Complex Angular Momentum Plane Research Articles

From Black Hole Spectral Instability to Stable Observables.

The quasinormal mode spectrum of black holes is unstable under small perturbation of the potential and has observational consequences in time signals. Such signals might be experimentally difficult to observe and probing this instability will be a technical challenge. Here, we investigate the spectral instability of time-independent data. This leads us to study the Regge poles (RPs), the counterparts to the quasinormal modes in the complex angular momentum plane. We present evidence that the RP spectrum is unstable but that not all overtones are affected equally by this instability. In addition, we reveal that behind this spectral instability lies an underlying structure. The RP spectrum is perturbed in such a way that one can still recover stable scattering quantities using the complex angular momentum approach. Overall, the study proposes a novel and complementary approach on the black hole spectral instability phenomena that allows us to reveal a surprising and unexpected mechanism at play that protects scattering quantities from the instability.

Exact quantization and analytic continuation

In this paper we give a streamlined derivation of the exact quantization condition (EQC) on the quantum periods of the Schrödinger problem in one dimension with a general polynomial potential, based on Wronskian relations. We further generalize the EQC to potentials with a regular singularity, describing spherical symmetric quantum mechanical systems in a given angular momentum sector. We show that the thermodynamic Bethe ansatz (TBA) equations that govern the quantum periods undergo nontrivial monodromies as the angular momentum is analytically continued between integer values in the complex plane. The TBA equations together with the EQC are checked numerically against Hamiltonian truncation at real angular momenta and couplings, and are used to explore the analytic continuation of the spectrum on the complex angular momentum plane in examples.

Numerical Regge pole analysis of resonance structures in state-to-state reactive differential cross sections

This is the third (and the last) code in a collection of three programs [Sokolovski et al. (2011), Akhmatskaya et al. (2014)] dedicated to the analysis of numerical data, obtained in an accurate simulation of an atom-diatom chemical reaction. Our purpose is to provide a detailed description of a FORTRAN code for complex angular momentum (CAM) analysis of the resonance effects in reactive angular scattering [for CAM analysis of integral reactive cross sections see [Akhmatskaya et al. (2014)]. The code evaluates the contributions of a Regge trajectory (or trajectories) to a differential cross section in a specified range of energies. The contribution is computed with the help of the methods described in [Dobbyn et al. (1999), Sokolovski and Msezane (2004), Sokolovski et al. (2007)]. Regge pole positions and residues are obtained by analytically continuing S-matrix element, calculated numerically for the physical integer values of the total angular momentum, into the complex angular momentum plane using the PADE_II program [Sokolovski et al. (2011)]. The code represents a reactive scattering amplitude as a sum of the components corresponding to a rapid “direct” exchange of the atom, and the various scenarios in which the reactants form long-lived intermediate complexes, able to complete several rotations before breaking up into products. The package has been successfully tested on the representative models, as well as the F + H2→ HF+H benchmark reaction. Several detailed examples are given in the text. Program summaryProgram title:DCS_ReggeCPC Library link to program files:https://doi.org/10.17632/gf4gm82n6m.1Licensing provisions: GPLv3Programming language:FORTRAN 90Supplementary material:PADE_II, MPFUN and QUADPACK packages, validation suites, script files, input files, readme files, Installation and User GuideNature of problem: The package extracts the positions and residues of resonance poles from numerical scattering data supplied by the user. This information is then used for the analysis of resonance structures observed in reactive differential cross sections.Solution method: The S-matrix element is analytically continued in the complex plane of either energy or angular momentum with the help of Padé approximation of type II. Resonance Regge trajectories are identified and their contributions to a differential cross section are evaluated at different angles and energies.Additional comments including restrictions and unusual features: Use of multiple precision MPFUN package.

Insight of the relation between Regge theory and Hadron dynamics

In the present contribution, efforts are made to develop an insight between two verticals of strong interaction phenomenology, namely the archaic Regge theory and an avant-garde Hadron dynamics. Even before the advent of quantum chromodynamics (QCD), Regge theory served as an effective tool in the understanding of the high energy limit of hadronic scattering processes and thereby Hadron dynamics. Regge theory provided a study mathematical framework based upon the theory of the complex angular momentum. It related the high energy behaviour of hadronic scattering processes to the singularities of the scattering amplitude in the complex angular momentum plane. With the advent of QCD, Regge formalism was however shunted to the background. Perhaps, the development of large accelerator centres such as CERN and BNL have led to the detection of large number of hadronic resonances and thus have rekindled interest in the Regge Theory specifically with the aid of Regge trajectories which are traced from these resonances. Theoretical and experimental modelling of Regge theory are now proving to be a successful venture.

Double logarithms in mathcal{N} \u2265 4 supergravity: weak gravity and Shapiro\u2019s time delay

A study of the double logarithmic in the center-of-mass energy, s, contributions to the four-graviton scattering amplitude is presented for four-dimensional mathcal{N} ≥ 4 supergravities. This includes a novel representation for the coefficients of the perturbative expansion based on exactly solvable recurrences. A review is given of the structure in the complex angular momentum plane for the t-channel partial wave singularities of the different amplitudes. Working in impact parameter representation, ρ, it is shown that the resummation of double logarithms makes gravity weaker in regions of small ρ and large s. This screening of the gravitational interaction at short distances in the double logarithmic sector of the amplitudes is more acute as the number of gravitinos in the theory increases. It brings corrections to the eikonal phase which can change the sign of the graviton’s deflection angle and generate regions with repulsive interaction. For very small impact parameters there appears a constant negative shift in both the eikonal phase and Shapiro’s time delay which is not large enough to generate causality violation.

Double-logarithms in mathcal{N} = 8 supergravity: impact parameter description & mapping to 1-rooted ribbon graphs

The set of double-logarithmic (DL) contributions (α t ln2s)n to the 4-graviton amplitude in mathcal{N} = 8 supergravity (SUGRA), with α being the gravitational coupling and (s, t) the Mandelstam invariants, is studied in impact parameter (ρ) representation. This sector of the amplitude shows interesting properties which shed light on the nature of quantum corrections in gravity. Besides having a convergent behaviour as s increases, which is not present in mathcal{N} < 4 SUGRA theories, there exists a critical line ρc(s) above which the Born amplitude prevails. The short distance region ρ < ρc(s) is dominated by the DL terms. As a consequence, when studied in terms of an eikonal approach in the forward limit, the scattering angle linked to the bending of the semiclassical trajectory of the graviton shows a transition from attractive gravity at large distances to a region at small ρ characterized by a repulsive DL contribution to the gravitational potential due to the gravitino content of the theory. In the complex angular momentum plane, this DL high energy asymptotics is driven by the rightmost pole singularity of a parabolic cylinder function. The resummation of DL quantum corrections in mathcal{N} = 8 SUGRA can be understood in terms of the counting of 1-rooted maps on orientable surfaces.

Regge pole description of scattering of scalar and electromagnetic waves by a Schwarzschild black hole

We revisit the problem of scalar and electromagnetic waves impinging upon a Schwarzschild black hole from complex angular momentum techniques. We focus more particularly on the associated differential scattering cross sections. We derive an exact representation of the corresponding scattering amplitudes by replacing the discrete sum over integer values of the angular momentum which defines their partial wave expansions by a background integral in the complex angular momentum plane plus a sum over the Regge poles of the $S$-matrix involving the associated residues. We show that, surprisingly, the background integral is numerically negligible for intermediate and high frequencies and, as a consequence, that the cross sections can be reconstructed in terms of Regge poles with very good agreement. We show in particular that, for large values of the scattering angle, a small number of Regge poles permits us to describe the black hole glory and that, by increasing the number of Regge poles, we can reconstruct very efficiently the differential scattering cross sections for small and intermediate scattering angles and therefore describe the orbiting oscillations. In fact, in the short-wavelength regime, the sum over Regge poles allows us to extract by resummation the information encoded in the partial wave expansion defining a scattering amplitude and, moreover, to overcome the difficulties linked to its lack of convergence due to the long-range nature of the fields propagating on the black hole. As a consequence, from asymptotic expressions for the lowest Regge poles and the associated residues based on the correspondence Regge poles -- ``surface waves'' propagating close to the photon sphere, we can provide an analytical approximation describing both the black hole glory and a large part of the orbiting oscillations. We finally discuss the role of the background integral for low frequencies.

Forward elastic scattering and Pomeron models

Recent data from LHC13 by the TOTEM Collaboration have indicated an unexpected decrease in the value of the $\rho$ parameter and a $\sigma_{tot}$ value in agreement with the trend of previous measurements at 7 and 8 TeV. These data at 13 TeV are not simultaneously described by the predictions from Pomeron models selected by the COMPETE Collaboration, but show agreement with the maximal Odderon dominance, as recently demonstrated by Martynov and Nicolescu. Here we present a detailed analysis on the applicability of Pomeron dominance by means of a general class of forward scattering amplitude, consisting of even-under-crossing leading contributions associated with single, double and triple poles in the complex angular momentum plane. We carry out fits to $pp$ and $\bar{p}p$ data in the interval 5 GeV - 13 TeV. The data set comprises all the accelerator data below 7 TeV and we consider two independent ensembles by adding either only the TOTEM data or TOTEM and ATLAS data at the LHC energy region. In the data reductions to each ensemble the uncertainty regions are evaluated with both one and two standard deviation ($\sim$ 68 \% and $\sim$ 95 \% CL, respectively). Besides the general analytic model, we investigate four particular cases of interest, three of them typical of outstanding models in the literature. We conclude that, within the experimental and theoretical uncertainties and both ensembles, the general model and three particular cases are not able to describe the $\sigma_{tot}$ and $\rho$ data at 13 TeV simultaneously. However, if the discrepancies between the TOTEM and ATLAS data are not resolved, one Pomeron model, associated with double and triple poles and with only 7 free parameters, seems not to be excluded by the complete set of experimental information presently available.

Soft pomerons and the forward LHC data

Recent data from LHC13 by the TOTEM Collaboration on σtot and ρ have indicated disagreement with all the Pomeron model predictions by the COMPETE Collaboration (2002). On the other hand, as recently demonstrated by Martynov and Nicolescu (MN), the new σtot datum and the unexpected decrease in the ρ value are well described by the maximal Odderon dominance at the highest energies. Here, we discuss the applicability of Pomeron dominance through fits to the most complete set of forward data from pp and p¯p scattering. We consider an analytic parameterization for σtot(s) consisting of non-degenerated Regge trajectories for even and odd amplitudes (as in the MN analysis) and two Pomeron components associated with double and triple poles in the complex angular momentum plane. The ρ parameter is analytically determined by means of dispersion relations. We carry out fits to pp and p¯p data on σtot and ρ in the interval 5 GeV–13 TeV (as in the MN analysis). Two novel aspects of our analysis are: (1) the dataset comprises all the accelerator data below 7 TeV and we consider three independent ensembles by adding: either only the TOTEM data (as in the MN analysis), or only the ATLAS data, or both sets; (2) in the data reductions to each ensemble, uncertainty regions are evaluated through error propagation from the fit parameters, with 90% CL. We argument that, within the uncertainties, this analytic model corresponding to soft Pomeron dominance, does not seem to be excluded by the complete set of experimental data presently available.

Complex angular momenta approach for scattering problems in the presence of both monopoles and short range potentials

It is analyzed the quantum mechanical scattering off a topological defect (such as a Dirac monopole) as well as a Yukawa-like potential(s) representing the typical effects of strong interactions. This system, due to the presence of a short-range potential, can be analyzed using the powerful technique of the complex angular momenta which, so far, has not been employed in the presence of monopoles (nor of other topological solitons). Due to the fact that spatial spherical symmetry is achieved only up to internal rotations, the partial wave expansion becomes very similar to the Jacob-Wick helicity amplitudes for particles with spin. However, since the angular-momentum operator has an extra "internal" contribution, fixed cuts in the complex angular momentum plane appear. Correspondingly, the background integral in the Regge formula does not decrease for large values of cos(Theta) (namely, large values of the Mandelstam variable s). Hence, the experimental observation of this kind of behavior could be a direct signal of non-trivial topological structures in strong interactions. The possible relations of these results with the soft Pomeron are shortly analyzed.

State-to-State F + H2 Reaction at Etrans = 0.04088 eV: QP Decomposition, Parametrized S Matrix Incorporating Regge Poles, and Uniform Asymptotic Complex Angular Momentum Analysis of the Angular Scattering

We report two new contributions for understanding the quantum dynamics of the benchmark state-to-state reaction, F + H2(vi, ji, mi) → FH(vf, jf, mf) + H, where (vi, ji, mi) and (vf, jf, mf) are the initial and final vibrational, rotational, and helicity quantum numbers, respectively. We analyze product differential cross sections (DCSs) for the transitions, 000 → 300, 000 → 310, and 000 → 320, at a translational energy of 0.04088 eV using the potential energy surface of Fu-Xu-Zhang. The two new contributions are as follows: (1) We exploit the recently introduced QP decomposition of J. N. L. Connor [ J. Chem. Phys . 2013 , 138 , 124310 ] to transform numerical partial-wave scattering (S) matrix elements for the three transitions into parametrized (analytic) formulas, in which all terms in the three parametrized S matrices have a direct physical interpretation. In particular, they contain the positions and residues of Regge poles in the first quadrant of the complex angular momentum (CAM) plane. We obtain very close agreement between the values of the parametrized and numerical S matrix elements. (2) We then apply a uniform asymptotic Watson/CAM theory, which allows a Regge pole to be close to a saddle point. It uses the parametrized S matrices and is applied to the partial wave series (PWS) representation for the scattering amplitude to understand structure in a DCS in terms of three contributing subamplitudes. We prove using this powerful CAM theory that resonance Regge poles contribute to the small-angle scattering in the DCSs for all three transitions, with the oscillations at larger angles arising from nearside-farside interference. We obtain very good agreement between the uniform asymptotic Watson/CAM DCSs and the corresponding PWS DCSs, except for angles close to the forward and backward directions, where (as expected) the Watson/CAM formulas become nonuniform.

Phenomenological theory of echo poles

Abstract In scattering theory the effect associated with the downward crossing of the phase-shift δ l ( k ) ( l being the orbital angular momentum and k the momentum) through δ l = π / 2 (mod π ) is called echo . In the standard nuclear theory (Breit–Wigner theory) the echo is described and evaluated in terms of scattering by an impenetrable sphere. However, this model holds only at sufficiently high energy, while it is inadequate at low energy. In this paper we show that the echo effect can be associated with two different regimes acting at low and high energy, respectively. At high energy the hard-sphere scattering model seems to describe appropriately the phenomenon. At low energy we propose a mechanism due to the exchange forces induced by the Pauli exclusion principle in the fermionic interaction, which leads to nonlocal potentials. These potentials admit for the scattering amplitude pole singularities in the fourth quadrant of the complex angular momentum plane. This paper analyzes the role played by this class of poles in the description of the low energy regime of echoes. A specific phenomenological analysis is performed, taking as typical example the α – α elastic scattering.

Numerical Regge pole analysis of resonance structures in elastic, inelastic and reactive state-to-state integral cross sections

We present a detailed description of a FORTRAN code for evaluation of the resonance contribution a Regge trajectory makes to the integral state-to-state cross section (ICS) within a specified range of energies. The contribution is evaluated with the help of the Mulholland formula (Macek et al., 2004) and its variants (Sokolovski et al., 2007; Sokolovski and Akhmatskaya, 2011). Regge pole positions and residues are obtained by analytically continuing S-matrix element, evaluated numerically for the physical values of the total angular momentum, into the complex angular momentum plane using the PADE_II program (Sokolovski et al., 2011). The code decomposes an elastic, inelastic, or reactive ICS into a structured, resonance, and a smooth, ‘direct’, components, and attributes observed resonance structure to resonance Regge trajectories. The package has been successfully tested on various models, as well as the F+H2→HF+H benchmark reaction. Several detailed examples are given in the text. Program summaryProgram title: ICS ReggeCatalogue identifier: AEST_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEST_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 71962No. of bytes in distributed program, including test data, etc.: 1405035Distribution format: tar.gzProgramming language: FORTRAN 90.Computer: Any computer equipped with a FORTRAN 90 compiler.Operating system: UNIX, LINUX.RAM: 256 MbClassification: 16.8.External routines: MPFUN (http://crd-legacy.lbl.gov/~dhbailey/mpdist/), QUADPACK http://www.netlib.org/quadpack/ (included in the distribution file)Subprograms used:Cat IdTitleReferenceAEHO_v1_0PADE IICPC 182 (2011) 448Nature of problem:The package extracts the positions and residues of resonance poles from numerical scattering data supplied by the user. This information is then used for the analysis of resonance structures observed in elastic, inelastic and reactive integral cross sections.Solution method:The S-matrix element is analytically continued in the complex plane of either energy or angular momentum with the help of Padé approximation of type II. Resonance Regge trajectories are identified and their contributions to the integral cross section are evaluated.Unusual features:Use of multiple precision MPFUN package.Additional comments:The following are included in the distribution file: PADE II, MPFUN and QUADPACK packages, validation suites, script files, input files, readme files, Installation and User GuideRunning time:From several minutes to hours depending on the number of energies involved, the precision level chosen and the number of iterations performed.

Identification of non-ordinary mesons from the dispersive connection between their poles and their Regge trajectories: The [formula omitted] resonance

We show how the Regge trajectory of a resonance can be obtained from its pole in a scattering process and analytic constraints in the complex angular momentum plane. The method is suited for resonances that dominate an elastic scattering amplitude. In particular, from the ρ(770) resonance pole in ππ scattering, we obtain its linear Regge trajectory, characteristic of ordinary quark–antiquark states. In contrast, the f0(500) pole—the sigma meson—which dominates scalar isoscalar ππ scattering, yields a nonlinear trajectory with a much smaller slope at the f0(500) mass. Conversely, imposing a linear Regge trajectory for the f0(500), with a slope of typical size, yields an elastic amplitude at odds with the data. This provides strong support for the non-ordinary nature of the sigma meson.

Extracting S-matrix poles for resonances from numerical scattering data: Type-II Padé reconstruction

We present a FORTRAN 77 code for evaluation of resonance pole positions and residues of a numerical scattering matrix element in the complex energy (CE) as well as in the complex angular momentum (CAM) planes. Analytical continuation of the S-matrix element is performed by constructing a type-II Padé approximant from given physical values (Bessis et al. (1994) [42]; Vrinceanu et al. (2000) [24]; Sokolovski and Msezane (2004) [23]). The algorithm involves iterative ‘preconditioning’ of the numerical data by extracting its rapidly oscillating potential phase component. The code has the capability of adding non-analytical noise to the numerical data in order to select ‘true’ physical poles, investigate their stability and evaluate the accuracy of the reconstruction. It has an option of employing multiple-precision (MPFUN) package (Bailey (1993) [45]) developed by D.H. Bailey wherever double precision calculations fail due to a large number of input partial waves (energies) involved. The code has been successfully tested on several models, as well as the F + H 2 → HF + H, F + HD → HF + D, Cl + HCl → ClH + Cl and H + D 2 → HD + D reactions. Some detailed examples are given in the text. Program summary Program title: PADE II Catalogue identifier: AEHO_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHO_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 19 959 No. of bytes in distributed program, including test data, etc.: 158 380 Distribution format: tar.gz Programming language: FORTRAN 77 Computer: Any computer equipped with a FORTRAN 90 compiler Operating system: UNIX, LINUX RAM: 256 Mb Classification: 16.8 External routines: NAG Program Library ( http://www.nag.co.uk/numeric/fl/FLdescription.asp) Nature of problem: The package extracts the positions and residues of resonance poles from numerical scattering data supplied by the user. The data can then be used for quantitative analysis of interference patterns observed in elastic, inelastic and reactive integral and differential cross sections. Solution method: The S-matrix element is analytically continued in the complex plane of either energy or angular momentum with the help of Padé approximation of type II. Resonance (complex energy or Regge) poles are identified and their residues evaluated. Unusual features: Use of multiple precision MPFUN package (Bailey (1993) [45]). (Distributed with the PADE II code.) Running time: From several seconds to several minutes depending on the precision level chosen and the number of iterations performed.

On Regge pole trajectories for a rational function approximation of Thomas–Fermi potentials

The positions of two Regge poles caused by the rational function approximate Thomas–Fermi potential (a screened attractive Coulomb potential) are studied in detail as a function of the scattering energy. The leading pole is traced from the right-hand (physical) complex angular momentum plane ℜ(ℓ) > −1/2 to the left-hand (unphysical) complex angular momentum plane ℜ(ℓ) < −1/2 as the scattering energy is increased indefinitely. The second Regge pole is located in the unphysical half-plane at all energies. In this study an exact numerical and an approximate semiclassical WKB methods are discussed in some detail, where the boundary conditions of the regular (physical) Schrödinger solution become important. Transitions in behavior of Regge poles are explained in terms of Stokes lines and turning points evolution. The major change in the direction of Regge trajectories is demonstrated as a transition in the topology of Stokes lines. This phenomenon is caused by switching from one pair of turning points to another pair as the third turning point approaches the Stokes line connecting to the original turning points.

Beauty, charm, and F L at HERA: New data vs. Early predictions

One of the well known effects of the asymptotic freedom is splitting of the leading-$\log$ BFKL pomeron into a series of isolated poles in complex angular momentum plane. Following our earlier works we explore the phenomenological consequences of the emerging BFKL-Regge factorized expansion for the small-$x$ charm ($F_2^c$) and beauty ($F_{2}^{b}$) structure functions of the proton. As we found earlier,the color dipole approach to the BFKL dynamics predicts uniquely decoupling of subleading hard BFKL exchanges from $F_2^c$ at moderately large $Q^{2}$. We predicted precocious BFKL asymptotics of $F_2^c(x,Q^2)$ with intercept of the rightmost BFKL pole $\alpha_{\Pom}(0)-1=\Delta_{\Pom}\approx 0.4$. High-energy open beauty photo- and electro-production probes the vacuum exchange at much smaller distances and detects significant corrections to the BFKL asymptotics coming from the subleading vacuum poles. In view of the accumulation of the experimental data on small-$x$ $F_{2}^{c}$ and $F_{2}^{b}$ we extended our early predictions to the kinematical domain covered by new HERA measurements.Our structure functions obtained in 1999 agree well with the determination of both $F_2^c$ and $F_2^b$ by the H1 published in 2006 but contradict to very recent (2008, preliminary) H1 results on $F_2^b$. We present also comparison of our early predictions for the longitudinal structure function $F_L$ with recent H1 data (2008) taken at very low Bjorken $x$. We comment on the electromagnetic corrections to the Okun-Pomeranchuk theorem.

The BFKL-Regge factorization and F 2 b , F 2 c , F L at HERA: physics implications of nodal properties of the BFKL eigenfunctions

The asymptotic freedom is known to split the leading-log BFKL pomeron into a series of isolated poles in the complex angular momentum plane. One of our earlier findings was that the subleading hard BFKL exchanges decouple from such experimentally important observables as small-x charm F2c, beauty F2b and the longitudinal structure functions of the proton at moderately large Q2. For instance, we predicted precocious BFKL asymptotics of F2c(x, Q2) with intercept of the rightmost BFKL pole aIP(0) − 1 = ΔIP ≈ 0.4. On the other hand, the small-x open beauty photo- and electro-production probes the vacuum exchange for much smaller color dipoles which entails significant subleading vacuum pole corrections to the small-x behavior. In view of the accumulation of the experimental data on small-xF2c and F2b we extend our 1999 predictions to the kinematical domain covered by new HERA measurements. Our parameter-free results agree well with the determination of F2c, FL and published H1 results F2b on but slightly overshoot the very recent (2008, preliminary) H1 results on F2b.

Surface plasmon polaritons and surface phonon polaritons on metallic and semiconducting spheres: Exact and semiclassical descriptions

We study the interaction of an electromagnetic field with a non-absorbing or absorbing dispersive sphere in the framework of complex angular momentum techniques. We assume that the dielectric function of the sphere presents a Drude-like behavior or an ionic crystal behavior modelling metallic and semiconducting materials. We more particularly emphasize and interpret the modifications induced in the resonance spectrum by absorption. We prove that ``resonant surface polariton are generated by a unique surface wave, i.e., a surface (plasmon or phonon) polariton, propagating close to the sphere surface. This surface polariton corresponds to a particular Regge pole of the electric part (TM) of the $S$ matrix of the sphere. From the associated Regge trajectory we can construct semiclassically the spectrum of the complex frequencies of the resonant surface polariton modes which can be considered as Breit-Wigner-type resonances. Furthermore, by taking into account the Stokes phenomenon, we derive an asymptotic expression for the position in the complex angular momentum plane of the surface polariton Regge pole. We then describe semiclassically the surface polariton and provide analytical expressions for its dispersion relation and its damping in the non-absorbing and absorbing cases. In these analytic expressions, we more particularly exhibit well-isolated terms directly linked to absorption. Finally, we explain why the photon-sphere system can be considered as an artificial (a ``plasmonic atom or ``phononic atom) and we briefly discuss the implication of our results in the context of the Casimir effect.

UNIFIED ALGEBRAIC TREATMENT OF RESONANCE

Energy resonance in scattering is usually investigated either directly in the complex energy plane (E-plane) or indirectly in the complex angular momentum plane (ℓ-plane). Another formulation complementing these two approaches was introduced recently. It is an indirect algebraic method that studies resonances in a complex charge plane (Z-plane). This latter approach will be generalized to provide a unified algebraic treatment of resonances in the complex E-, ℓ-, and Z-planes. The complex scaling (rotation) method will be used in the development of this approach. The resolvent operators (Green's functions) are formally defined in these three spaces. Bound states spectrum and resonance energies in the E-plane are mapped onto a discrete set of poles of the respective resolvent operator on the real line of the ℓ- and Z-planes. These poles move along trajectories as the energy is varied. A finite L2 basis is used in the numerical implementation of this approach. Stability of poles and trajectories against variation in all computational parameters is demonstrated. Resonance energies for a given potential are calculated and compared with those obtained by other studies.