Complete Green Function Research Articles

Functional Integral Method for Potential Scattering Amplitude in Quantum Mechanics

The functional integral method can be used in quantum mechanics to find the scattering amplitude for particles in the external field. We obtained the potential scattering amplitude from the complete Green function in the corresponding external field through solving the Schrodinger equation, after being separated from the poles on the mass shell, which takes the form of an eikonal (Glauber) representation in the high energy region and the small scattering angles. Considering specific external potentials such as the Yukawa or Gaussian potential, we found the corresponding differential scattering cross-sections.

Internal Tide Generation Using Green Function Analysis: To WKB or Not to WKB?

AbstractAn established analytical technique for modeling internal tide generation by barotropic flow over bottom topography in the ocean is the Green function–based approach. To date, however, for realistic ocean studies this method has relied on the WKB approximation. In this paper, the complete Green function method, without the WKB approximation, is developed and tested, and in the process, the accuracy of the WKB approximation for realistic ridge geometries and ocean stratifications is considered. For isolated Gaussian topography, the complete Green function approach is shown to be accurate via close agreement with the results of numerical simulations for a wide range of height ratios and criticality; in contrast, the WKB approach is found to be inaccurate for small height ratios in the subcritical regime and all tall topography that impinges on the pycnocline. Two ocean systems are studied, the Kaena and Wyville Thomson Ridges, for which there is again excellent agreement between the complete Green function approach and numerical simulations, and the WKB approximate solutions have substantial errors. This study concludes that the complete Green function approach, which is typically only modestly more computationally expensive than the WKB approach, should be the go-to analytical method to model internal tide generation for realistic ocean ridge scenarios.

Topographic scattering of the low-mode internal tide in the deep ocean

We investigate the role of deep-ocean topography in scattering energy from the large spatial scales of the low-mode internal tide to the smaller spatial scales of higher modes. The complete Green function method, which is not subject to the restrictions of the WKB approximation, is used for the first time to study the two-dimensional scattering of a mode-1 internal tide incident on subcritical and supercritical topography of any form in arbitrary stratifications. For an isolated Gaussian ridge in a uniform stratification, large amplitude critical topography is the most efficient at mode-1 scattering and small amplitude topography scatters with an efficiency on the order of 5–10%. In a nonuniform stratification with a pycnocline, the results are qualitatively the same as for a constant stratification, albeit with the key features shifted to larger height ratios. Having validated these results by direct comparison with the results of nonlinear numerical simulations, and in the process demonstrated that WKB results are not appropriate for reasonable ocean predictions, we proceed to use the Green function approach to quantify the role of topographic scattering for the region of the Pacific Ocean surrounding the Hawaiian Islands chain. To the south, the Line Islands ridge is found to scatter ∼40% of a mode-1 internal tide coming from the Hawaiian Ridge. To the north, realistic, small-amplitude, rough topography scatters ∼5–10% of the energy out of mode 1 for transects of length 1000–3000 km. A significant finding is that compared to large extents of small-amplitude, rough topography a single large topographic feature along the path of a mode-1 internal tide plays the dominant role in scattering the internal tide.

The second act of hydro: small perturbations

Hydrodynamical description of the ‘Little Bang’ in heavy ion collisions is surprisingly successful: here we systematically study propagation of small perturbations treated hydrodynamically. Using analytic description of the expanding fireball known as the ‘Gubser flow’, we proceed to linearized equations for perturbations. As all variables are separated and all equations solved (semi)analytically, we can collect all the harmonics and reconstruct the complete Green function of the problem, even in the viscous case. Applying it to the power spectrum, we found acoustic minimum at m = 7 and maximum at m = 9, which remarkably have some evidence for both in the data. We estimate effective viscosity and the size of the perturbation from the fit to power spectrum. The shape of the two-point correlator is also reproduced remarkably well. At the end, we argue that independent perturbations are local, and thus harmonics phases are correlated.

Fate of the initial state perturbations in heavy ion collisions. III. The second act of hydrodynamics

The hydrodynamical description of the ``Little Bang'' in heavy-ion collisions is surprisingly successful, mostly owing to the very small viscosity of the quark-gluon plasma. In this paper we systematically study the propagation of small perturbations, also treated hydrodynamically. We start with a number of known techniques allowing for the analytic calculation of the propagation of small perturbations on top of the expanding fireball. The simplest approximation is the ``geometric acoustics,'' which substitutes the wave equation by mechanical equations for the propagating ``phonons.'' Next we turn to the case in which variables can be separated, where one can obtain not only the eikonal phases but also the amplitudes of the perturbation. Finally, we focus on the so-called Gubser flow, a particular conformal analytic solution for the fireball expansion, on top of which one can derive closed equations for small perturbations. Perfect hydrodynamics allows all variables to be separated and all equations to be solved in terms of known special functions. We can thus collect the analytical expression for all the harmonics and reconstruct the complete Green's function of the problem. In the viscous case the equations still allow for variable separation, but one of the equations has to be solved numerically. Summing all the harmonics we show real-time perturbation evolution, observing the viscosity-induced changes in the spectra and the correlation functions. The calculated angular shape of the correlation function is remarkably similar to the shape emerging from the experimental data, for sufficiently large viscosity. We predict a minimum at $m\ensuremath{\sim}7$ and maximum at $m\ensuremath{\sim}9$ harmonics, which also have some experimental evidence for it. We conclude that local ``hot spots'' in the initial state are the only visible origin of the observed correlations.

Retrieval of Moho-reflected shear wave arrivals from ambient seismic noise

Theoretical studies on ambient seismic noise (ASN) predict that complete Green's function between seismic stations can be retrieved from cross correlation. However, only fundamental mode surface waves emerge in most studies involving real data. Here we show that Moho-reflected body wave (SmS) and its multiples can be identified with ASN for station pairs near their critical distances in the short period band (1–5 s). We also show that an uneven distribution of noise sources, such as mining activity and wind–topography interaction, can cause surface wave precursors, which mask weaker body wave phases.

Reply to comment by M. S. Bos on “Anomalous ocean load tide signal observed in lake‐level variations in Tierra del Fuego”

[1] The comments of Bos [2010] are valuable and certainly contribute to an improved understanding of the tidal interactions in Tierra del Fuego. They are based on a thorough examination of the subject and emphasize the scientific relevance of this investigation. [2] Bos [2010, Figure 2] presents modelled lake-tide predictions that differ from those given in our article. The reason for this difference consists essentially in incorrect phase lags for the direct effect as given by Richter et al. [2009, Figure 2a]. Due to inconsistencies between the used ETERNA and TASK-2000 software tools an unnoticed time reference shift of one hour was introduced which resulted in the about 30° too large phase lags. [3] However, our modelled load tides (indirect effect) are not affected thereby. Still, the load tides predicted by Bos [2010] are about 10% larger in amplitude compared to our results. This is most likely explained by the application of different load models for the Magellan Strait (e.g., our Magellan Strait model produces a slight reduction in the total load tide amplitude in contrast to the statement by Bos). An impact of accounting for the complete global tidal load model rather than our limited domain in the order of 0.2 mm, 15° as stated by Bos seems to us questionable. Test computations had shown that using the tidal loading predictions of M. S. Bos and H.-G. Scherneck (Ocean tide loading provider, 2009, available at http://www.oso.chalmers.se/∼loading/) (implying convolution over the complete global model; FES2004 ocean tide model, Gutenberg-Bullen earth model) changes the load tide contributions at the three tide gauge sites by no more than 0.01 mm and 1.5°, respectively (crustal deformation effect only). [4] A corrected version of Richter et al.'s [2009] Figure 2a is displayed in Figure 1. It reveals now a smaller difference between observed and modelled lake tide signals. However, the remaining misclosure still exceeds the uncertainties of the observed signal and also the 3σ interval (which has been added in Figure 1). Here, σ represents the site-specific standard deviation of the M2 amplitude and M2 phase as determined from a 12-months lake-level observation record. For site B, the σ were derived directly as the standard deviation of the M2 tidal parameters of three independent 12-month records (σ12B). For sites A and C, for which only one year of observations is available, the respective σ were derived from the standard deviations of the M2 parameters of two independent 6-month records, scaled by the ratio σ12B/σ6B between the standard deviations of the 12-month solution and two 6-month solutions (σ6B) at site B (assuming that the gain in accuracy by increasing the record length from 6 to 12 months is equal at all three sites). The geometrical relation between the vectors (i.e., relative phase relation) continues to suggest that this significant difference between observation and model is to be attributed to an underestimation of the modelled load tide amplitudes. A satisfactory agreement would require a scaling of the modelled load tide amplitudes by about 120%. [5] The article was aimed to present our lake-tide observations and their deviation from the model predictions. Our considerations concerning the elastic earth model were only one general interpretation/speculation in an attempt to explain the obtained differences. We did, however, not pretend to infer a realistic elastic earth model from the tidal parameters. For such an undertaking our approach (constant scaling of the complete Green's function, no consideration of vertical crustal structure, etc.) would indeed be too simplistic. Therefore, the discussion of this topic by Bos [2010, Figure 1] represents a valuable extension of knowledge. [6] Thus, thanks to the comment by Bos [2010] the deviation of the modelled load tide from the observations has been reduced from 30% to 20%. Nevertheless, a 20% amplification of the theoretical load tides is still important compared to other published observational results. And, in our opinion, it is still difficult to explain it by uncertainties of the observations and the load (ocean tide) model only. On the other hand, a substantial amplification of elastic loading effects has been observed also in other regions in South America [Bevis et al., 2004]. However, we agree with Bos that the differences between the observations in Lago Fagnano and the predictions of the internet ocean tide loading provider should be interpreted with care. At present, the authors are continuing the investigations of tidal interactions in Tierra del Fuego and the results will be made public to the scientific community in due time.

Persistence in advection of a passive scalar

We consider the persistence phenomenon in advected passive scalar equation in one dimension. The velocity field is random with the v(k,omega)v(-k,-omega) approximately mid R:kmid R:;{-(2+alpha)} . In the presence of the nonlinearity the complete Green's function becomes G;{-1}=-iomega+Dk2+Sigma . We determine Sigma self-consistently from the correlation function which gives Sigma approximately k;{beta} , with beta=(1-alpha)2 . The effect of the nonlinear term in the equation in the O(;{2}) is to replace the diffusion term due to molecular viscosity by an effective term of the form Sigma_{0}k;{beta} . The stationary correlator for this system is [sech(T2)];{1beta} . Using the self-consistent theory we have determined the relation between beta and alpha . Finally, the independent interval approximation (IIA) method is used to determine the persistent exponent.

Fano interference in mesoscopic rings with quantum dots

AbstractWe propose a general formalism for describing the coexistence of coherent and resonant transport in hybrid mesoscopic structures. The approach is based on Landauer‐Büttiker formula for the electronic transmittance and on an old formula of Feshbach. The latter gives the complete Green function of coupled subsystems in terms of effective Green functions of the disconnected parts and provides informations about the individual contributions of each subsystem to transport. Motivated by the experiments of Kobayashi et al.(Phys. Rev. Lett. 88, 256806 (2002)) and Holleitner et al. (Phys. Rev. Lett. 87, 256802 (2001)) we apply the formalism to study transport in Aharonov‐Bohm interferometers (ABI) containing one or two coupled two‐dimensional quantum dots (QD). In the single dot case, we reproduce and explain the magnetic field control of the Fano interference and investigate the interaction effects in a self‐consistent approach. In the double dot case, we obtain the charging diagrams and establish precise criteria for the observation of mesoscopic Fano effect and AB oscillations. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A very broad‐band stochastic source model used for near source strong motion prediction

The simulation of near‐source ground motion requires the calculation of ground motion in a very broad‐band frequency range (typically from 10 s to 0.1 s). However, the calculation of complete Green's function is too time consuming for high‐frequency simulations. We present a method to obtain very broad‐band synthetic seismograms, combining the kinematic spectral source model of Bernard et al., [1996], and an algorithm, based on the wavenumber technique of Bouchon and Aki, [1977]. High‐frequency radiations are evaluated with ray theory and a broad‐band dislocation defined on a finely discretized fault plane. The low‐frequency radiations due to the near‐ and intermediate‐fields are calculated with the complete field algorithm on a low‐wavenumber filtered dislocation. This dislocation is defined on a coarse grid over the fault, hence significantly reducing the computing time. We apply this method to compute the ground motion at a station located near an extended fault, where the near‐, intermediate‐ and far‐field radiations are important.

The radiation modes of a vertically varying half-space: a new representation of the complete Green's function in terms of modes

SUMMARY We prove that the complete Green’s function for an isotropic vertically varying halfspace can be written in a dyadic form as a sum on the usual normal modes plus an integral over radiation modes. The radiation modes form the basis on which to represent the body waves, which are not trapped in the structure. The expressions for their eigenfunctions, the so-called improper eigenfunctions, are found for both the SH and the P-SV wave operators. We prove that they are orthogonal to each other and to the normal-mode eigenfunctions. In a numerical example, we present improper eigenfunctions, and compare complete wavefield seismograms synthesized by the mode method and by the discrete wavenumber integration method.

Green function for multilayers: Light scattering in planar cavities.

A compact and transparent plane-wave expansion of the complete Green function for an absorbing multilayered system is derived. The plane-wave solutions involved match, in the appropriate limit, the space-mode functions recently employed for quantization of the radiation field in a planar cavity and make this Green function particularly convenient for consideration of optical processes in realistic cavities and multilayers in general. This is illustrated by considering classically light scattering on a molecule in a planar cavity. It is demonstrated that the cross sections derived, besides generally describing light scattering on molecules embedded in a multilayer, correctly reproduce the effects of field confinement on light scattering in a planar cavity as predicted recently by the quantum-mechanical theory of Cairo et al. [Phys. Rev. Lett. 70, 1413 (1993)]. It is also pointed out that scattering into waves guided by the cavity lossy (metallic) mirrors may become an effective process in low-order microcavities. As a by product, a classical approach to spontaneous emission in absorbing planar cavities emerges from these considerations.

Transition operators in electromagnetic-wave diffraction theory. II. Applications to optics.

This paper is the second of a series. The first [G. E. Hahne, Phys. Rev. A 45, 7526 (1992)] developed a general theory of the transition operator approach to diffraction of time-harmonic electromagnetic waves from fixed obstacles, such that the response of the obstacle, denoted by \ensuremath{\Omega}, to an impinging electromagnetic signal with wave number ${\mathit{k}}_{0}$ was simulated by nonlocal, homogeneous Leontovich (i.e., impedance) boundary conditions on the obstacle's surface, which surface is called \ensuremath{\partial}\ensuremath{\Omega}. Moreover, the exterior region, called ${\mathrm{\ensuremath{\Omega}}}^{\mathrm{ex}}$, was presumed to be unbounded empty space, and has an electromagnetic response that can be expressed in terms of the so-called radiation impedance operator, denoted Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$; Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ is a certain invertible, linear functional operator that maps the space of complex tangent-vector fields on \ensuremath{\partial}\ensuremath{\Omega} into itself.The matching of the limiting tangential electric and magnetic fields on \ensuremath{\partial}\ensuremath{\Omega} yielded a functional-operator expression for the transition operator and thereby a formal reduction to quadratures of the entire direct-scattering problem. This paper is intended to serve as an illustration and elaboration of the formalism presented in the first paper. After a brief recapitulation of the theory, the following topics are dealt with. First, an infinite sum in terms of vector spherical harmonics is obtained for Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ for \ensuremath{\partial}\ensuremath{\Omega} a sphere. Second, a quasiplanar approximation, based directly on the exact result for planar \ensuremath{\partial}\ensuremath{\Omega}, is obtained for Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ when \ensuremath{\Vert}${\mathit{k}}_{0}$\ensuremath{\Vert} is large compared to the local surface curvatures of \ensuremath{\partial}\ensuremath{\Omega} and attention can be restricted to small neighborhoods on \ensuremath{\partial}\ensuremath{\Omega}; when augmented by the method of stationary phase, this approximation is shown to lead to the familiar physical optics method for smooth, convex surfaces. Third, the latter method, supported by further interventions of the method of stationary phase, is applied in a well-established manner to secure the results of geometrical optics for the complete Green's function for the time-harmonic Maxwell field in the presence of a smooth-surfaced, convex, perfectly electrically conducting obstacle.One feature of the latter computations is that the original source currents from which the free-space Maxwell Green's function is constructed are presumed to generate ordinary outgoing (``causal'') electromagnetic waves for electric current sources, and purely ingoing (``anticausal'') waves for magnetic current sources (i.e., sinks) of electromagnetic radiation. Fourth, a formal construction is derived for mapping the Leontovich boundary conditions on an inner surface into a set of Leontovich boundary conditions on an outer surface, in the circumstance that a layer comprised of a material medium, which has possibly nonuniform and nonsymmetric tensors representing its constitutive properties, fills the domain between the surfaces; the construction presumes that a Green's function is available for the corresponding Maxwell equations in the medium when the medium is extended appropriately to fill all space. The main body of the paper concludes with a brief discussion of directions of possible future work and applications. The first appendix shows how to apply the method of stationary phase in the present context, in particular for the mixed case that the original radiation source is anticausal and the response currents generated in the obstacle are causal sources of radiation. A second appendix develops an acoustic (Helmholtz equation) analog to the fourth topic mentioned above, and exhibits an interplay of the theory with symplectic transformations in the case that the principle of reciprocity holds; it is noted that a symplectic connection also exists in the electromagnetic case when the propagation medium's constitutive properties are such that reciprocity holds.

Resonant state expansion of the resolvent.

An analytic method of generating resonant state expansions from the standard completeness relation of nonrelativistic quantum mechanics is described and shown to reproduce the generalized completeness relations, earlier derived, involving resonant states. The method is then applied to the expansion of the resolvent (the complete Green's function), the symmetry properties of which seem to be destroyed if a conventional application of the completeness relations is made. These forms of expansions have a continuum term which contains symmetry-restoring contributions and can therefore never vanish identically, nor can it be neglected. The symmetry-conserving form of the expansion has a set of discrete terms which are identical in form to those of the Mittag-Leffler series for the resolvent. In addition, it contains a continuum contribution which in some cases vanishes identically, but in general does not. We illustrate these findings with numerical applictions in which the potential (a square well) is chosen so as to permit analytic evaluation of practically all functions and quantities involved.

Scattering from slot near conducting wedge using hybrid method of moments/geometrical theory of diffraction: TE case

An efficient solution for the scattering of a transverse electric plane wave from a slot near a wedge has been implemented using the hybrid method of moments/geometrical theory of diffraction. Comparisons with the complete Green function solution show that the hybrid technique is accurate even for small spacings.

Transition operators in electromagnetic-wave diffraction theory: General theory.

A formal theory is developed for the scattering of time-harmonic electromagnetic waves from impenetrable immobile obstacles with given linear, homogeneous, and generally nonlocal boundary conditions of Leontovich (impedance) type for the wave of the obstacle's surface. The theory is modeled on the complete Green's function and the transition (T) operator in time-independent formal scattering theory of nonrelativistic quantum mechanics. An expression for the differential scattering cross section for plane electromagnetic waves is derived in terms of certain matrix elements of the T operator for the obstacle.

Transition operators in acoustic-wave diffraction theory. I. General theory.

The objective of this paper is the establishment of a formal theory of the scattering of time-harmonic acoustic scalar waves from impenetrable, immobile obstacles; the time-independent formal scattering theory of nonrelativistic quantum mechanics, in particular the theory of the complete Green's function and the transition (T) operator, provides the model. The quantum-mechanical approach is modified to allow the treatment of acoustic-wave scattering with imposed boundary conditions of impedance type on the surface \ensuremath{\partial}\ensuremath{\Omega} of an impenetrable obstacle. With ${\mathit{k}}_{0}$ as the free-space wave number of the signal, a simplified expression is obtained for the ${\mathit{k}}_{0}$-dependent T operator for a general case of homogeneous impedance boundary conditions for the acoustic wave on \ensuremath{\partial}\ensuremath{\Omega}. All the nonelementary operators that enter the expression for the T operator are formally simple, rational algebraic functions of a certain invertible linear operator Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}$, which is called the radiation impedance operator, and which maps any sufficiently well-behaved, complex-valued function on \ensuremath{\partial}\ensuremath{\Omega} into another such function on \ensuremath{\partial}\ensuremath{\Omega}. The nonlocal operators Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}$ and Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}^{\mathrm{\ensuremath{-}}1}$ are defined only implicitly, in that Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}^{\mathrm{\ensuremath{-}}1}$ is the operator that maps the limiting-value function on \ensuremath{\partial}\ensuremath{\Omega} of an outgoing-wave solution to the scalar Helmholtz equation into the uniquely corresponding limiting-normal-derivative function, and Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}$ does the inverse operation. Previous appearances in the literature of these operators, or their analogs for other elliptic, linear partial-differential equation systems, are cited. An analytical study of the dominant singularities of the operators (considered as two-point kernels on a smooth \ensuremath{\partial}\ensuremath{\Omega}), and of their behavior in the geometrical acoustics limit, is the subject of a second paper [G. E. Hahne, following paper, Phys. Rev. A 43, 1990 (1991)].

Optimum strong‐motion array geometry for source inversion—II

AbstractOptimum strong‐motion array geometry for source inversions is again determined for each of three types of earthquake faults: strike‐slip, dip‐slip and offshore subduction thrust. The method is the same as employed in a previous study;1 however, use of a complete Green's function in an elastic half‐space provides better results for engineering practice. It is found that the complete Green's function is capable of stabilizing the accuracy of an inversion solution obtained using theoretical seismograms, regardless of the differences in array configuration. The optimum strong‐motion array for a strike‐slip fault is characterized by stations well distributed in azimuth, while the optimum array for a dip‐slip event has stations arranged in a grid‐shaped form. The array geometries obtained here are grossly similar to those in the previous study,1 which were derived using only the far‐field S waves, and are more consistent with those proposed at the 1978 International Workshop on Strong‐Motion Earthquake Instrument Arrays.2.

Transition-metal impurities in semiconductors and heterojunction band lineups.

The empirical rule connecting transition-metal levels in semiconductors with band-edge offsets at semiconductor heterojunctions is theoretically justified by a complete Green's-function calculation. It is shown from a detailed analysis that the transition-metal levels are effectively pinned to the average dangling-bond energy, the latter being at the same time interpreted as the charge neutrality level which controls the band-edge offset of a heterojunction.

Sagittal elastic waves in cubic superlattices

Using a recent extension of the surface Green function matching (SGFM) method, we calculate bulk bands for sagittal waves in superlattices made of alternating layers of two different elastic cubic media. The secular equation, involving a surface projection of the superlattice Green function, is written in terms of the bulk Green functions for each medium. This is applied to the calculation of dispersion relations for GaAs-AlAs and Al-W superlattices, two systems exhibiting appreciable differences in their gap patterns. The analysis can be readily used to construct the complete Green function of the system, hence to calculate any desired spectral function. Application is made to study the spatial dependence of the spectral strength (local density of states) corresponding to various representative modes in Al-W superlattices, separating out the longitudinal from the total strength.