Method Of Analytic Continuation Research Articles

Ionisation and excitation probabilities of a hydrogen atom suddenly released from penetrable confinement

ABSTRACTThe problem of the stability of the hydrogen atom when it is released from a confining environment is studied. The stability is analysed in terms of excitation and ionisation probabilities of the final state for different initial states. A spherical confining cavity of finite barrier height with inner radius and thickness Δ with the nuclear position at its centre has been considered. The ionisation probability presents different sharply peaked, non-symmetric local maxima as a function of the confinement size. This behaviour is related to the energy of the initial confined state that presents several maxima and minima in a kink-like structure as a function of the confinement size. The physical origin of these effects has been explained in terms of tunnelling and re-tunnelling of the atomic states. The sudden approximation and the analytic continuation method have been employed.

Maximum quantum entropy method

Maximum entropy method for analytic continuation is extended by introducing quantum relative entropy. This new method is formulated in terms of matrix-valued functions and therefore invariant under arbitrary unitary transformation of input matrix. As a result, the continuation of off-diagonal elements becomes straightforward. Without introducing any further ambiguity, the Bayesian probabilistic interpretation is maintained just as in the conventional maximum entropy method. The applications of our generalized formalism to a model spectrum and a real material demonstrate its usefulness and superiority.

On quasi-satellite periodic motion in asteroid and planetary dynamics

Applying the method of analytical continuation of periodic orbits, we study quasi-satellite motion in the framework of the three-body problem. In the simplest, yet not trivial model, namely the planar circular restricted problem, it is known that quasi-satellite motion is associated with a family of periodic solutions, called family $f$, which consists of 1:1 resonant retrograde orbits. In our study, we determine the critical orbits of family $f$ that are continued both in the elliptic and in the spatial model and compute the corresponding families that are generated and consist the backbone of the quasi-satellite regime in the restricted model. Then, we show the continuation of these families in the general three-body problem, we verify and explain previous computations and show the existence of a new family of spatial orbits. The linear stability of periodic orbits is also studied. Stable periodic orbits unravel regimes of regular motion in phase space where 1:1 resonant angles librate. Such regimes, which exist even for high eccentricities and inclinations, may consist dynamical regions where long-lived asteroids or co-orbital exoplanets can be found.

Phantom holomorphic projections arising from Sturm’s formula

We construct Siegel Poincare series of weight three and genus two outside their natural domain of convergency by the method of analytic continuation. In contrast to genus two Poincare series of higher weights greater three, the Poincare series thus obtained do not define holomorphic cuspforms of weight three. In fact, additional nonholomorphic phantom parts show up and we are able to describe them explicitly in terms of holomorphic forms of weight one. As a consequence of this result on Poincare series, in the case under consideration we can also show that Sturm’s operator not only projects to the holomorphic discrete series of weight three, as one would expect, but also gives rise to an additional phantom projection. For genus two the weight three case is distinguished, giving the first example for this new kind of phenomenon.

Diagnosing Fractionalization from the Spin Dynamics of Z_{2} Spin Liquids on the Kagome Lattice by Quantum MonteCarlo Simulations.

Based on large-scale quantum MonteCarlo simulations, we examine the dynamical spin structure factor of the Balents-Fisher-Girvin kagome lattice spin-1/2 model, which is known to harbor an extended Z_{2} quantum spin liquid phase. We use a correlation-matrix sampling scheme combined with a stochastic analytic continuation method to resolve the spectral functions of this anisotropic quantum spin model with a three-site unit cell. Based on this approach, we monitor the spin dynamics throughout the phase diagram of this model, from the XY-ferromagnetic region to the Z_{2} quantum spin liquid regime. In the latter phase, we identify a gapped two-spinon continuum in the transverse scattering channel, which is faithfully modeled by an effective spinon tight-binding model. Within the longitudinal channel, we identify gapped vison excitations and exhibit indications for the translational symmetry fractionalization of the visons via an enhanced spectral periodicity.

On the phase and group velocities of optical solitons

The method of analytical continuation of the dispersion parameters on the complex plane is analyzed. On this basis the new general formulae for phase and group velocities of the soliton-like pulses are obtained.

Two-Photon Exchange Correction in Elastic Lepton–Proton Scattering

We present the dispersion relation approach based on unitarity and analyticity to evaluate the two-photon exchange contribution to elastic electron-proton scattering. The leading elastic and first inelastic $\pi N$ intermediate state contributions are accounted for in the region of small momentum transfer $Q^2 < 1~\mathrm{GeV}^2$ based on the available data input. The novel methods of analytical continuation allow us to exploit the MAMI form factor data and the MAID parameterization for the pion electroproduction amplitudes as input in the calculation. The results are compared to the recent CLAS, VEPP-3 and OLYMPUS data as well as to the full two-photon exchange correction in the near-forward approximation, which is based on the Christy and Bosted unpolarized structure functions fit. Additionally, predictions are given for a forthcoming muon-proton scattering experiment.

Shape resonances of Be− and Mg− investigated with the method of analytic continuation

The regularized method of analytic continuation is used to study the low-energy negative ion states of beryllium (configuration 2$s^2\varepsilon p\; ^2P$) and magnesium (configuration 3$s^2\varepsilon p\; ^2P$) atoms. The method applies an additional perturbation potential and it requires only routine bound-state multi-electron quantum calculations. Such computations are accessible by most of the free or commercial quantum chemistry software available for atoms and molecules. The perturbation potential is implemented as a spherical Gaussian function with a fixed width. Stability of the analytic continuation technique with respect to the width and with respect to the input range of electron affinities is studied in detail. The computed resonance parameters $E_r$=0.282 eV, $\Gamma$=0.316 eV for the $2p$ state of Be$^-$ and $E_r$=0.188 eV, $\Gamma$=0.167 for the $3p$ state of Mg$^-$, agree well with the best results obtained by much more elaborated and computationally demanding present-day methods.

The Crepant Transformation Conjecture for toric complete intersections

Let X and Y be K-equivalent toric Deligne–Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier–Mukai transformation between the K-groups of X and Y, via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on X and Y: they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne–Mumford stacks and toric complete intersections, and the Mellin–Barnes method for analytic continuation of hypergeometric functions.

A numerical test of the topographic bias

Abstract In 1962 A. Bjerhammar introduced the method of analytical continuation in physical geodesy, implying that surface gravity anomalies are downward continued into the topographic masses down to an internal sphere (the Bjerhammar sphere). The method also includes analytical upward continuation of the potential to the surface of the Earth to obtain the quasigeoid. One can show that also the common remove-compute-restore technique for geoid determination includes an analytical continuation as long as the complete density distribution of the topography is not known. The analytical continuation implies that the downward continued gravity anomaly and/or potential are/is in error by the so-called topographic bias, which was postulated by a simple formula of L E Sjöberg in 2007. Here we will numerically test the postulated formula by comparing it with the bias obtained by analytical downward continuation of the external potential of a homogeneous ellipsoid to an inner sphere. The result shows that the postulated formula holds: At the equator of the ellipsoid, where the external potential is downward continued 21 km, the computed and postulated topographic biases agree to less than a millimetre (when the potential is scaled to the unit of metre).

Study of Rare Semileptonic Bc+→D+νν- Decay in the Light-Cone Quark Model

We study the exclusive semileptonic rare Bc+→D+νν¯ decay in the framework of light-cone quark model. The transition form factors f+(q2) and fT(q2) are evaluated in the timelike region using the analytic continuation method in q+=0 frame. The analytic solutions of these form factors are compared with the results obtained from the double pole parametric form. The branching ratio for Bc+→D+νν¯ decay is calculated and compared with the other theoretical model predictions. The predicted results in this model can be tested at the LHCb experiments in near future which will help in testing the unitarity of CKM quark mixing matrix, thus providing an insight into the phenomenon of CP violation.

Handling Analysis and Defining Conditions of Dangerous-Sfe Divergent Stability Loss of a Two-Link Road Train Nonlinear Model

Road train steady motion mode divergent stability loss compiles with the critical according to A.M. Liapunov case of a single zero root. That said both safe and dangerous stability loss scenarios are possible according to N.N. Bautin. Dangerous stability loss is followed with a semi-trailer intensive drifting even in case of linear motion. Analyzing the reasons of such system behavior requires developing new effective analytical approaches towards defining safe-dangerous articulated vehicle divergent stability loss because direct methods for finding corresponding Liapunov indexes may appear ineffective in the analytical form being excessively cumbersome. The work presents a formalized approach to analyzing safe stability loss conditions the essence of which is in defining conditions when bifurcation set structure rearrangement occurs in linear motion critical speed small neighborhood. The kind of approach has been tested by the authors when analyzing single unit vehicle stability. Analytical relations are presented defining road train configuration following circular paths with constant Ackermann angle; consideration of analytical results accuracy evaluation is performed based on comparing to the results received with numerical analytic parameter continuation method; analytical relations are received corresponding to safe linear motion mode stability loss (in the sense of N.N. Bautin). The work develops methods of analyzing two-link vehicle non-linear model two-parameter steady modes manifold stability.

Direct detection of metal-insulator phase transitions using the modified Backus-Gilbert method

The detection of the (semi)metal-insulator phase transition can be extremely difficult if the local order parameter which characterizes the ordered phase is unknown. In some cases, it is even impossible to define a local order parameter: the most prominent example of such system is the spin liquid state. This state was proposed to exist in the Hubbard model on the hexagonal lattice in a region between the semimetal phase and the antiferromagnetic insulator phase. The existence of this phase has been the subject of a long debate. In order to detect these exotic phases we must use alternative methods to those used for more familiar examples of spontaneous symmetry breaking. We have modified the Backus-Gilbert method of analytic continuation which was previously used in the calculation of the pion quasiparticle mass in lattice QCD. The modification of the method consists of the introduction of the Tikhonov regularization scheme which was used to treat the ill-conditioned kernel. This modified Backus-Gilbert method is applied to the Euclidean propagators in momentum space calculated using the hybrid Monte Carlo algorithm. In this way, it is possible to reconstruct the full dispersion relation and to estimate the mass gap, which is a direct signal of the transition to the insulating state. We demonstrate the utility of this method in our calculations for the Hubbard model on the hexagonal lattice. We also apply the method to the metal-insulator phase transition in the Hubbard-Coulomb model on the square lattice.

Nearly Deconfined Spinon Excitations in the Square-Lattice Spin- 1/2 Heisenberg Antiferromagnet

We study the dynamic spin structure factor of the spin-$1/2$ square-lattice Heisenberg antiferromagnet and of the $J$-$Q$ model (with 4-spin interactions $Q$ and Heisenberg exchange $J$). Using an improved method for stochastic analytic continuation of imaginary-time correlation functions computed with QMC simulations, we can treat the sharp ($\delta$-function) contribution from spinwaves (magnons) and a continuum at higher energy. The results for the Heisenberg model agree with neutron scattering experiments on Cu(DCOO)$_2$$\cdot$4D$_2$O, where a broad spectral-weight continuum at $q=(\pi,0)$ was interpreted as deconfined spinons. Our results at $(\pi,0)$ show a similar reduction of the magnon weight and a large continuum, while the continuum is much smaller at $q=(\pi/2,\pi/2)$ (as also seen experimentally). Turning on $Q$, we observe a rapid reduction of the $(\pi,0)$ magnon weight to zero, well before the deconfined quantum phase transition into a spontaneously dimerized state. We re-interpret the picture of deconfined spinons at $(\pi,0)$ in the experiments as nearly deconfined spinons---a precursor to deconfined quantum criticality. To further elucidate the picture of a fragile $(\pi,0)$-magnon in the Heisenberg model and its depletion in the $J$-$Q$ model, we introduce an effective model in which a magnon can split into two spinons that do not separate but fluctuate in and out of the magnon space (in analogy with the resonance between a photon and a particle-hole pair in the exciton-polariton problem). The model reproduces the $(\pi,0)$ and $(\pi/2,\pi/2)$ features of the Heisenberg model. It can also account for the rapid loss of the $(\pi,0)$ magnon with increasing $Q$ and a remarkable persistence of a large magnon pole at $q=(\pi/2,\pi/2)$ even at the deconfined critical point.

Green's function for a point heat source embedded in an infinite body with two circular elastic inclusions

• The analytical continuation technique and the alternation method are used. • The derived series solution is rapidly convergent. • The interfacial normal stresses become negative with a higher temperature. • Our present proposed method is efficient and general. A general solution for a point heat source interacting with two circular inclusions embedded in an infinite solid is presented in this work. Based on the method of analytical continuation in conjunction with the alternating technique, the solutions to the heat conduction and thermoelastic problems are derived. A rapidly convergent series solution for both the temperature and stress field, which is expressed in terms of an explicit general term of the corresponding homogeneous complex potential, is obtained in an elegant form. As a numerical illustration, the constant contour values of the temperature and stresses are displayed to provide a better understanding of both heat and stress flows for different locations of a point heat source. Numerical results of interfacial stresses are also shown in graphic form and discussed in detail. The solution obtained in this study can be treated as Green's functions which enable us to formulate an integral equation for the crack problem associated with two circular inclusions.

Analytic-continuation approach to the resummation of divergent series in Rayleigh-Schrödinger perturbation theory

The method of analytic continuation is applied to estimate eigenvalues of linear operators from finite order results of perturbation theory even in cases when the latter is divergent. Given a finite number of terms ${E}^{(k)},k=1,2,\ensuremath{\cdots}M$ resulting from a Rayleigh-Schr\odinger perturbation calculation, scaling these numbers by ${\ensuremath{\mu}}^{k}$ ($\ensuremath{\mu}$ being the perturbation parameter) we form the sum $E(\ensuremath{\mu})={\ensuremath{\sum}}_{k}{\ensuremath{\mu}}^{k}{E}^{(k)}$ for small $\ensuremath{\mu}$ values for which the finite series is convergent to a certain numerical accuracy. Extrapolating the function $E(\ensuremath{\mu})$ to $\ensuremath{\mu}=1$ yields an estimation of the exact solution of the problem. For divergent series, this procedure may serve as resummation tool provided the perturbation problem has a nonzero radius of convergence. As illustrations, we treat the anharmonic (quartic) oscillator and an example from the many-electron correlation problem.

Ionization probability of the hydrogen atom suddenly released from confinement

AbstractThe problem of the stability of a confined atom when it is extracted from the confining cavity has been investigated, modeled by a spherical hard wall potential. The ionization probability when the atom is released from confinement has been obtained. The dependence of the ionization probability on the confinement radius and on the quantum numbers of the initial confined state has been studied. The probability density function of the ionization energy of the ejected electron has been obtained for the different cases considered. The oscillatory structure of this distribution function, with a principal maximum located in the neighborhood of the energy of the initial state and minima very close to zero has been elucidated. The sudden approximation has been applied and the analytic continuation method has been used to calculate the different stationary states.

Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum.

We consider a zero-temperature one-dimensional system of bosons interacting via the soft-shoulder potential in the continuum, typical of dressed Rydberg gases. We employ quantum MonteCarlo simulations, which allow for the exact calculation of imaginary-time correlations, and a stochastic analytic continuation method, to extract the dynamical structure factor. At finite densities, in the weakly interacting homogeneous regime, a rotonic spectrum marks the tendency to clustering. With strong interactions, we indeed observe cluster liquid phases emerging, characterized by the spectrum of a composite harmonic chain. Luttinger theory has to be adapted by changing the reference lattice density field. In both the liquid and cluster liquid phases, we find convincing evidence of a secondary mode, which becomes gapless only at the transition. In that region, we also measure the central charge and observe its increase towards c=3/2, as recently evaluated in a related extended Bose-Hubbard model, and we note a fast reduction of the Luttinger parameter. For two-particle clusters, we then interpret such observations in terms of the compresence of a Luttinger liquid and a critical transverse Ising model, related to the instability of the reference lattice density field towards coalescence of sites, typical of potentials which are flat at short distances. Even in the absence of a true lattice, we are able to evaluate the spatial correlation function of a suitable pseudospin operator, which manifests ferromagnetic order in the cluster liquid phase, exponential decay in the liquid phase, and algebraic order at criticality.

Delayed feedback control of self-mobile cavity solitons in a wide-aperture laser with a saturable absorber.

We investigate the spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subjected to time-delayed optical feedback. Using a combination of analytical, numerical, and path continuation methods, we analyze the bifurcation structure of stationary and moving cavity solitons and identify two different types of traveling localized solutions, corresponding to slow and fast motion. We show that the delay impacts both stationary and moving solutions either causing drifting and wiggling dynamics of initially stationary cavity solitons or leading to stabilization of intrinsically moving solutions. Finally, we demonstrate that the fast cavity solitons can be associated with a lateral mode-locking regime in a broad-area laser with a single longitudinal mode.

Elastic Green’s Function in Anisotropic Bimaterials Considering Interfacial Elasticity

The two-dimensional elastic Green’s function is calculated for a general anisotropic elastic bimaterial containing a line dislocation and a concentrated force while accounting for the interfacial structure by means of a generalized interfacial elasticity paradigm. The introduction of the interface elasticity model gives rise to boundary conditions that are effectively equivalent to those of a weakly bounded interface. The equations of elastic equilibrium are solved by complex variable techniques and the method of analytical continuation. The solution is decomposed into the sum of the Green’s function corresponding to the perfectly bonded interface and a perturbation term corresponding to the complex coupling nature between the interface structure and a line dislocation/concentrated force. Such construct can be implemented into the boundary integral equations and the boundary element method for analysis of nano-layered structures and epitaxial systems where the interface structure plays an important role.