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The Sturm–Liouville equation on the half-line [0,∞) is considered with a potential which decays exponentially at infinity. It is known that Green's function has an analytic continuation into part of the unphysical sheet im λ . Here it is shown that analytic continuation into the whole of this sheet is possible for certain types of decay. The methods are constructive, and they also provide a new and rigorous way of locating resonances.

The analytic continuation of the Jost solution into a sector of the unphysical sheet is established for analytic potentials with power decay. The method also yields resonance-free regions which, in the case of super-exponential decay, are confirmed to be the best possible of their kind by the known asymptotic distribution of resonances.

We consider the analytic continuation of solutions to the nonlinear partial differential equation ∂∂tmu=Ft,x,∂∂tj∂∂xαuj+|α|⩽mj⩽m−1 in the complex domain. Suppose a solution u(t,x) is known to be holomorphic in the domain {(t,x)∈C×Cn;|x

We consider general multiple zeta-functions of multi-variables, including both Barnes multiple zeta-functions and Euler–Zagier sums as special cases. We prove the meromorphic continuation to the whole space, asymptotic expansions, and upper bound estimates. These results are expected to have applications to some arithmetical L -functions (such as of Hecke and of Shintani). The method is based on the classical Mellin–Barnes integral formula.

We present a way to use Bayesian statistical inference and the principle of maximum entropy to analytically continue imaginary-time quantum Monte Carlo data. We supply the details that are lacking in the seminal literature but are important for the motivated reader to understand the assumptions and approximations embodied in these methods. First, we summarize the general relations between quantum correlation functions and spectral densities. We then review the basic principles, formalism, and philosophy of Bayesian inference and discuss the application of this approach in the context of the analytic continuation problem. Next, we present a detailed case study for the symmetric, infinite-dimension Anderson Hamiltonian. We chose this Hamiltonian because the qualitative features of its spectral density are well established and because a particularly convenient algorithm exists to produce the imaginary-time Green's function data. Shown are all the intermediate steps of data and solution qualification. The importance of careful data preparation and error propagation in the analytic continuation is discussed in the context of this example. Then, we review the different physical systems and physical quantities to which these, or related, procedures have been applied. Finally, we describe other features concerning the application of our methods, their possible improvement, and areas for additional study.

Stable sampling formula using basic hypergeometric series for reconstructing analytic functions from exponentially spaced samples is considered. Criterion for selecting regularizing parameter and error estimates are obtained.

It is shown how a particular correlation function, computed quantum mechanically or semiclassically as a function of real-time, can be analytically continued to obtain thermal rate constants (or other thermally averaged quantities) over a wide range of temperature. The correlation function is first evaluated at a reference temperature, and then analytically continued to obtain information at other desired temperatures. The method is applied to the calculation of the thermal rate constant for the Cl+H2→HCl+H (J=0) reaction using the flux correlation function formalism, and it is seen to be capable of obtaining accurate rate constants over a broad temperature range.

We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two- particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically., 25 pages, latex, 6 ps files, uses epsf.tex. Typos corrected

This paper investigates axisymmetric shapes of an incompressible inviscid fluid bubble in a uniform flow. Approximate solutions are constructed for the governing nonlinear boundary value problem using the domain perturbation method. Bifurcation and turning points are located by applying a special type of Hermite-Pade approximation.

By using a calculus based on Brownian bridge measures, it is shown that under mild assumptions on V (e.g. V is in the Kato class) the fundamental solution (FS) q ( t,x,y ) for the heat equation ∂ t u = 1 2 Δ − V)u can be represented by the Feynman-Kac formula. Furthermore, it has an analytic continuation in t over C + , where C + ={z ∈ C , Re z > 0} , and q (e + i t,x,y ) can be expressed via Wiener path integrals. For small e > 0 it can be considered as an approximation of the FS for the Schrodinger equation ∂ t ψ=i( 1 2 Δ − V)ψ . We also give an estimate of q ( t,x,y ) for t ∈ C + .

We present a new proof of the Nehari-Gilbert theorem that deals with the location of singularities of Jacobi series expansions. Our method is based on a general result of independent interest extending a theorem of Bony and Schapira concerning the global domain of real analyticity of solutions to certain linear partial differential equations to equations with singular coefficients.

The need to calculate the spectral properties of a Hermitian operatorHfrequently arises in the technical sciences. A common approach to its solution involves the construction of the Green's function operatorG(z) = z?H]?1in the complexzplane. For example, the energy spectrum and other physical properties of condensed matter systems can often be elegantly and naturally expressed in terms of the Kohn?Sham Green's functions. However, the nonanalyticity of resolvents on the real axis makes them difficult to compute and manipulate. The Herglotz property of a Green's function allows one to calculate it along an arc with a small but finite imaginary part, i.e.,G(x+iy), and then to continue it to the real axis to determine quantities of physical interest. In the past, finite-difference techniques have been used for this continuation. We present here a fundamentally new algorithm based on the fast Fourier transform which is both simpler and more effective.

We present an analytic continuation of a polynomial representation of the full, interacting time-independent Green function, thereby enabling the use of negative, imaginary absorbing potentials to shorten the grid necessary to treat scattering problems. The approach retains the clean separation of the energy and Hamiltonian dependences characteristic of our earlier orthogonal polynomial representation of the operator ( E - H +i0 + ) −1 . This treatment, combined with our time-independent wavepacket Lippmann-Schwinger equation method, leads to a computational approach in which all of the energy dependence resides in known analytical expansion coefficients. The Hamiltonian operator appears as the argument of other orthogonal polynomials. These act solely on an initial wavepacket which provides a “universal source” of scattered waves, independent of the particular energies of interest. This energy independence, combined with highly truncated grids, results in an extremely efficient procedure for scattering calculations.

In this paper we consider the dependence of Schrodinger equation eigenvalue problems on the coupling-constant parameters in the potential. We show that unless great care is taken, analytic continuation in these parameters can lead to surprising and paradoxical conclusions. Careful analysis of the analytical properties shows that such eigenvalue problems can have elaborate internal structure; these problems can incorporate several different eigenvalue problems joined together. For example, the anharmonic oscillator, whose potential is V(x)=a2x6−3ax2, is actually four different eigenvalue problems combined into one.

Obtaining dynamical characteristics of lattice field models amounts to analytic continuation of the simulated Matsubara Green's functions into the real frequency domain. This is in general an ill-posed problem with the statistical errors and the correlations in the simulation data imposing additional difficulties on its solution. We present a technique to extract dynamical correlation functions, using singular value decomposition. Prior information on the range of support of the expected solution and optimal sampling of the simulation data are incorporated in the analytic continuation procedure. Results are presented for the spectrum, density of states and dynamical spin susceptibility in the Hubbard and Heisenberg models. A generalisation of the method to the spectrum of QCD is proposed.

Real variationally based large-order perturbation theory (LOPT), implemented via the perturbational-variational Rayleigh-Ritz (PV-RR) matrix formalism, and combined with analytic continuation into the complex plane by shifted-origin (SO) perturbation series, has been applied to compute accurately with Pade approximants the complex Stark eigenvalues (resonances) of numerous excited states of hydrogenic ions over a range of field strengths. The problem of hydrogenic excited-state zeroth-order degeneracy in spherical coordinates is eliminated by the novel procedure of introducing a new SO unperturbed Hamiltonian with nondegenerate eigenfunctions. The SO PV-RR LOPT procedure is not restricted to one-electron systems.

A recently introduced variant of the stabilization/analytic continuation method is used to determine the resonance energies and widths for a model potential scattering problem. This method is shown to give accurate resonance parameters in which stabilization is accomplished by application of a positively charged sphere.

When in the Birkhoff series the leading contributions of the main resonances (given by the continued fraction expansion of the linear frequency) are retained, the Fourier spectrum of the conjugation function is given by geometric series. Any Fourier coefficient corresponding to a main resonance is therefore approximated by a single pole in ϱ2, where ϱ is the distance from the origin in the normal coordinates, and the residue turns out to be exponentially small in 1/|ϱ| and proportional to the resonance width. In the ϱ2 plane the poles are on the real axis and their signs alternate since at any pole the winding number is resonant. The poles located on the real ϱ axis correspond to true resonances with island structure in the phase plane, the poles on the imaginary ϱ axis to virtual resonances with no counterpart in the phase plane. When the next to leading contribution of the main resonances are included, any Fourier coefficient is given by a series of poles with exponentially small residues. This singularity pattern is consistent with the asymptotic properties of the Birkhoff series and suggest that the Pade approximants should be used to continue them analytically. This is fully confirmed by numerical results.

It is proved that the small scale yielding solution cannot serve as the elastic solution of a well-posed elastoplastic problem for plane stress loading conditions under the Tresca yield condition. In this study the small scale yield stresses are substituted into the Tresca yield criterion to obtain a prescribed elastoplastic boundary. A plastic stress function for a perfectly plastic material is sought which satisfies both equilibrium and the yield condition across the elastic-plastic interface. An intermediate integral is found for the governing partial differential equation which is a nonlinear second-order equation of the Monge-Ampere class. From a complete solution of the intermediate integral, the initial value problem for the plastic stress function is solved analytically. The plastic stress field that is determined from this function exhibits discontinuities which necessitate its rejection as a valid solution. A uniqueness theorem eliminates the possibility of an alternate solution.

We study the Lucas zeta function defined by using the Lucas sequence which is a generalization of the Fibonacci sequence. This zeta function can be meromorphically continued to the whole complex plane, and in a special case, it has “trivial zeros” like the Riemann zeta function. Analogues of Dirichlet’s L -functions are also investigated.

In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $\zeta_{E,q}(s)$ in terms of nested series of $\zeta_{E,q}(n)$ is derived. Finally we introduce the new concept of the dynamics of analytically continued $q$-Euler numbers and polynomials., Comment: 5 pages

Journal URL: http://www.sciencedirect.com/science/journal/00092614

Journal URL: http://www.sciencedirect.com/science/journal/00092614