearly independent solutions of (1 .I) which are analytic at z = 0. This procedure will be applicable whenever D is as given above, i.e., D is an n x n diagonal matrix with some combination of zeros, ones and twos on the diagonal. Actually, no restriction need be placed on the size of the diagonal entries in D since, due to a result of H. L. Turrittin [7], the rank of a linear differential system can be reduced to rank one at the expense of increasing the dimension of the system. Rank one corresponds to di = 2, i = l,..., n. The techniques which we use are primarily based on the work of Grimm and Hall [2], the results of Hall [4], and the Cesari-Hale alternative problem technique (see Hale [3]).

A semi-classical theory, derived from the Feynman path-integral formalism by applying the saddle point method, is formulated for elastic and inelastic heavy ion reactions. This theory is a natural extension of earlier semi-classical methods in the sense that classical, real, trajectories are extended to complex ones by an analytic continuation of Hamilton's equations. It describes properly the role played by the complex interaction and the resulting absorptive, refractive and diffractive phenomena. The connection with the WKB-approximation is discussed. A comparison between the results of three types of calculation is made: exact (quantum mechanical), semi-classical and the earlier semi-classical approach based on real trajectories. This shows the quantitative power of the semi-classical method presented here, not only for the elastic and inelastic cross sections but it also reproduces the quantum mechanical phase shifts, reflection coefficients and excitation coefficients in quite some detail, contrary to the earlier semi-classical calculations.

The conformal field theory of analytic differentials on a general Riemann surface is formulated. All the determinants and correlation functions are expressed in terms of theta-functions.

It is shown that the light-cone operator expansion of the product of two SU(3) ⊗ SU(3) currents or divergences, the requirement of maximal analyticity of the second kind, and the mathematical method of analytic continuation can be used to shed light on the validity and meaning of the Ward-Takahashi identities, of the Bjorken-Johnson-Low limit and in general of sum rules derived from current algebra either at equal time or on the light cone. In particular we give a definite content to the PCAC hypothesis in terms of the ‘dimension’ of the axial divergence and suggest some specific values for this ‘dimension’. It is also shown that the requirement of maximal analyticity of the second kind and gauge invariance impose a non-trivial condition on the Cornwall-Corrigan-Norton sum rule.

The steady-state equation for N -group neutron transport in slab geometry is written as an integral equation. A spectral analysis is made of the integral operator and related to the criticality problem. The method depends on a representation for the resolvent kernel for a subcritical slab and on analytic continuation in a complex parameter to characterize eigenvalues in terms of singularities of the resolvent. The analytic continuation is based on a bifurcation analysis of some nonlinear matrix integral equations whose solutions provide a matrix Wiener-Hopf factorization of the Fourier transform of the kernel of the transport operator.

Trilinear invariant forms over spaces transforming under the so-called elementary representations of SL (2, C) (obtained from the principal series by analytic continuation in the representation parameters) are studied with regard to their analyticity properties in the representation parameters. The method is based on a natural one-one correspondence between the invariant forms and invariant separately homogeneous distributions (called kernels of the forms) in three complex two-dimensional non-zero vectors. There exists a family Ψ of kernels of forms with analytic dependence on the representation parameters (Ψ being unique up to a family of complex multiples dependent on the parameters). Also associated kernels obtained by differentiating Ψ in the parameters are studied.

The point we try to make is that in an indirect science like elementary particle physics, it is not sufficient to have a specific description of the world brought by some happy inspiration, but rather it is necessary to optimize among large classes of (preferably among all) possible logically equivalent “revelations”. Indeed, although the leading concepts of which every description of nature makes use should bear a very close relation to the experimentally accessible data, in those situations when the basic laws are inherited from other fields, their concepts may prove to be very remote from experiment, and to “measure” them one might have to go through wildly unstable inverse problems (ill posed problems in the Hadamard sense). Moreover, the instabilities of the inverse laws become especially dangerous when the corresponding “direct laws” are too smooth, as it happens in particle physics whenever we try to cling to classical concepts (Lagrangians, interaction terms, etc.) which were purposedly chosen to produce “good” classical physics laws.

Proofs are given that the quasidiscrete energy levels for the Stark effect can be obtained via quasiclassical minimization of inter-related spherically symmetrical hamiltonians. Implicit formulas, expressed just in terms of algebraic equations, are written down to first 1/ N order. The analytic continuation of such formulas towards resonance energies is also established. General symmetry properties, suitable parametrizations, and limiting cases are discussed.

The chiral anomaly in heterotic strings with full and partial modular invariance in D = 2n + 2 dimensions is calculated. The boundary terms which were present in previous calculations are shown to be cancelled in the modular-invariant case by contact terms, which can be obtained by an appropriate analytic continuation. The relation to the low-energy field theory is explained. In theories with partial modular invariance, an expression for the anomaly iis obtained and shown to be non-zero in general.

The standard map is a nonintegrable discrete time analog of the vertical pendulum. Detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency. The functional dependence of the coordinate q on the canonical angle variable θ is analtyically continued into the complex θ-plane, where natural boundaries are found at constant absolute values of Im θ. The boundaries represent the appearance of chaotic motion in the complex plane. When the domain of analyticity shrinks to zero, the KAM invariant curve is destroyed. Two independent numerical methods with Fourier analysis in the angle variable were used, one based on a variation-annihilation method and the other on a double expansion. The results were further checked by direct solution of the complex equations of motion. The numerically simpler, but intrinsically complex, semipendulum and semistandard map are also studied. We conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or KAM surfaces of general nonintegrable systems with analytic Hamiltonians.

For a given summability method, the Okada theorem describes a domain, into which an arbitrary power series can be analytically continued, if such a domain is known for the geometric series. In this paper, Okada's theorem is extended to more general methods of analytic continuation. This results is applied to a special rational approximation, the so-called Pade-type approximation.

A hybrid of dimensional and analytic regularization is used to regulate and uncover a Meijer's G -function representation for a class of massless, divergent Feynman integrals in an axial gauge. Integrals in the covariant gauge belong to a subclass and those in the light-cone gauge are reached by analytic continuation. The method decouples the physical ultraviolet and infrared singularities from the spurious axial gauge singularity but regulates all three simultaneously. For the axial gauge singularity, the new analytic method is more powerful and elegant than the old principal value prescription, but the two methods yield identical infinite as well as regular parts.

Analytic continuation has been proposed as a way to understand metastable states in statistical physics. In that view, the imaginary part of the continued free energy is associated with the decay rate of the metastable system. In this article we show how the relation has been confirmed on a simple stochastic system, spin flip dynamics for the Curie-Weiss ferromagnet.

On the basis of the complex variable method for determining the stationary two-dimensional thermal stresses, the thermal stresses in an orthotropic rectangular plate with a rigid ribbonlike inclusion under a steady state temperature field is considered. The solution is found by the analytic continuation argument and the modified mapping-collocation technique. Numerical results indicate a dependence of the orthotropic stress intensity factors on the thermal, elastic and geometrical constants over a certain parameter range

The basis set calculation of Siegert eigenvalues, which provides the energies and lifetimes of electronic resonance states, has been carried out for a one-dimensional model system using a trial wavefunction which asymptotically behaves as a complex gaussian function. The high accuracy of the results demonstrates that the asymptotic form of the wavefunction is not important when the complex gaussian exponent is chosen by a stationary condition rather than by an asymptotic boundary condition. This indicates that molecular resonances can be computed using relatively small gaussian orbital basis sets. For the resonances studied here, we also show that the modifications in the basis-set calculation of Siegert eigenvalues proposed by Yaris, Lovett and Winkler lead to accurate results only when all of the basis functions go to zero at the origin and when an accurate estimate of the resonance position is already available. In addition, application of the analytic continuation of stabilization graphs to this system is shown to yield results of limited accuracy.

It is shown that analytic continuation on n, the number of components of the order parameter, yields phase transitions at finite temperatures for one dimensional systems with short range forces, as soon as n < 1 (for zero dimensional systems, as soon as n < 0). The critical exponents are found to satisfy the scaling laws, with ν = η − 1 for d = 1 and −2

A study is made of theories having the unusual analyticity properties recently proposed by Lee and Wick. A prescription is given for setting up a covariant perturbation-theory expansion of the scattering amplitudes, based on Feynman graphs. It is found that the presence of a complex pole in the upper half plane of the physical sheet leads to points of non-analyticity in the physical region, such that the values of the amplitude to either side of the point are not related by analytic continuation. It is shown how this is compatible with unitarity. The nature of the non-analyticity is not fully determined by unitarity. Neither, in the case of the more complicated graphs, is it fully determined by the perturbation-theory prescription, and some extra constraint must be imposed on the theory to remove the ambiguity. It is shown that the prescription of Lee and Wick has an exactly similar ambiguity, but for their prescription different results are obtained in different Lorentz frames. An estimate is made of the extent to which the theory violates causality, and is found to be too small to measure.

A diagonal form for the continuation of a multiparticle unitarity integral to complex angular momentum is given involving the “Froissart-Gribov” continuations of multiparticle amplitudes to complex helicity and angular momentum defined in a previous paper. The form for the continuation of the three-particle unitarity integral is considered in detail and shown to converge and satisfy the Carlson condition on the asymptotic behaviour under quite general conditions.

A report is made on a new attempt to determine analytic continuation of scattering amplitudes, which consists of evaluating iteratively on a computer the usual dispersion relation and its inverted one. Its application to the pp forward amplitude shows that this method is practical and powerful. Its imaginary part is shown to have a large dip just above the p p threshold, which makes its high-energy real part nearly twice that determined experimentally ignoring spin-flip.

A Lagrangian formalism is constructed for a symmetric, traceless tensor (spinor-tensor) of rank N which describes a multiplet with spin 0, 1, 2, …, N( 1 2 , 3 2 , …, N + 1 2 ) . From physical quantities that are obtainable from this formalism we derive the corresponding quantities for an infinite multiplet with integer (half-integer) spin by means of an analytic continuation which changes the original field into a basis of an infinite-dimensional, unitary representation of the Lorentz group. Feynman rules for infinite multiplets are determined in this way. Within this formalism mass spectra of infinite multiplets are always of the Majorana type, and the difficulties related to the spin-statistics theorem and spacelike solutions do not arise.

We have studied an analytic continuation of three-particle production partial wave amplitudes ƒ jm in complex values of total angular momentum j. In contrast to the two-particle case, there is no continuation of ƒ jm which decreases in the whole right half-plane of j. Nevertheless, at fixed values of pair energies of three particles it is possible to construct a unique continuation increasing slowly enough in the right half j-plane. If the amplitudes ƒ jm are studied in the whole three-particle physical region, it is necessary to consider at least six different analytic functions of j. This fact is a generalization of signature for the case of amplitudes ƒ jm . Each of the six continuations satisfies a simple unitarity condition in one of the pair energies at complex j. The integration domain in these unitarity conditions includes, generally speaking, the integration over nonphysical values of angles. These properties were obtained while investigating the simplest Feynman diagrams. However, as it follows from the derivation they are to be of a general character. The three-particle contribution to the unitarity condition for elastic scattering amplitude at complex j is also considered. This enables us to study the origin of Mandelstam branch points. However, we can write down the exact expression for the three-particle contribution into the unitarity condition only in the case of some simplest diagrams.

In using FESR at fixed momentum transfer an analytic continuation of the amplitude into the uhysical region near to threshold is necessary. We show how direct channel Regge poles may be used to facilatate this and give a detailed application to the π N amplitudes.

This paper is a report of some studies leading to a new mathematical description of the condensation point for a simple class of models of first-order phase transitions. The paper consists of three main parts. In the first part it is pointed out that, although the conventional droplet model of condensation predicts that the free energy has an essential singularity at the condensation point, this singularity is so weak as to be experimentally unobservable. Furthermore, the analytic continuation of the free energy beyond the singularity describes a metastable phase according to the assumptions of the model. The second part of the paper is devoted to the study of a soluble functional integral that exhibits an essential singularity similar to that found in the droplet model. A method is developed for computing the singular properties of such integrals in cases where it is not possible to evaluate the integrals exactly. In the third part of the paper this method is applied to a simple model of a ferromagnet at temperatures well below the Curie point. Most of the really characteristic features of the droplet model are recovered in this calculation. The detailed results have a bearing on problems of phase coexistence, surface energies, and possibly even condensation rates.

Following the Cutkosky theory of representation of data by analytic functions, a probability functional is introduced which takes account of the different statistical errors in the determination of the real and imaginary parts of a function along its boundary of analyticity. With this probability the most probable function is found and the mean value and standard deviation of any functional are determined in terms of this function. These results are applied to the problem of continuation of a function off its boundary and a modified dispersion relation derived which defines the best continuation possible given the form of the probability function.

We prove that a set of nonsingular free solutions of Maxwell's equations forms a representation of the group obtained by analytic continuation of the Poincare group to complex values of the group parameters, and that a set of singular solutions forms a representation of the group obtained by analytic continuation of the conformal group to complex values of the group parameters. These results are obtained by constructing a theory governing 2 × 2 complex matrix fields defined for complex values of position and time; the equations of this theory are invarient with respect to complex Poincare transformations and complex conformal transformations, but the set of nonsingular solutions is in one-to-one correspondence with a set of nonsingular solutions of Maxwell's equations, and a similar correspondence exists for the singular solutions. Certain collections of solutions of Maxwell's equations for the field of a current form representations of these complex groups if both magnetic and electric currents are permitted, in which case complex transformations provide a natural connection between electric and magnetic charge. A class of complex transformations also yield natural relations between sources moving slower than light and sources moving faster than light.

Synopsis The question is to what extent the usual concepts apply to the general form of the S matrix envisaged in Part I, viz., an S that is not holomorphic in the upper half plane, but for which the stationary scattering states are still a complete set. The answer is that it is again possible to find the explicit expansion of an arbitrary function with respect to this set, so that the evolution of an arbitrary initial state can be calculated. Emission can be described by choosing a superposition of stationary states in such a way that the wave function vanishes initially. Resonance levels and excited states, however, can only be uniquely defined if an analytic continuation of S is possible.

The structure functions of deep inelastic scattering and of deep inelastic annihilation of a current in the go 3 theory are studied in the ladder approximation. To this aim an integral equation which is the scaling limit of the classical ABFST equation is used. By solving this equation in a particular case we obtain in closed form the sum of all ladder diagrams with all massless rungs except for the first and the last ones which are given arbitrary masses. The solution is a hypergeometric function in two variables and its analyticity properties in the scaling variable ω, for fixed values of the external mass, are studied. The structure function for annihilation turns out to be, in this case, the analytic continuation of the scattering structure function. Going beyond this particular solution we show in general from the integral equation that the average multiplicity of the secondary particles in the annihilation region approaches a finite limit for increasing q 2 , q μ being the four momentum of the current. Implications of this result for e + e − annihilation into hadrons are discussed.

An investigation was made of scattering-amplitude expansions in functions characterized by the poles of the analytic continuation of the S-matrix elements in the complex angular plane. An analytical expression is given for the one pole contribution to the scattering amplitude.

A new inversion formula for the Stieltje's transform is derived and used for the analytic continuation of nucleon form factors. For given experimental accuracy the extent to which the spectral functions may be determined is analysed and with present data it is clear that only a very poor resolution may be obtained. An estimate of the experimental accuracy required to resolve the rho contribution is given.

We prove that the energy levels of an arbitrary anharmonic oscillator ( x 2 m and in any finite number of dimensions) are determined uniquely by their Rayleigh-Schrodinger series via a (generalized) Borel summability method. To use this method for computations, one must make an analytic continuation which we accomplish by (a rigorously unjustified) use of Pade approximants in the case of p 2 + x 2 + β x 4 . The numerical results appear to be better than with the direct use of Pade approximants.

This is the first of the series of papers in which it is shown that the iteration of the kernel in the Lippmann-Schwinger equation is the basic manipulation by which a meaningful result is obtained for any finite strength of the potential, no matter whether the Born series converges or not. In order that this statement may be true the potential must be free from a long range tail and a strong singularity. In this paper the conditions for a generalized potential, which can be nonlocal and/or energy-dependent, to have these properties when transformed into the configuration space are discussed directly in terms of its matrix element in momentum space. These conditions are interpreted as equivalent to the restriction that the potential decreases faster than 1 r 3 at a large distance and is less singular than 1 r 2 at the origin. It follows from these conditions that all the individual terms of the Born series as well as the traces of the kernel and its iteration to an arbitrarily higher order are bounded.

The anomalous discontinuities in the 1-channel, called for in pion- deuteron elastic scattering, are derived. The methods used start with a choice of the deuteron mass and of the energy such that the discontinuities are given by physical unitarity. Analytic continuation in the deuteron mass leads to the emergence of the anomalous thresholds and provides the formulas for the anomalous discontinuities. The representation of the physical amplitude is obtained by subsequent andalytic continuation in the energy. (auth)

The Goldstone linked cluster expansion is used to determine the energy as an analytic function of the coupling constant. This function is many-valued and describes the various energy levels of the system. The energy of each level can be obtained from the Goldstone expansion by continuing it analytically along a properly chosen path in the complex plane. The Brueckner ladder approximation 3) is shown to be an approximation to an analytic continuation along a path which always leads to the normal state — the state in which no binding occurs.

The momentum space analytic function I(Z1, Z2, Z3; ϱ) corresponding to the configuration space function.

The double-pomeron coupling strength in the dual resonance model is found in both the inclusive and exclusive regions by comparison with experiments. Double-pomeron coupling occurs in inclusive experiments in the Mueller diagram for the central plateau region. Its strength can also be bounded from its non-observation in the two-particle to four-particle exclusive experiments. The dual resonance model is used to perform the analytic continuation of a six-point amplitude between these regions. The results show that the coupling strength for two forward pomerons in the exclusive region must be less than 1 300 of that in the inclusive region. This is experimental evidence for substantial forward double-pomeron decoupling in exclusive processes.

Nonequivalent results are found for the two-particel quasibound state in the s-d exchange and the Anderson models. In the latter the pole disappears for a T o ∼ 1 3 T k and for a moment μ ∼ 10−3 μB. In the former the quasi-bound state is not pure singlet and hence possesses a moment. These new results depend on proper analytic continuation in the study of the Green function poles.

A new, cutoff-free approach to singular scattering equations in momentum space is developed which applies to four-fermion interactions and singular potentials. Generalized function perturbative solutions are confronted with solutions nonanalytic in the coupling constant. New techniques for dealing with singular interactions in momentum space in terms of analytic continuation procedures are developed which furnish finite, cutoff-free results at each step of approximation.

The lattice Green's function for the simple cubic lattice (γ = 1) and tetragonal lattice at an arbitrary point (l, m, n) I(a;l,m,n;γ) = 1 π 3 ∫∫ ∫ o π cos lx cos my cos nx a−iϵ−γ cos x− cos y− cos x dxdydz is evaluated, assuming a ⩾ 0, γ ⩾ 0 without loss of generality. The integral I(a; l, m, n; γ) which has singularities at a = ± γ ± 1 ± 1, is expressed in all regions of (a, γ), i.e., for (i) a > 2 + γ, (ii) 2 − γ > a > γ(γ 2), (iv) a a (0 2, in terms of Kampe de Feriet function by the method of the analytic continuation using the Mellin-Barnes type integral. The numerical values are shown in figures. The high temperature susceptibilities of the Heisenberg model of the ferro- and antiferromagnets are calculated using the results of I(a; l, m, n; γ), showing a shift from three to two dimensions and that from three to one dimensions. The correlation function of the isotropic ferromagnet is calculated and the critical index ν is observed to be 1.

A rapid method for calculating partial-wave scattering amplitudes over a range of energies is presented. Approximate values of the partial-wave Fredholm determinant are computed at a number of points in the complex energy plane from a single matrix tridiagonalization and are continued to the real axis via point-wise rational fraction analytic continuation. For two model problems accurate scattering information is obtained over a wide range of energies, and resonances are easily located by noting the zeros of the real part of the determinant.