Riemann Sheets Research Articles

The role of a kernel in complex langevin systems

The role of a kernel in the complex Langevin equation is investigated in detail theoretically and numerically. We have applied this scheme to Minkowski stochastic quantization and the numerical simulation of simple polynomial models. It is shown that the kernel can stabilize the solution of the Langevin equation. Solutions in different Riemann sheets are found and criteria are given to select the physical one.

Observable effects of poles and shadow poles in coupled-channel systems.

Using separable potentials in a coupled-channel Lippmann-Schwinger equation, we investigate the motion of poles on the different Riemann sheets and their observable effects as the coupling strength is varied. Only in the weak-coupling limit can one determine in which uncoupled channel the pole originated by the sheet on which it lies. Cusp structure versus resonance peak as one crosses a threshold is found to depend on the distance of the pole from the physical sheet. Resonance poles in each uncoupled channel can produce double-loop Argand plots similar to those seen in \ensuremath{\pi}N ${P}_{11}$ amplitude analysis.

Propagating, damped, and leaky surface waves on the corrugated traction-free boundary of an elastic half-space

The dispersion relation for surface waves on the corrugated boundary (periodic in one direction and constant in the other) of an elastic half-space is derived using a modal approach, and the result is shown to be equivalent to that derived by the null-field approach. The dispersion relation is solved numerically for roots on all Riemann sheets for varying corrugation heights, frequencies, and angles of propagation. Many of the roots move on several sheets, and this leads to zero, one, or two roots residing on the physical sheet.

Nonuniqueness of families of periodic solutions in a four dimensional mapping

If we follow a family of periodic solutions along a closed path in a parameter space of two dimensions we may not return to the original solution when the parameters return to the original values. We study such nonuniqueness phenomena in simple and double period familis. Nonuniqueness appears if a closed path in the parameter space goes around a critical point. In some cases we find Riemann sheets in the same way as in multiply valued functions. In other cases the connections of various families change in a complicated way around the critical point. All these phenomena are explained analytically. At the critical point there is a ‘collision of bifurcations’. The changes of the connections of various families at such collisions of bifurcations are studied in some detail.

Series analysis of multivalued functions.

A recurrent problem in mathematical physics, for example, in the theory of critical phenomena, is the need to study the structure of physically interesting functions at a branch point of a complex structure usually called a ``confluent singularity.'' In such a neighborhood the function is necessarily multivalued. In addition, the value of such a function is sometimes required on a branch cut or even off the first Riemann sheet. Our approach to this problem is inspired by the Riemann (global) monodromy theorem and consists of using series expansions to form integral approximants (special case of Hermite-Pad\'e approximants) to represent multivalued functions on multiple Riemann sheets. We prove an analogous local-monodromy-theorem, functional-representation results. We further identify the important ``separation property'' and use it to prove a convergence theorem for horizontal sequences of integral approximants. We make an extensive numerical investigation, using horizontal, diagonal, and constrained diagonal sequences, and find that these methods give excellent results on a wide variety of test functions of rather complex structure.

The double many‐body expansion of potential energy surfaces from interacting 2S atoms

AbstractThe double many‐body expansion (DMBE) method has been applied to the many‐valued ground‐state potential energy surfaces from three and four interacting 2S atoms. New forms are suggested where the degeneracy manifold of the two Riemann sheets of the potential surface can be made identical to that predicted from the London equation that is correct to first‐order. Considerable flexibility can be introduced through adjustable coefficients to fit ab initio data, whereas the potential energy terms can be chosen to maintain the correct analytic properties at the conical intersection. Hence, the new forms may be used on the extrapolation of the potential energy from the lowest‐ to the upper‐Riemann sheets. Long‐range effects are semiempirically described from the general formalism of the DMBE method.

Pole structure of the J pi =3/2(+) resonance in 5He.

The S-matrix pole structure of the ${\mathrm{J}}^{\mathrm{\ensuremath{\pi}}}$${=(3/2}^{+}$ resonance in $^{5}\mathrm{He}$ is obtained from R-matrix parameters for the system. Two poles are found on different unphysical Riemann sheets. One is a conventional resonance that is primarily responsible for structure in the n-\ensuremath{\alpha} total cross section. The other is a ``shadow'' pole that contributes to the large values of the cross section for the reaction $^{3}\mathrm{H}$(d,n${)\mathrm{}}^{4}$He at low energies. It is the first experimental evidence for the existence of shadow poles in nuclear and particle physics. The two-pole structure of the resonance accounts for different peak positions and widths in the amplitudes for the n-\ensuremath{\alpha} total cross section and for the $^{3}\mathrm{H}$(d,n) reaction cross section.

Fusion-energy reaction 3H

We have extended our past measurements of the $^{2}\mathrm{H}$(t,\ensuremath{\alpha})n reaction near the low-energy ${(3/2)}^{+}$ resonance by measuring eight more data points over the lab deuteron energy range 80--116 keV. This was accomplished by bombarding a tritium gas target with deuterons, in contrast to the previous measurements in which a deuterium gas target was bombarded with tritons. The present data are accurate to 1.6%. The results of including the present data in a simple two-channel, two-level, R-matrix analysis and also in a large three-channel, multilevel, R-matrix analysis are presented. The resonance is characterized by giving the S-matrix poles from the R-matrix analyses. Of interest is the discovery that both analyses give two resonance poles on different (unphysical) Riemann sheets, one of them being a so-called ``shadow pole.'' This is the first experimental observation of a shadow pole in nuclear and particle physics. Maxwellian reactivities up to a plasma temperature of 20 keV are presented.

On the singularity analysis of intersecting separatrices in near-integrable dynamical systems

It has been recently proved by S. L. Ziglin that transversal intersections of separatrices (invariant manifolds) in near-integrable Hamiltonian systems of two degrees of freedom imply the existence of multi-valued solutions with infinitely many Riemann sheets. Ziglin's theorem is illustrated here on a simple example and then extended and applied to non-Hamiltonian, analytic flows x ̇ =f( x) + εg( x,t) , with x≡(x,y) and g( x,t)=g( x,t+2π) , which for ε = 0 possess the Painlevé property. On the other hand, the theoretically expected logarithmic singularities for ε ≠ 0 are obtained explicitly in solutions near the intersecting separatrices. Thus, we conjecture that dynamical systems with the Painlevé property, can have no separatrix intersections and hence no strange attractors, etc. These singularities are then numerically located and found to form certain very interesting “chimney” patterns in the complex t-plane, on which they accumulate densely. The upper part of the chimneys (away from the Re t axis) is asymptotically quite insensitive to changes in parameter values or initial conditions. The singularity pattern itself, however, becomes “ denser” and each chimney is seen to gradually “collapse” towards the Re t axis, as the amplitude of the driving term increases and the motion becomes more chaotic.

An exact solutin for incompressible flow through a two-dimensional Laval nozzle

A careful examination of the variation of the velocity along the centerline and the contour of a Laval nozzle in the physical plane shows that either the upper or the lower half of the Laval nozzle assumes the same form of a slitted thick airfoil with tandem trailing edges. These two airfoils lie on different Riemann sheets in the hodograph plane. The interior of the airfoil is then mapped onto an infinite strip in the complex potential plane. Making use of these results, we obtained an exact solution for the incompressible potential flow through a two-dimensional Laval nozzle. The solution is applicable for nozzles with any given contraction ratio mexpansion rations, and throat wall radius R*. As examples of the method, various nozzle contours, the velocity distribution of the flow, and the locations of the fluid particles at different time intervals are presented.

Singularities in the scattering of a very slow electron by a weakly polar molecule

A simple one-electron potential with a dipole tail is used to study the tracks of poles of the scattering amplitude in the complex momentum plane, close to the origin, and for states with a large s-wave component. Several Riemann sheets have to be considered, because the dipole tail leads to a branch point at the origin. On the 'physical' sheet, there is a single track in the upper half-plane; it follows the positive imaginary axis down to the origin and stops there. On other Riemann sheets, there are two distinct tracks in the lower half-plane, which are mirror images of one another in the negative imaginary axis. They meet the track on the physical sheet at the origin. The tracks in the lower half-plane exist only if the dipole moment is less than 1.19 D, and collapse to the negative imaginary axis if the dipole moment vanishes.

Fast generation of three-dimensional computational boundary-conforming periodic grids of C-type

A fast computer program, GRID3C, has been developed to generate multilevel three-dimensional, C-type, periodic, boundary conforming grids for the calculation of realistic turbomachinery and propeller flow fields. The technique is based on two analytic functions that conformally map a cascade of semi-infinite slits to a cascade of doubly infinite strips on different Riemann sheets. Up to four consecutively refined three-dimensional grids can be automatically generated and permanently stored on four different computer tapes. Grid nonorthogonality is introduced by a separate coordinate shearing and stretching performed in each of three coordinate directions. The grids can be easily clustered closer to the blade surface, the trailing and leading edges and the hub or shroud regions by changing appropriate input parameters. Hub and duct (or outer free boundary) can have different axisymmetric shapes. A vortex sheet of arbitrary thickness emanating smoothly from the blade trailing edge is generated automatically by GRID3C. Blade cross-sectional shape, chord length, twist angle, sweep angle, and dihedral angle can vary in an arbitrary smooth fashion in the spanwise direction. Input coordinates must be Cartesian, while the output grid coordinates can be Cartesian or cylindrical.

Unstable-particle scattering and an analytic quasi-unitary isobar model

An explicit model is constructed for the scattering of unstable particles such as $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\pi}$, ${K}^{*}\ensuremath{\pi}\ensuremath{\rightarrow}{K}^{*}\ensuremath{\pi}$, and $\ensuremath{\rho}K\ensuremath{\rightarrow}\ensuremath{\rho}K$. It has correct analyticity properties and satisfies quasi-two-body unitarity. The essential ingredient of the approach is a Chew-Mandelstam function for unstable particles in which the particle-resonance branch cuts are located in second Riemann sheets of the three-particle complex energy plane. The model shall be applied in future studies of final states such as $\ensuremath{\rho}\ensuremath{\pi}$, ${K}^{*}\ensuremath{\pi}$, and $\ensuremath{\rho}K$.

Problems pertaining to the inclusive cross section

The consequences of the appearance of anomalous thresholds in the 3-to-3 amplitude are analyzed. It is shown that they are the result of constraints imposed by unitarity and can produce different asymptotic behavior on different Riemann sheets for the six-point function. The implications for the inclusive cross section in the fragmentation and pionization regions are discussed.

Complex roots of the Stoneley-wave equation

Abstract The equation governing elastic waves propagating along a solid-solid interface is found to have sixteen (16) independent roots on its eight (8) associated Riemann sheets. The range of existence (in terms of material parameters) for the real root corresponding to the propagation of Stoneley waves has long been known. It is found that outside this range there are two types of behavior. If the material of greater density has a velocity slightly greater than that of the material of lesser density, the unattenuated Stoneley waves make a transition to attenuated Interface waves, i.e., they leak energy away from the interface as they propagate along it. If the more dense material has a velocity more than about three times that of the less dense, then the Interface-wave root disappears and energy is propagated along the interface as Rayleigh waves. This Rayleigh-wave propagation is associated with a different root of the fundamental equation. On the other hand, if the material of greater density has a velocity much lower than that of the material of lower density (a case that is difficult to find physically), then no energy will be propagated along the interface at all. This result was unexpected. Some rather interesting behavior of the 16 roots was noted as the physical parameters were varied over a wide range. In addition to the normal collisions between pairs of roots, and between individual roots and branch points (with attendant Riemann sheet jumping), it was found that some roots go through the point at infinity and return with a change in sign. At least one unexpected case of a multiple root was found. Another case was noted in which a pair of complex roots change quadrants in the complex phase-velocity plane, leading to a discontinuity in root type. Finally, it was noted that, in a cyclic variation of the material parameters, it is possible to choose a path such that the roots, when followed individually, will not return to their original values. In fact, as many as five cycles in parameter space can be accomplished before the roots return. All this strange mathematical behavior seems to have no physical significance, but has been presented to increase understanding of the general behavior of the dispersion relations associated with elastic-wave propagation.

Two-Spin-Wave Resonances and the Analyticity of thetMatrix in the Heisenberg Ferromagnet

We consider the scattering of two spin waves in a simple cubic Heisenberg ferromagnet with nearest-neighbor interaction at zero temperature. The analytic properties of the integrals relevant to the bound-state problem are examined and the solutions to the bound-state conditions are located in the complex energy ($\ensuremath{\omega}$) plane for the first time. It is found that, as a function of $\ensuremath{\omega}$, there are two Riemann sheets near the bottom of the two-spin-wave band. For total wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ less than the threshold value for bound states, the existence of the $d$-wave resonant states discovered by Boyd and Callaway is reaffirmed. Furthermore, we confirm the observation of Boyd and Callaway that no $s$-wave scattering resonance exists.

Direct-Channel Reggeization of Strong-Interaction Scattering Amplitudes. I

The construction of Reggeized strong-interaction amplitudes is discussed in a series of three papers. In this paper (Paper I), the amplitude is expanded in a partial-wave series with the individual partial-wave amplitudes represented as products over direct-channel Regge-pole contributions. Unitarity bounds automatically hold for direct-channel processes. Such representations are shown to account for the essential features of both elastic and inelastic scattering, in terms of the Regge trajectories on several Riemann sheets. It is shown that difficulties usually associated with rising trajectories are naturally avoided in this formulation.

Current algebra and π-π scattering lengths. A dispersion relation approach

The usual simple linear expansion of the scattering amplitude ins, t andu is used to continue on to the mass shell but not to physical values ofs. The continuation to physicals. values is made using a linear combination of the partial-wave dispersion relations on the first and second Riemann sheets, into which any desired effects of the unitarity cut may easily be inserted. A complicated spectrum ofT=0 andT=2 scattering lengths is obtained. Many of the qualitative features of the solution are in agreement with other calculations in which some account has been taken of the unitarity cut.

Inelastic Levinson's theorem, CDD singularities, and multiple resonance poles

Some effects of inelastic channels on elastic partial wave amplitudes are discussed. A Levinson's theorem for both the real part of the phase shift and the total phase of the elastic amplitude is derived. The CDD singularities required to make the elastic amplitude calculated by single-channel inelastic N D equations agree with the many-channel calculation without CDD singularities are fully characterized in both the η and R methods. Finally, a simple explanation of multiple resonance poles on different Riemann sheets of the amplitude is given in terms of the analyticity properties of η.

Radiation from a center-fed cylindrical antenna surrounded by a plasma sheath

The radiation field of a plasma-clad cylindrical antenna fed across an infinitesimal gap has been derived. The limitations on the validity of the far-field pattern have been stated. In the vicinity of the antenna, but in the region far from the gap, an expression is derived for the surface wave contribution which is the major source of field strength in this region. Analysis of the contour integration and discussion of the location of the various types of poles of the integrand on their appropriate Riemann sheets are presented. Numerical computations of far-field patterns and surface wave fields are presented graphically.