Riemann Sheets Research Articles

Supersymmetric quantum mechanics living on topologically non-trivial Riemann surfaces

Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators H (±) is chosen antilinear. Secondly, both these components of a super-Hamiltonian $$ \mathcal{H} $$ are defined along certain topologically non-trivial complex curves r (±)(x) which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map $$ \mathcal{T} $$ between ‘tobogganic’ partner curves r (+)(x) and r (−)(x) is emphasized.

Acoustic mode waves and individual arrivals excited by a dipole source in fluid-filled boreholes

An approach of separating individual wave arrivals for a dipole logging is presented. The branch points are treated as a type of logarithm and the Riemann surface structure of the multivalued function is studied, that is, the displacement potential within the borehole. Based on the theoretical analysis, the complex poles contributing to the wave field on various Riemann sheets are investigated in detail for the case of a fast formation. It is shown that poles on Riemann sheet (0,0) are real and form branches of modes with dispersion. Mathematically, it is demonstrated that the flexural mode has no cutoff frequency, which is different from the traditional point of view. Poles on other relevant Riemann sheets are complex and form many branches on the complex frequency-wavenumber plane. Further investigation on the pole and branch cut contributions indicates that the vertical branch cut integration method has limitations in separating wave arrivals. By properly taking into account the complex poles on various Riemann sheets together with branch cut integrations, wave arrivals are separated from the full waveforms effectively for both the fast and slow formation models. Specially, there are complex poles on Riemann sheet (0,−1) in the vicinity of the compressional branch cut for a slow formation with a large Poisson’s ratio, which have small imaginary parts and contribute a lot to the p-wave arrival.

Leaky modes and their contributions to the compressional head wave in a borehole excited by a dipole source

A method of studying the contributions of leaky modes to the wave field is presented based on the analysis of the Riemann surface structure of the characteristic function, and the sensitivities of contributions to various factors of interest are examimed. Numerical results show that their contributions to the compressional head wave are related to the distributions of complex poles on (−1, −1) and (0, −1) Riemann sheets on the frequency-wavenumber (ω-k) plane. For fast formations, their contributions are small, while for slow formations with large Poisson’s ratio, their contributions are large because of those complex poles with small imaginary parts near the compressional vertical branch cut. The decaying factor of the contributions of leaky modes is approximately proportional to 1/distance2.

Accurate evaluation of Green’s functions in a layered medium by SDP-FLAM

Based on local Taylor expansions on the complex plane, a method for fast locating all modes (FLAM) of spectral-domain Green’s Functions in a planar layered medium is developed in this paper. SDP-FLAM, a combination of FLAM with the steepest descent path algorithm (SDP), is employed to accurately evaluate the spatial-domain Green’s functions in a layered medium. According to the theory of complex analysis, the relationship among the poles, branch points and Riemann sheets is also analyzed rigorously. To inverse the Green’s functions from spectral to spatial domain, SDP-FLAM method and discrete complex image method (DCIM) are applied to the non-near field region and the near filed region, respectively. The significant advantage of SDP-FLAM lies in its capability of calculating Green’s functions in a layered medium of moderate thickness with loss or without loss. Some numerical examples are presented to validate SDP-FLAM method.

Extraction of resonances from meson-nucleon reactions

We present a pedagogical study of the commonly employed speed-plot (SP) and time-delay (TD) methods for extracting the resonance parameters from the data of two-particle coupled-channels reactions. Within several exactly solvable models, it is found that these two methods find poles on different Riemann sheets and are not always valid. We then develop an analytic continuation method for extracting nucleon resonances within a dynamical coupled-channel formulation of $\ensuremath{\pi}N$ and $\ensuremath{\gamma}N$ reactions. The main focus of this paper is on resolving the complications from the coupling with the unstable $\ensuremath{\pi}\ensuremath{\Delta}$, $\ensuremath{\rho}N$, and $\ensuremath{\sigma}N$ channels, which decay into $\ensuremath{\pi}\ensuremath{\pi}N$ states. By using the results from the considered exactly solvable models, explicit numerical procedures are presented and verified. As a first application of the developed analytic continuation method, we present the nucleon resonances in some partial waves extracted within a recently developed coupled-channels model of $\ensuremath{\pi}N$ reactions. The results from this realistic $\ensuremath{\pi}N$ model, which includes $\ensuremath{\pi}N$, $\ensuremath{\eta}N$, $\ensuremath{\pi}\ensuremath{\Delta}$, $\ensuremath{\rho}N$, and $\ensuremath{\sigma}N$ channels, also show that the simple pole parametrization of the resonant propagator using the poles extracted from SP and TD methods works poorly.

Fold Points and Singularities in Hall MHD Differential–Algebraic Equations

We consider the singularity crossing phenomenon in differential-algebraic equations (DAEs) of Hall MHD systems in one spatial dimension. The Hall MHD DAEs have singularities with impasse points, pseudoequilibrium points, or singularity-induced bifurcation (SIB) points. The pseudoequilibrium and SIB points allow for smooth transitions between the plus (supersonic) and minus (subsonic) Riemann sheets. Within the singular pseudoequilibrium points, there may exist only one analytic trajectory crossing the sonic curve (as in the case of SIB point), two analytic and two other trajectories of lower degree of smoothness in the case of pseudosaddle points, or two analytic and an uncountable number of trajectories of lower smoothness in the case of singular pseudonodes. In this paper, we show examples of singular points in Hall MHD systems described by DAEs and explain the singularity (also called the sonic or forbidden curve) crossing phenomenon by using the recent developments in the qualitative analysis of DAEs.

An efficient numerical algorithm for stability testing of fractional-delay systems

This paper presents a numerical algorithm for BIBO stability testing of a certain class of the so-called fractional-delay systems. The characteristic function of the systems under consideration is a multi-valued function of the Laplace variable s which is defined on a Riemann surface with finite number of Riemann sheets where the origin is a branch point. The stability analysis of such systems is not straightforward because there is no universally applicable analytical method to find the roots of the characteristic equation on the right half-plane of the first Riemann sheet. The proposed method is based on the Rouche’s theorem which provides the number of the zeros of a given function in a given simple closed contour. One advantage of the proposed method over previous works is that it gives the number and the location of the unstable poles. The algorithm has a reliable result which is illustrated by several examples.

0 + and 1 + heavy mesons in heavy chiral unitary approach

In terms of the heavy chiral unitary approach, the D-s0* (2317), D-s1* (2460) and D-0* (2308) resonances are discussed within the molecular state model. By examining the poles of the full unitary coupled-channel scattering amplitude on appropriate Riemann sheets, the former two states can be well reproduced, but not the third state. If one believes that the third state and first state are corresponding states in the non-strange and strange sectors, respectively, and they are all dominated by the molecular structure, there should exist one wide-width state at about 2.1 GeV and one narrow-width state at about 2.44 GeV, which are associated with the D-0*(2308) state. Meanwhile, we predict possible B-0*(5536) [B-1*(5581)] and B-0*(5819) [B-1*(5877)] states in the non-strange sector as the corresponding states of the strange B-s0* (5725) [B-s1* (5778)] ones.

Dynamically generated 1+ heavy mesons

By using a heavy chiral unitary approach, we study the S wave interactions between heavy vector meson and light pseudoscalar meson. By searching for poles of the unitary scattering amplitudes in the appropriate Riemann sheets, several 1(+) heavy states are found. In particular, a D*K bound state with a mass of 2.462 +/- 0.010 GeV which should be associated with the recently observed D-s1 (2460) state is obtained. In the same way, a B*(K) over bar boundstate (B-s1) with mass of 5.778 +/- 0.007 GeV in the bottom sector is predicted. The spectra of the dynamically generated D-1 and B-1 states in the I = 1/2 channel are also calculated. One broad state and one narrow state are found in both the charmed and bottom sectors. The coupling constants and decay widths of the predicted states are further investigated. (c) 2007 Elsevier B.V. All rights reserved.

Resolving the wave vector in negative refractive index media

We address the general issue of resolving the wave vector in complex electromagnetic media including negative refractive media. This requires us to make a physical choice of the sign of a square root imposed merely by conditions of causality. By considering the analytic behavior of the wave vector in the complex plane, it is shown that there are a total of eight physically distinct cases in the four quadrants of two Riemann sheets.

Avoided level crossings and singular points

Avoided crossings of discrete states as well as of resonance states are traced back to the existence of branch points (BPs) which are singular points in the complex energy plane. They cannot be identified with exceptional points. At a BP width bifurcation occurs, different Riemann sheets evolve and the levels do not cross anymore when the system is still further opened. The BPs are physically meaningful: they influence the dynamics of open as well as of closed quantum systems. The geometric phase that arises by encircling a BP, is different from the phase that appears by encircling a diabolic point. This is found to be true even for the two BPs into which the diabolic point is unfolded by opening the system.

Low lying axial-vector mesons as dynamically generated resonances

We make a theoretical study of the $s$-wave interaction of the nonet of vector mesons with the octet of pseudoscalar mesons starting from a chiral invariant Lagrangian and implementing unitarity in coupled channels. By looking for poles in the unphysical Riemann sheets of the unitarized scattering amplitudes, we get two octets and one singlet of axial-vector dynamically generated resonances. The poles found can be associated with most of the low lying axial-vector resonances quoted by the Particle Data Group: ${b}_{1}(1235)$, ${h}_{1}(1170)$, ${h}_{1}(1380)$, ${a}_{1}(1260)$, ${f}_{1}(1285)$, and two poles to the ${K}_{1}(1270)$ resonance. We evaluate the couplings of the resonances to the $VP$ states and the partial decay widths in order to reinforce the arguments in the discussion.

Avoided level crossings, diabolic points, and branch points in the complex plane in an open double quantum dot

We study the spectrum of an open double quantum dot as a function of different system parameters in order to receive information on the geometric phases of branch points in the complex plane (BPCP). We relate them to the geometrical phases of the diabolic points (DPs) of the corresponding closed system. The double dot consists of two single dots and a wire connecting them. The two dots and the wire are represented by only a single state each. The spectroscopic values follow from the eigenvalues and eigenfunctions of the Hamiltonian describing the double dot system. They are real when the system is closed, and complex when the system is opened by attaching leads to it. The discrete states as well as the narrow resonance states avoid crossing. The DPs are points within the avoided level crossing scenario of discrete states. At the BPCP, width bifurcation occurs. Here, different Riemann sheets evolve and the levels do not cross anymore. The BPCP are physically meaningful. The DPs are unfolded into two BPCP with different chirality when the system is opened. The geometric phase that arises by encircling the DP in the real plane, is different from the phase that appears by encircling the BPCP. This is found to be true even for a weakly opened system and the two BPCP into which the DP is unfolded.

Collective modes of an anisotropic quark-gluon plasma: II

We continue our exploration of the collective modes of an anisotropic quark gluon plasma by extending our previous analysis to arbitrary Riemann sheets. We demonstrate that in the presence of momentum-space anisotropies in the parton distribution functions there are new relevant singularities on the neighboring unphysical sheets. We then show that for sufficiently strong anisotropies that these singularities move into the region of spacelike momentum and their effect can extend down to the physical sheet. In order to demonstrate this explicitly we consider the polarization tensor for gluons propagating parallel to the anisotropy direction. We derive analytic expressions for the gluon structure functions in this case and then analytically continue them to unphysical Riemann sheets. Using the resulting analytic continuations we numerically determine the position of the unphysical singularities. We then show that in the limit of infinite contraction of the distribution function along the anisotropy direction that the unphysical singularities move onto the physical sheet and result in real spacelike modes at large momenta for all "out-of-plane" angles of propagation.

Inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

An inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions is derived by introducing an affine parameter to avoid constructing Riemann sheets. A one-soliton solution simpler than that in the literature is obtained, which is a breather and degenerates to a bright or dark soliton as the discrete eigenvalue becomes purely imaginary. The solution is mapped to that of the modified nonlinear Schrödinger equation by a gaugelike transformation, predicting some sub-picosecond solitons in optical fibers.

The Proper Current Spectra of an Open Integrated Microstrip Waveguide

Continuous EM field spectrum of an integrated open waveguide structure is identified as the branch cut contribution to singularity expansion of those fields in the complex axial transform plane. Those singularities in that plane include poles associated with the guiding structure and branch points contributed by layered background environments. The manner in which singularities in background environments manifest themselves as branch points in the complex axial transform plane is reviewed. Using a spectral microstrip current profile [1], the EM fields in the background layer environments of the integrated open waveguide are recovered from the spectral domain by integration contour deformation on the different Riemann sheets of the complex ζ-plane. During the integration both around branch cuts and along real axis, singularities in the complex transverse transform plane migrate in a complicated manner. EM fields are computed through those two methodologies are numerically validated. Extensive numerical results show the nature of EM fields physically in the open waveguide structure, which can be decomposed into discrete and continuous spectra in the layered background environments.

The Proper EM Field Spectra of an Open Integrated Microstrip Waveguide

Continuous EM field spectrum of an integrated open waveguide structure is identified as the branch cut contribution to singularity expansion of those fields in the complex axial transform plane. Those singularities in that plane include poles associated with the guiding structure and branch points contributed by layered background environments. The manner in which singularities in background environments manifest themselves as branch points in the complex axial transform plane is reviewed.Using a spectral microstrip current profile [1], the EM fields in the background layer environments of the integrated open waveguide are recovered from the spectral domain by integration contour deformation on the different Riemann sheets of the complex ζ-plane. During the integration both around branch cuts and along real axis, singularities in the complex transverse transform plane migrate in a complicated manner. EM fields are computed through those two methodologies are numerically validated.Extensive numerical results show the nature of EM fields physically in the open waveguide structure, which can be decomposed into discrete and continuous spectra in the layered background environments.

Laser induced surface modes at water–elastic and poroelastic solid interfaces

A study of acoustic waves propagating along a fluid–solid plane interface, induced by thermoelastic excitation with a laser beam, is performed. The displacement of the interface in the time domain is predicted with an integration over frequency and on different Riemann sheets. The prediction, which includes the effect of the bulk waves induced by the thermoelastic excitation, can be performed at small distances from the heated region. For a bulk velocity in the fluid smaller than the shear wave velocity in the solid (water–glass configuration), and between the velocity of the shear and the compressional wave (water–plexiglass configuration), the poles that contribute to the displacement are identified by comparing the results of the integration in the different paths. A first analysis of the laser induced surface modes for water–porous ceramics configurations is performed in the context of the Biot theory. Measurements are presented.

The Riemann surface of the Chiral Potts model free energy function

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice “inversion relation” method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site T pq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of T pq . In terms of the t p ,t q variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N−1)-dimensional lattice (for N>2). The function T pq is meromorphic on this surface: we obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N+4)/4] parameters to be determined.

Linear [formula omitted] model in the Gaussian wave functional approximation II: analyticity of the S-matrix and the effective potential/action

We show an explicit connection between the solution to the equations of motion in the Gaussian functional approximation [I. Nakamura, V. Dmitrašinović, Prog. Theor. Phys. 106 (2001) 1195] and the minimum of the (Gaussian) effective potential/action of the linear Σ model, as well as with the N/D method in dispersion theory. The resulting equations contain analytic functions with branch cuts in the complex mass squared plane. Therefore the minimum of the effective action may lie in the complex mass squared plane. Many solutions to these equations can be found on the second, third, etc. Riemann sheets of the equation, though their physical interpretation is not clear. Our results and the established properties of the S-matrix in general, and of the N/D solutions in particular, guide us to the correct choice of the Riemann sheet. We count the number of states and find only one in each spin-parity and isospin channel with quantum numbers corresponding to the fields in the Lagrangian, i.e., to Castillejo–Dalitz–Dyson (CDD) poles. We examine the numerical solutions in both the strong and weak coupling regimes and calculate the Källén–Lehmann spectral densities and then use them for physical interpretation.