document: the methods used to calculate the energy and the lifetime of a resonance state are numerous @xcite and , in some cases , has been put forward over strong foundations  @xcite . 
   however , the analysis of the numerical results of a particular method when applied to a given problem is far from direct . the complex scaling ( complex dilatation ) method @xcite , when the hamiltonian @xmath5 allows its use , reveals a resonance state by the appearance of an isolated complex eigenvalue on the spectrum of the non - hermitian complex scaled hamiltonian , @xmath6 @xcite . 
   of course in an actual implementation the rotation angle @xmath7 must be large enough to rotate the continuum part of the spectrum beyond the resonance s complex eigenvalue . 
   moreover , since most calculations are performed using finite variational expansions it is necessary to study the numerical data to decide which result is the most accurate . to worsen things the variational basis sets usually depend on one ( or more ) non - linear parameter . for bound states 
   the non - linear parameter is chosen in order to obtain the lowest variational eigenvalue . for resonance states things 
   are not so simple since they are embedded in the continuum . 
   the complex virial theorem together with some graphical methods @xcite allows to pick the best numerical solution of a given problem , which corresponds to the stabilized points in the @xmath7 trajectories @xcite . 
   other methods to calculate the energy and lifetime of the resonance , based on the numerical solution of complex hamiltonians , also have to deal with the problem of which solutions ( complex eigenvalues ) are physically acceptable . 
   for example , the popular complex absorbing potential method , which in many cases is easier to implement than the complex scaling method , produces the appearance of nonphysical complex energy stabilized points that must be removed in order to obtain only the physical resonances @xcite . 
   the aforementioned issues explain , at some extent , why the methods based only in the use of real @xmath2 variational functions are often preferred to analyze resonance states . 
   these techniques reduce the problem to the calculation of eigenvalues of real symmetric matrices @xcite . 
   of course , these methods also have its own drawbacks . 
   one of the main problems was recognized very early on ( see , for example , the work by hol@xmath8ien and midtdal @xcite ) : if the energy of an autoionizing state is obtained as an eigenvalue of a finite hamiltonian matrix , which are the convergence properties of these eigenvalues that lie in the continuum when the size of the hamiltonian matrix changes ? but in order to obtain resonance - state energies it is possible to focus the analysis in a global property of the variational spectrum : the density of states ( dos)@xcite , being unnecessary to answer this question . 
   the availability of the dos allows to obtain the energy and lifetime of the resonance in a simple way , both quantities are obtained as least square fitting parameters , see for example @xcite . despite its simplicity , the determination of the resonance s energy and width based in the dos is far from complete . 
   there is no a single procedure to asses both , the accuracy of the numerical findings and its convergence properties , or which values to pick between the several `` candidates '' that the method offers @xcite . 
   recently , pont _ et al _ @xcite have used _ 
   finite size scaling _ 
   arguments @xcite to analyze the properties of the dos when the size of the hamiltonian changes . 
   they presented numerical evidence about the critical behavior of the density of states in the region where a given hamiltonian has resonances . 
   the critical behavior was signaled by a strong dependence of some features of the density of states with the basis - set size used to calculate it . 
   the resonance energy and lifetime were obtained using the scaling properties of the density of states . 
   however , the feasibility of the method to calculate the resonance lifetime laid on the availability of a known value of the lifetime , making the whole method dependent on results not provided by itself . 
   the dos method relies on the possibility to calculate the ritz - variational eigenfunctions and eigenvalues for many different values of the non - linear parameter @xmath9 ( see kar and ho   @xcite ) . for each basis - set size , @xmath3 , used , there are @xmath3 variational eigenvalues @xmath10 
   . each one of these eigenvalues can be used , at least in principle , to compute a dos , @xmath11 , resulting , each one of these dos in an approximate value for the energy , @xmath12 , and width , @xmath13 , of the resonance state of the problem . if the variational problem is solved for many different basis - set sizes , 
   there is not a clear cut criterion to pick the `` better '' result from the plethora of possible values obtained . 
   this issue will be addressed in section  [ model ] .    in this work , in order to obtain resonance energies and lifetimes , we calculate all the eigenvalues for different basis - set sizes , and present a recipe to select adequately certain values of @xmath3 , and one eigenvalue for each @xmath3 elected , that is , we get a series of variational eigenvalues @xmath14 . 
   the recipe is based on some properties of the variational spectrum which are discussed in section  [ some - properties ] . 
   the properties seem to be fairly general , making the implementation of the recipe feasible for problems with several particles . actually , because we use scaling properties for large values of @xmath3 , the applicability of the method for systems with more than three particles could be restricted because the difficulties to handle large basis sets . 
   the set of approximate resonance energies , obtained from the density of states of a series of eigenvalues selected following the recipe , shows a very regular behaviour with the basis set size . 
   this regular behaviour facilitates the use of finite size scaling arguments to analyze the results obtained , in particular the extrapolation of the data when @xmath15 . 
   the extrapolated values are the most accurate approximation for the parameters of the resonance state that we obtain with our method . 
   this is the subject of section  [ recipe ] , where we present results for models of one and two particles . 
   following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths in section  [ golden - rule ] . using the scaling function that characterizes the behaviour of the approximate energies as a guide 
   , it is possible to find a very good approximation to the resonance width since , again , the data generated using our prescription seems to converge when @xmath4 . 
   finally , in section  [ discusion ] we summarize and discuss our results . 
   when one is dealing with the variational spectrum in the continuum region , some of its properties are not exploited to obtain more information about the presence of resonances , usually the focus of interest is the stabilization of the individual eigenvalues . 
   the stabilization is achieved varying some non - linear variational parameter . if @xmath9 is the inverse characteristic decaying length of the variational basis functions , then the spectrum of the kinetic energy scales as @xmath16 , moreover , for potentials that decay fast enough , the spectrum of the whole hamiltonian _ also _ scales as @xmath16 for large ( or small ) enough values of @xmath9 ( see appendix ) . 
   this is so , since the variational eigenfunctions are @xmath2 approximations to plane waves _ except _ when @xmath9 belongs to the stabilization region . 
   when @xmath9 belong to the stabilization region of a given variational eigenvalue , say @xmath17 , then @xmath18 ( where @xmath0 is the resonance energy ) and the variational eigenfunction @xmath19 has the localization length of the potential well . 
   we intend to take advantage of the changes of the spectrum when @xmath9 goes from small to large enough values . 
   the variational spectrum satisfies the hylleras - undheim theorem or variational theorem : if @xmath3 is the basis set size , and @xmath20 is the @xmath21 eigenvalue obtained with a variational basis set of size @xmath3 , then @xmath22 actually , since the threshold of the continuum is an accumulation point , then for small enough values of @xmath9 and a given @xmath23 there is always a @xmath24 such that    @xmath25    for the kinetic energy variational eigenvalues , and for fixed @xmath26 , and @xmath27 , if the ordering given by equation  ( [ ordering ] ) holds for some @xmath9 then it is true for all @xmath9 . 
   of course this is not true for a hamiltonian with a non zero potential that support resonance states . 
   so , we will take advantage of the variational eigenvalues such that for @xmath9 small enough satisfies equation  ( [ ordering ] ) but , for @xmath9 large enough @xmath28    despite its simplicity , the arguments above give a complete prescription to pick a set of eigenstates that are particularly affected by the presence of a resonance . 
   choose @xmath3 and @xmath29 arbitrary , and then look for the smaller values of @xmath30 and @xmath27 such that the two inequalities , equations . 
   ( [ ordering ] ) and ( [ reverse ] ) are fulfilled . 
   so far , all the examples analyzed by us show that if the inequalities are satisfied for some @xmath30 and @xmath27 then they are satisfied too by the eigenvalues @xmath31 , for @xmath32 . 
   to illustrate how our prescription works we used two different model hamiltonians . 
   the first model , due to hellmann  @xcite , is a one particle hamiltonian that models a @xmath33-electron atom . 
   the second one is a two particle model that has been used to study the low energy and resonance states of two electrons confined in a semiconductor quantum dot  @xcite . 
   the details of the variational treatment of both models will be kept as concise as possible . 
   the one particle model has been used before for the determination of critical nuclear charges for @xmath33-electron atoms  @xcite , it also gives reasonable results for resonance states in atomic anions  @xcite and continuum states  @xcite . 
   the interaction of a valence electron with the atomic core is modeled by a one - particle potential with two asymptotic behaviours . 
   the potential behaves correctly in the regions where electron is far from the atomic core ( @xmath34 electrons and the nucleus of charge @xmath35 ) and when it is near the nucleus . 
   the hamiltonian , in scaled coordinates @xmath36 , is @xmath37 where @xmath38 and @xmath39 is a range parameter that determines the transition between the asymptotic regimes , for distances near the nucleus @xmath40 and in the case @xmath41 the nucleus charge is screened by the @xmath34 localized electrons and @xmath42 . 
   another advantage of the potential comes from its analytical properties . 
   in particular this potential is well behaved and the energy of the resonance states can be calculated using complex scaling methods . 
   so , besides its simplicity , the model potential allows us to obtain the energy of the resonance by two independent methods and check our results .    the two particle model that we considered describes two electrons interacting via the coulomb repulsion and confined by an external potential with spherical symmetry . 
   we use a short - range potential suitable to apply the complex scaling method . 
   the hamiltonian @xmath5 for the system is given by    @xmath43    where @xmath44 , @xmath45 the position operator of electron @xmath46 ; @xmath47 and @xmath48 determine the range and depth of the dot potential . 
   after re - scaling with @xmath47 , in atomic units , the hamiltonian of equation  ( [ hamiltoniano ] ) can be written as @xmath49 where @xmath50 .    the variational spectrum of the two particle model , equation  ( [ hamil2part ] ) , and all the necessary algebraic details to obtain it , has been studied with great detail in reference  @xcite so , until the end of this section , we discuss the variational solution of the one particle model given by equation  ( [ hamil ] ) . 
   the discrete spectrum and the resonance states of the model given by equation  ( [ hamil ] ) can be obtained approximately using a real @xmath51 truncated basis set @xmath52 to construct a @xmath53 hamiltonian matrix @xmath54 . 
   we use the rayleigh - ritz variational method to obtain the approximations @xmath55 
   @xmath56    for bound states this functions are variationally optimal . 
   the functions @xmath57 are    @xmath58    and @xmath59 are the associated laguerre polynomials of @xmath60 order and degree @xmath61 . the non - linear parameter @xmath9 is used for eigenvalue stabilization in resonance analysis  @xcite . note that @xmath9 plays a similar role that the finite size of the box in spherical box stabilization procedures  @xcite , as stated by kar _ et . 
   @xcite . 
   resonance states are characterized by isolated complex eigenvalues , @xmath62 , whose eigenfunctions are not square - integrable . 
   these states are considered as quasi - bound states of energy @xmath0 and inverse lifetime @xmath1 . for the hamiltonian equation  ( [ hamil ] ) , 
   the resonance energies belong to the positive energy range @xcite .    using the approximate solutions of hamiltonian  ( [ hamil ] ) we analyze the dos method @xcite that has been used extensively to calculate the energy and lifetime of resonance states , in particular we intend to show that 1 ) the dos method provides a host of approximate values whose accuracy is hard to assess , and 2 ) 
   if the dos method is supplemented by a new optimization rule , it results in a convergent series of approximate values for the energy and lifetime of resonance states . 
   the dos method relies on the possibility to calculate the ritz - variational eigenfunctions and eigenvalues for many different values of the non - linear parameter @xmath9 ( see kar and ho   @xcite ) . 
   the localized dos @xmath63 can be expressed as @xcite @xmath64 since we are dealing with a numerical approximation , we calculate the energies in a discretization @xmath65 of the continuous parameter @xmath9 . 
   in this approximation , equation  ( [ densidad_sin_suma ] ) can be written as @xmath66 where @xmath67 is the @xmath30-th eigenvalue of the @xmath53 matrix hamiltonian with @xmath68 and @xmath39 fixed .    in complex scaling methods 
   the hamiltonian is dilated by a complex factor @xmath69 . as was pointed out long time ago by moiseyev and coworkers  @xcite , the role played by @xmath9 and @xmath70 
   are equivalent , in fact , our parameter @xmath9 corresponds to @xmath71 . 
   besides , the dos attains its maximum at optimal values of @xmath9 and @xmath0 that could be obtained with a self - adjoint hamiltonian without using complex scaling methods  @xcite . 
   so , locating the position of the resonance using the maximum of the dos is equivalent to the stabilization criterion used in complex dilation methods that requires the approximate fulfillment of the complex virial theorem @xcite . 
   the values of @xmath72 and @xmath73 are obtained performing a nonlinear fitting of @xmath63 , with a lorentzian function ,    @xmath74}.\ ] ]    one of the drawbacks of this method results evident : for each pair @xmath75 there are several @xmath11 , and since each @xmath11 provides a value for @xmath76 and @xmath77 one has to choose which one is the best . 
   kar and ho @xcite solve this problem fitting all the @xmath11 and keeping as the best values for @xmath0 and @xmath1 the fitting parameters with the smaller @xmath78 value . 
   at least for their data the best fitting ( the smaller @xmath78 ) usually corresponds to the larger @xmath30 . 
   this fact has a clear interpretation , if the numerical method approximates @xmath0 with some @xmath79 , where @xmath3 is the basis set size of the variational method , a large @xmath30 means that the numerical method is able to provide a large number of approximate levels , and so the continuum of positive - energy states is `` better '' approximated .    in a previous work @xcite 
   we have shown that a very good approximation to the energy of the resonance state is obtained considering just the energy value where @xmath11 attains its maximum . 
   we denote this value as @xmath76 . 
   figure  [ prefig1 ] shows the approximate resonance energy @xmath76 for different basis set size @xmath3 , where @xmath29 is the index of the variational eigenvalue used to calculate the dos . 
   we used the values @xmath80 and @xmath81 corresponding to the ones used before  @xcite in the analysis of @xmath82 resonances . 
   the figure  [ prefig1 ] also shows the value calculated using complex scaling . 
   it is clear that the accuracy of all the values shown is rather good ( all the values shown differ in less than 6@xmath83 ) , and that larger values on @xmath84 provide better values for the resonance energy . 
   these facts are well known from previous works , _ 
   i.e. _ almost all methods to calculate the energy of the resonance give rather stable and accurate results for @xmath0 
   . however , the practical importance of this fact is reduced : these are uncontrolled methods , so the accuracy of the values obtained from the dos can not be assessed ( without a value independently obtained ) and these values do not seem to converge to the value obtained using complex scaling when @xmath3 is increased and @xmath84 is kept fixed . 
   there is another fact that potentially could render the whole method useless : for small or even moderate @xmath29 , the values @xmath76 become _ unstable _ ( see figure  [ prefig1 ] ) when @xmath3 is large enough . 
   this last point has been pointed previously  @xcite . in the problem that we are considering 
   is rather easy to obtain a large number of variational eigenvalues in the interval where the resonances are located , allowing us to calculate @xmath76 up to @xmath85 , but this situation is far from common see , for example , references @xcite . 
   so far we have presented only results about the behaviour of the one particle hamiltonian , from now on we will discuss both models , equations  ( [ hamil ] ) and [ hamil2part ] . 
   it is known that the variational eigenvalues @xmath17 do not present crossings when they are calculated for some fixed values of @xmath3 , _ 
   i.e _ the variational spectrum is non - degenerate for any finite hamiltonian matrix . 
   as a matter of fact the avoided crossings between successive eigenvalues in the variational spectrum are the watermark of a resonance . 
   an interesting feature emerges when the variational spectrum for many different basis set sizes @xmath3 are plotted together versus the parameter @xmath9 . besides the places where @xmath86 attains its minimum value , which correspond to the stabilization points , there are some gaps which correspond to crossings between eigenvalues obtained with different basis set sizes , see figure  [ prefig2 ] . 
   moreover , the crossings corresponds to eigenvalues with different index @xmath29 , and are the states that satisfy the inequalities equations  ( [ ordering]),and ( [ reverse ] ) . 
   it is worth to remark that the main features shown by figure  [ prefig2 ] are independent of the number of particles of the hamiltonian and the particular values of the threshold of the continuum . 
   figure  [ bundle2p ] shows the behaviour of the variational eigenvalues obtained for the two particle hamiltonian equation  ( [ hamil2part ] ) . in this case 
   the ionization threshold is not the asymptotic value of the potential , but it is given by the energy of the one particle ground state . 
   the resonance state came from the two - particle ground state that becomes unstable and enters into the continuum of states when the quantum dots becomes `` too small '' to accommodate two electrons . for more details about the model , 
   see reference  @xcite . 
   the left panel of figure  [ prefig3 ] shows the behaviour of the maximum value of the dos , @xmath87 , for the one particle hamiltonian , obtained for different basis - set sizes and fixed @xmath29 ( in this case @xmath85 ) , and the @xmath88 obtained choosing a `` bundle '' of states that are linked by a crossing , these states have @xmath89 and @xmath90 respectively . from our numerical data , 
   the maximum value of the dos scales with the basis - set size following two different prescriptions . for @xmath29 
   fixed , @xmath91 , with @xmath92 , while when the pair @xmath93 is chosen from the set of pairs that label a bundle of states @xmath94 , with @xmath95 . 
   in particular , for @xmath85 we get that @xmath96 , and @xmath97 when @xmath98 .    of course we can pick sets of states that are not related by a crossing . 
   for instance , we also picked sets with a simple prescription as follows : choose a given initial pair @xmath99 and form a set of states with the states labeled by @xmath100 and so on . 
   figure  [ prefig3](a ) shows two examples obtained choosing @xmath101 , @xmath102 and @xmath103 and @xmath104 both with @xmath105 . 
   quite interestingly , the data in figure  [ prefig3 ] show that the scaled maxima of the dos for a bundle and two different sets seem to converge to the _ same _ value when @xmath106 , but only for the bundle the scaling function is @xmath107 . 
   the advantage obtained from picking those eigenvalues @xmath17 that belong to a given bundle is still more evident when the corresponding dos and @xmath0 are calculated . 
   the right panel of figure  [ prefig3 ] shows the @xmath0 obtained from the dos whose maxima are shown in the left panel . 
   it is rather evident that these values now seem to converge , besides , the extrapolation to @xmath4 results in a more accurate approximate value for @xmath0 . in contradistinction 
   , the values for @xmath0 corresponding to a fixed index @xmath29 ( the values shown in the figure [ prefig3 ] correspond to @xmath85 ) do not seem to converge anywhere close to the value obtained using complex rotation . 
   figure  [ er2par ] shows the resonance energies obtained from the bundles of states shown in figure  [ bundle2p ] for the two - particle model . since the numerical solution of this model is more complicated than the solution of the one - particle model the number of approximate values is rather reduced . 
   however , it seems that the data also supports a linear scaling of @xmath76 with @xmath108 . 
   many real algebra methods to calculate resonance energies use a golden rule - like formula to calculate the resonance width . in this section 
   we will use the formula and stabilization procedure proposed by tucker and truhlar @xcite that we will describe briefly for completeness . 
   this projection formula seems to work better for one - particle models . for two - particle models 
   its utility has been questioned  @xcite , so to analyze the width of the resonance states of the quantum dot model we fitted the corresponding dos using equation  ( [ lorentz ] ) . 
   the method of tucker and truhlar @xcite is implemented by the following steps . choose a basis @xmath109 where @xmath9 is a non - linear parameter . 
   diagonalize the hamiltonian using up to @xmath3 functions of the basis . 
   look for the stabilization value @xmath110 and its corresponding eigenfunction @xmath111 which are founded for some value @xmath112 . define the projector @xmath113 where @xmath114 is the normalized projection of @xmath111 onto the basis @xmath115 for any other @xmath9 . 
   diagonalize the hamiltonian @xmath116 in the basis @xmath115 , again as a function of @xmath9 , and find a value @xmath117 of @xmath9 such that @xmath118 where @xmath119 denotes eigenvalue @xmath84 of the projected hamiltonian for the scale factor @xmath117 , and @xmath120 is the corresponding eigenfunction .    with the previous definitions and quantities , the resonance width @xmath1 is given by @xmath121 where @xmath122.\ ] ]    despite some useful insights , the procedure sketched above does not determine all the intervening quantities , for instance there are many solutions to equation  ( [ seg - estabili ] ) and , of course 
   , the stabilization method provides several good candidates for @xmath111 and @xmath112 . 
   we are able to avoid some of the indeterminacies associated to the tucker and truhlar procedure using a bundle of states associated to a crossing , so @xmath111 and @xmath112 are given by any of the eigenfunctions associated to a bundle and @xmath123 comes from the stabilization procedure . then we construct projectors @xmath124 where @xmath19 is one of the variational eigenstates that belong to a bundle of states . with the projectors 
   @xmath125 we construct hamiltonians @xmath126 , and find the solutions to the problem @xmath127 since there is not an a priori criteria to choose one particular solution of equation  ( [ tercera - estabili ] ) we show our numerical findings for several values of @xmath84 . 
   figure  [ prefig4 ] shows the behaviour of the resonance width calculated with equation  ( [ golden2 ] ) , where we have used @xmath128 as @xmath111 and @xmath129 , where @xmath130 and @xmath131 .    despite 
   that the different sets corresponding to different values of @xmath84 do not converge to any definite value , for @xmath3 large enough all the sets scale as @xmath132 , with @xmath133 . since the resonance energy scales as @xmath107 , at least when a bundle of states with a crossing is chosen to calculate approximations ( see figure  [ prefig3 ] ) , we suggest that the right scaling for @xmath1 is given by @xmath134 . 
   of course for a given basis size , particular variational functions , stabilization procedures and so on , we can hardly expect to find a proper set of @xmath1 whose scaling law would be @xmath107 . instead of this 
   we propose that the data in the right panel can be fitted by @xmath135 then the best approximation for the resonance width is obtained fitting the curve and selecting the @xmath136 as @xmath137 for @xmath138 the closest value to one . 
   as pointed in reference  @xcite , the projection technique to calculate the width of a resonance can be implemented if a suitable form of the projection operator can be found . 
   as this procedure is marred by several issues we used the dos method to obtain the approximate widths of a resonance state of the two particle model . 
   figure  [ gama2par ] shows the widths calculated associated to the energies shown in figure  [ er2par ] , the parameters of the hamiltonian are exactly the same . 
   there is no obvious scaling function that allows the extrapolation of the data but , even for moderate values of @xmath3 , it seems as the data converge to the value obtained using complex scaling . 
   in this work we analyzed the convergence properties of real @xmath51 basis - set methods to obtain resonance energies and lifetimes . 
   the convergence of the energy with the basis - set size for bound states is well understood , the larger the basis set the better the results and these methods converge to the exact values for the basis - set size going to infinite ( complete basis set ) . 
   this idea is frequently applied to resonance states . 
   the increase of the basis - set size in some commonly used methods does not improve the accuracy of the value obtained for the resonance energy @xmath0 , as showed in figure  [ prefig1 ] . 
   this undesirable behavior comes from the fact that the procedure is not variational as in the case of bound states . 
   moreover , the exact resonance eigenfunction does not belong to the hilbert space expanded by the complete basis set . in this work we presented a prescription to pick a set or bundle of states that has linear convergence properties for small width resonances . 
   this procedure is robust because the choice of different bundles results in very similar convergence curves and energy values . 
   in fact , in the method described here , the pairs @xmath93 of the bundles play the role of a second stabilization parameter together with the variational parameter @xmath9 . of a second we tested the method in others one and two particle systems and the general behavior of them 
   is the same . 
   the results are very good in all cases leading to an improvement in the calculation of the resonance energies . 
   nevertheless we have to note that the method could no be applied in cases where two or more resonance energies lie very close because the overlapping bundles . 
   the lifetime calculation is more subtle . 
   the use of golden - rule - like formulas , as we applied here , always give several possible outcomes for the width @xmath139 , corresponding to different pseudo - continuum states @xmath140 . 
   the projection technique , equation  ( [ golden2 ] ) , is not the exception and it is not possible to select _ a priori _ which value of @xmath141 is the most accurate . 
   the linear convergence of the dos with basis - set size suggests that the scaling in the lifetime value , in accordance with the energy scaling , should be linear . 
   regrettably , the projection method gives discrete sets of values which can not be tuned to obtain an exact linear convergence . 
   our recipe is to choose the set @xmath139 whose scaling is closest to the linear one , then the best estimation for the resonance width is obtained from extrapolation . 
   many open questions remain on the analysis of the different convergence properties of resonance energy and lifetime . 
   the method presented here to obtain the resonance energy from convergence properties works very well , but the appearance of bundles in the spectrum is not completely understood . even there is not a rigorous proof , the numerical evidence supports the idea that the behaviour of the systems studied here is quite general . 
   in this appendix we give arguments that support our assumptions on the scaling of the eigenenergies with the basis - set parameter @xmath9 . 
   we present our argument for one body hamiltonians , but it is straightforward to generalize to more particles with pair interactions decaying fast enough at large distances .      1 . 
   let @xmath142 be an @xmath143 matrix with all its matrix elements having the form @xmath144 , where @xmath145 , then if @xmath146 the eigenvalues of @xmath142 scales with @xmath147 : @xmath148=0\;\rightarrow det[a(1)-\frac{\lambda(\eta)}{f(\eta)}\,i]=0 \rightarrow \lambda(1)=\frac{\lambda(\eta)}{f(\eta ) } \,.\ ] ] 2 . 
   let @xmath149 @xmath143 be symmetric matrices with @xmath150 , and @xmath151 the eigenvalues of @xmath149 and @xmath152 respectively , in nondecreasing order , then , by the minimax principle @xcite      consider a spherical one - particle potential with compact support , @xmath154 if @xmath155 , and finite , @xmath156 ( both conditions could be relaxed , but we adopt them for simplicity ) . 
   let the basis - set functions be of the form @xmath157 with @xmath158 and @xmath159 , then the coefficients take the form @xmath160 .        and 
   then , by equation ( [ asa ] ) , all the eigenvalues of the kinetic energy have the same scaling with @xmath16 . 
   we have to show that , in both limits , @xmath162 and @xmath163 , for all the potential matrix elements hold @xmath164 , and then , by the wielandt - hoffman theorem @xcite , the eigenenergies are a perturbation of the eigenvalues of the kinetic energy . 
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summary: the resonance states of one- and two - particle hamiltonians are studied using variational expansions with real basis - set functions . 
   the resonance energies , @xmath0 , and widths , @xmath1 , are calculated using the density of states and an @xmath2 golden rule - like formula . 
   we present a recipe to select adequately some solutions of the variational problem . 
   the set of approximate energies obtained shows a very regular behaviour with the basis - set size , @xmath3 . 
   indeed , these particular variational eigenvalues show a quite simple scaling behaviour and convergence when @xmath4 . following the same prescription to choose particular solutions of the variational problem 
   we obtain a set of approximate widths . using the scaling function that characterizes the behaviour of the approximate energies as a guide 
   , it is possible to find a very good approximation to the actual value of the resonance width .