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horn and weinstein@xcite and horn et al@xcite proposed the @xmath0expansion for the calculation of the ground state energy of quantum  mechanical models . it is the taylor expansion about @xmath1 of a monotonically decreasing function @xmath2 that leads to the ground state energy when @xmath3 provided that the chosen reference function exhibits a nonzero overlap with the ground state . the coefficients of such maclaurin series are known as connected moments or cummulants . since the extrapolation of the @xmath0expansion towards @xmath3 by means of pad approximants@xcite did not appear to produce encouraging results , cioslowski@xcite proposed an exponential series and stubbins@xcite compared it with other extrapolation approaches . on matching the @xmath0expansion and the exponential series at origin one has to solve a system of nonlinear equations . in order to bypass this problem cioslowski@xcite developed a systematic algorithm for obtaining just the parameter related to the energy by removing all the other variables in the nonlinear equations . the resulting approach is known as connected moments expansion ( cmx ) . from the properties of the pad approximants knowles@xcite derived a compact an ellegant explicit expression for the approximants to the energy in terms of the connected moments . since the cmx approximants may exhibit singularities@xcite other authors proposed improved approaches like the alternate moments expansion ( amx)@xcite and the generalized moments expansion(gmx)@xcite . although these methods may avoid the singularities in the cmx approximants they do not seem to improve the convergence properties of the sequences of approximants and commonly their results are not more accurate than the cmx ones .    a recent analysis of the convergence properties of the cmx required the calculation of all the variables in the cmx ansatz@xcite . for that reason , amore and fernndez@xcite proposed a systematic method for the full solution of the cmx nonlinear equations motivated by an earlier discussion@xcite of the connected  moments polynomial approach@xcite . nonlinear equations like the cmx ones are an old problem in applied mathematics and were first solved by prony@xcite and later weiss and mcdonough@xcite proposed an alternative approach based on pad approximants . in fact , prony s method is well known in numerical analysis@xcite and has been widely applied to a variety of problems in chemistry and engineering@xcite . further inspection of the procedure developed by amore and fernndez@xcite has revealed that it is closely related to prony s method , although the main result derived by the former authors does not appear in the discussions of the latter method@xcite . the main purpose of this paper is to show the connection between prony s method and the procedure developed by amore and fernndez . in sec . [ sec : prony ] we show how to solve a particular set of nonlinear equations by means of prony s method and derive the main result of amore and fernndez@xcite . in sec . [ sec : generating ] we discuss the application of that method to the extrapolation of the generating functions for the moments and connected moments . in sec . [ sec : examples ] we test the general analytical results on simple models and show that they are useful for a discussion of the convergence properties of the cmx as well as for the approximate calculation of the autocorrelation function . in what follows we discuss the solution of the system of @xmath4 equations @xmath5 for the @xmath4 unknowns @xmath6 and @xmath7 , where @xmath8 are known real numbers and @xmath9 an integer . prony s method is based on the construction of the polynomial@xcite @xmath10 where the polynomial roots @xmath7 , and thereby the polynomial coefficients @xmath11 , are unknown . it follows from @xmath12 that the coefficients @xmath11 are solutions to the linear system of equations @xmath13 if the matrix @xmath14 with elements @xmath15 , @xmath16 , @xmath17 is nonsingular then the solution is unique . once we have the polynomial coefficients @xmath11 we obtain its roots @xmath7 and then solve the resulting system of linear equations ( [ eq : main_eqs ] ) ( for example for @xmath18 ) for the remaining unknowns @xmath6 . the technique just outlined is known since 1795@xcite and is commonly called prony s method@xcite . weiss and mcdonough@xcite proposed an alternative way of solving the nonlinear equations by means of a partial fraction expansion of pad approximants , and in what follows we describe a different strategy for obtaining the roots @xmath7 developed by amore and fernndez@xcite . the starting point of this procedure is the homogeneous system of @xmath19 linear equations @xmath20 with @xmath19 unknowns @xmath21 . there will be nontrivial solutions ( @xmath22 ) only if the determinant vanishes : @xmath23 under such conditions we define @xmath24 so that equations ( [ eq : secular_eqs ] ) become @xmath25 that leads to @xmath26 , @xmath27 . therefore , it follows from @xmath28 that the roots of the determinant ( [ eq : secular_det ] ) are exactly the roots of the polynomial @xmath29 of prony s method . we have now at least three alternative procedures for solving the nonlinear equations ( [ eq : main_eqs ] ) . first , the original prony s method that consists of solving the system of linear equations ( [ eq : linear_pj ] ) for the coefficients of the polynomial @xmath29 , then calculating its roots @xmath7 , and finally solving the resulting system of linear equations ( [ eq : main_eqs ] ) for the remaining unknowns @xmath6 . second , the partial fraction expansion of the pad approximants proposed by weiss and mcdonough@xcite . third , our approach that consists of obtaining the polynomial @xmath29 from the determinant  ( [ eq : secular_det ] ) . the choice of either of them is probably a matter of taste . we find our technique quite straightforward and it has revealed a most interesting connection between the exponential expansion of the generating function for the moments and the rayleigh  ritz method in the krylov space@xcite .    prony s method was developed for fitting a series of exponential functions to given numerical data@xcite . in what follows we apply it to a completely different problem so that the name prony s method refers only to the strategy for solving the nonlinear equations ( [ eq : main_eqs ] ) . the general equations developed in the preceding section prove useful for the analysis of the cmx@xcite . we first consider the moments  generating function@xcite @xmath30 where @xmath31 is the hamiltonian operator of the system and @xmath32 is an arbitrary reference state . the coefficients of its maclaurin series @xmath33 are the moments @xmath34 . for simplicity we assume that the spectrum of @xmath31 is discrete @xmath35 where @xmath36 , and without loss of generality we choose the eigenvectors of @xmath31 to be orthonormal : @xmath37 . under such conditions the generating function exhibits the exponential behaviour @xmath38 where the expansion coefficients @xmath39and the energies @xmath40 are unknown . we can calculate them approximately by means of the method developed in the preceding section and the ansatz @xmath41 where @xmath42 and @xmath43 are the approximations to @xmath44and @xmath40 , respectively . if we require that the maclaurin expansion of this ansatz yields the first moments @xmath45 exactly we are left with the nonlinear equations ( [ eq : main_eqs ] ) with @xmath46 , @xmath47 and @xmath48 .    it is has been proved that matching those expansions at origin is equivalent to the application of the rayleigh ritz variational method in the krylov space ( rrk)@xcite . note that the determinant ( [ eq : secular_det ] ) becomes the secular determinant of the rrk . the rrk solutions @xmath49 satisfy @xmath50 and the application of prony s method in the way just outlined yields @xmath51 . the cmx is based on the generating function @xmath52 that is monotonically decreasing@xcite and the coefficients of its maclaurin series are the connected moments @xmath53 : @xmath54 the recurrence relation@xcite @xmath55 yields the connected moments ( or cummulants ) @xmath56 in terms of the moments @xmath45 . the main interest in @xmath2 is that it provides a size consistent approach to the ground state energy@xcite @xmath57 when @xmath58 . in order to carry out this extrapolation cioslowski@xcite proposed the exponential  series ansatz @xmath59 where the unknown parameters @xmath60 are supposed to be real and positive and @xmath61 is the approximation to @xmath62 . matching this expression with the @xmath0expansion ( [ eq : t_exp ] ) leads to the set of equations @xmath63 and @xmath64 we can solve the set of @xmath4 equations ( [ eq : i_j_a_j_b_j ] ) by means of the method of the preceding section with @xmath65 and @xmath66 . in this case the exponential parameters @xmath60 , @xmath27 are the roots of the pseudo  secular determinant @xmath67 once we have them we solve @xmath19 of the resulting @xmath4 linear equations ( [ eq : i_j_a_j_b_j ] ) for the coefficients @xmath42 , @xmath27 , and then we obtain @xmath61 from equation  ( [ eq : a0_i1-an ] ) . cioslowski@xcite and knowles@xcite developed remarkable strategies for obtaining @xmath61 without calculating the other parameters explicitly . we have just shown that the explicit calculation of all the parameters in the exponential  series ansatz ( [ eq : e^(n)(t ) ] ) is quite straightforward . in order to test the equations developed in the preceding section and show their usefulness in the analysis of the cmx we first consider an exactly solvable model : the dimensionless harmonic oscillator @xmath68 as a trial function we choose one of the examples discussed in earlier papers@xcite    @xmath69    for which @xmath70 , @xmath71 . ( the normalization factor is irrelevant to present purposes ) one advantage of this example is that it is not difficult to obtain the generating function exactly    @xmath72    note that @xmath73 because @xmath74 in agreement with the discussion in sec . [ sec : generating ] .    by means of the general results of the preceding section we can easily calculate the generating function @xmath2 approximately in two ways : as @xmath75 from equation ( [ eq : z_n(t ) ] ) and directly from equation ( [ eq : e^(n)(t ) ] ) . fig  [ fig : etho2a ] shows that @xmath76 approaches the exact generating function @xmath2 for all @xmath77 as @xmath19 increases . on the other hand , the approximate expression @xmath78 given by eq . ( [ eq : e^(n)(t ) ] ) is an unsatisfactory approximation to the exact @xmath2 as shown in fig . [ fig : etho2b ] ( except for sufficiently small values of @xmath0 ) . this situation does not appear to improve as @xmath19 increases . it is found that @xmath79 for @xmath80 because in both cases there is one negative root @xmath60 associated to a negative coefficient @xmath42 . for example , @xmath81 , @xmath82 , @xmath83 , @xmath84 , @xmath85 , and @xmath86 for @xmath87 . note that the curves in both figures are directly comparable in the sense that the calculation of either @xmath88 and @xmath78 requires exactly the same number of moments @xmath45 ( @xmath89 ) . the behaviour of @xmath78 for larger @xmath19 may be different ; for example , for @xmath90 there are two negative roots and @xmath91 . the anomalous behaviour of @xmath78 is due to the poles of @xmath2 in the complex @xmath0plane . note that eq . ( [ eq : e(t)_ho_exact ] ) exhibits three real poles in the @xmath92plane and the closest to the origin is located at @xmath93 . it is reasonable to assume that the cmx ansatz @xmath94 should approach the taylor series about @xmath95 for @xmath2 when the approach is successful . however , this expansion does not converge for @xmath96 which is necessary for matching the @xmath0expansion and exponential series at @xmath1 . this problem was discussed by amore et al@xcite by means of two  level models and we see it here again for the harmonic oscillator . those authors have shown that the cmx approximants for the harmonic oscillator with the trial function ( [ eq : phi(x)_ho ] ) converge towards the second  excited state @xmath97 . from the approximate expressions @xmath94 with @xmath98 we obtain @xmath99 , respectively , in agreement with those results . this example shows that the cmx approximants may converge to a meaningful result even when the ansatz @xmath78 ( on which the approach is based ) is an unacceptable global approximation to @xmath2 . knowles@xcite argued that if the connected moments @xmath100 are positive for all @xmath101 then the @xmath102 are real and positive . later , massano et al@xcite and mancini et al@xcite suggested that knowles argument may not be valid and that it is the hadamard determinants constructed from the @xmath100 that should be positive . the example just analysed supports the latter conclusion because the first connected moments are all positive ( @xmath103 , @xmath104 , @xmath105 , @xmath106 , @xmath107 , @xmath108 , @xmath109 ) but one of the roots is negative as shown above . we realize that the general formulas derived in sec . [ sec : prony ] are most useful for this kind of analysis . in fact , the calculation of the roots @xmath60 by means of present formulas is almost as easy as the calculation of the hadamard determinants and the former provide a much clearer indication of the suitability of the ansatz @xmath78 as an approach for @xmath2 .    by means of the example discussed above we do not want to convey the wrong impression that the cmx is suitable for the calculation of excited  state energies because it is quite difficult to choose an appropriate trial function for this purpose ( except when we can exploit the symmetry of the problem to make the trial function orthogonal to the ground state@xcite ) . the main interest in that exactly solvable problem is to show that we may predict an anomalous behaviour of the cmx by means of the roots @xmath110 . if they are not real and positive it is reasonable to suspect that the trial function may not be suitable and that it is convenient to choose another one . for example , if @xmath111 all the roots @xmath60 are positive and the resulting @xmath78 is a remarkably good approximation to @xmath2 for all @xmath112 even at the low orders @xmath113 . finally , we explore the possibility of calculating of the correlation function @xmath114 where @xmath115 is the initial state normalized to unity and @xmath116 is the state at time @xmath0 . in particular we concentrate on the real function @xmath117 .    as a first example we consider the harmonic oscillator ( [ eq : h_ho ] ) and the trial function @xmath118 the exact correlation function is    @xmath119    fig .  [ fig : zitho ] shows results for @xmath120 . there is no doubt that @xmath121 converges towards @xmath117 as @xmath19 increases . the approximation is satisfactory within the first period but the errors accumulate as @xmath0 increases . this fact is not surprising because the approximate expression is constructed from the maclaurin series for @xmath122 . the second example is the anharmonic oscillator @xmath123 and the same trial function ( [ eq : phi(0)_ho ] ) . [ fig : zitaho ] shows that the approximate results for @xmath120 follow the same trend as in the case of the harmonic oscillator . at first sight it may be surprising that the convergence rate for the anharmonic oscillator appears to be greater than for the harmonic one . the reason is that the chosen trial state exhibits a greater overlap with the ground state of the former . note that the magnitude of the coefficient @xmath61 of @xmath124 tells us that @xmath125 is approximately 0.943 for the harmonic oscillator and 0.981 for the anharmonic one . as a two  dimensional example we consider the anharmonic oscillator @xmath126 where @xmath127 , and the trial function @xmath128 fig . [ fig : zitpe ] shows results for @xmath129 that look quite similar to those for the one  dimensional models . one does not expect that it may be possible to construct @xmath130 from its taylor expansion about @xmath1 in all the cases . the problems discussed above are simple examples of single well oscillators . in the case of the double  well oscillator @xmath131 and the trial state localized in one of the wells @xmath132 the approximants to @xmath117 do not appear to converge although the results for @xmath133 look reasonable . if , on the other hand , we locate the initial state on top of the barier ( for example , eq . ( [ eq : phi(0)_ho ] ) ) then the results are as accurate as those for the single wells . the main goal of this paper is to show that  prony s method provides the full solution of the cmx equations in a straightforward and simple way . we have outlined the connection between prony s method and the procedure developed recently by amore and fernndez@xcite . although both approaches are equivalent and prony s one is known since long ago it seems that the pseudo  secular determinants obtained by amore and fernndez are not available elsewhere .    in spite of the fact that the harmonic oscillator with the trial function ( [ eq : phi(x)_ho ] ) was chosen as an illustrative example in two earlier papers@xcite we think that present discussion based on the exact generating function ( [ eq : e(t)_ho_exact ] ) is clearer and provides more information about the behaviour of the cmx ansatz @xmath78 . in particular , we have shown that this simple problem clearly illustrates that knowles conclusion about the sign of the connected moments@xcite is not valid as argued before by massano et al@xcite and mancini et al@xcite . it is clear that the roots @xmath102 provide a clear indication of the success of the cmx and the method developed by amore and fernndez@xcite , as well as prony s method , make their calculation straightforward . both procedures may be applied to the calculation of the autocorrelation function for simple quantum  mechanical models . if the approach converges then one obtains a reasonable analytic expression for the autocorrelation function within the first period of oscillation .                                                 for the harmonic oscillator ( line ) and approximants @xmath134 with @xmath135 ( squares , blue ) , @xmath87 ( filled squares , red ) , @xmath136 ( circles , green ) and @xmath137 ( filled circle , black).,width=340 ]
we show that prony s method provides the full solution to the nonlinear equations of the connected  moments expansion ( cmx ) . 
 knowledge of all the parameters in the cmx ansatz is useful for the analysis of the convergence properties of the approach . 
 prony s method is also suitable for the calculation of the correlation function for simple quantum  mechanical models .