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the negative positronium ion ps@xmath0 is a bound state of two electrons and a positron .
it is the simplest bound three body system from the theoretical point of view , since it does not contain a hadronic nucleus .
it provides an important testing ground for quantum electrodynamics ( qed ) , which should be able to describe this purely leptonic bound state with high precision .
because of the @xmath1 annihilation , ps@xmath2 is unstable , with a lifetime of about four times that of para - positronium .
it decays predominantly into two or three photons , with the one - photon decay possible but extremely rare .
it is weakly bound and has no excited states in the discrete spectrum @xcite ( for a discussion of resonances , see @xcite ) .
following its prediction by wheeler in 1946 @xcite and experimental observation in 1981 by mills @xcite , the positronium ion has been subject to much theoretical study .
its non - relativistic bound state energy , decay rate , branching ratios of various decay channels , and polarizabilities have been computed accurately using variational methods @xcite .
recently , intense positronium sources have become available , opening new possibilities for experimental studies of ps@xmath2 @xcite . the measured decay rate @xcite agrees with the theoretical prediction .
improved measurements of the decay rate , the three - photon branching ratio , and the binding energy have been proposed @xcite .
a challenge in the theoretical study of this three - body system is that its wave function is not known analytically , even if only the coulomb interaction is considered . since all particle masses and magnitudes of their charges are equal , it is not possible to use the born - oppenheimer approximation .
so far all precise theoretical predictions of ps@xmath2 properties have relied on variational calculations .    in the present paper
we explore a different approach to computing the wave function and the binding energy of ps@xmath2 .
we use dimensional scaling ( ds ) method , in which the dimensionality of space @xmath3 is a variable .
we focus on the limits @xmath4 and @xmath5 . a precise result for @xmath6
may be obtained by interpolating between the two limits using perturbation theory in @xmath7 .
the advantage of ds is that the two limits of the schrdinger equation often have relatively simple solutions .
full inter - particle correlation effects are included at every order in the perturbation expansion in @xmath7 .
more information about dimensional scaling and further references can be found in @xcite .
it is important to note that the dimensional limits considered here are not physical in the sense that the form of the potential energy is taken to be @xmath8 , regardless of the dimension .
a physical limit of a system would use an appropriate coulomb potential that is the solution of a @xmath3-dimensional poisson equation .
for example for @xmath9 it is linear , logarithmic for @xmath10 , and depends on charge separation as @xmath11 for @xmath12 .
since we are ultimately interested in @xmath6 physics , it is useful to fix the potential to be the @xmath6 coulomb interaction .
the @xmath4 limit used here offers the additional simplification that after coordinate and energy rescaling the potential takes form @xmath13 , which can be formally replaced by a dirac delta function @xcite .
we find that the ds provides a useful complement to the variational method . in the future
, it can be employed to independently check matrix elements of operators needed in precise studies of ps@xmath2 .
this paper is organized as follows . in section  [ sec : d1 ] we consider the @xmath9 limit of the ps@xmath0 system .
we solve the schrdinger equation numerically to find an eigenvalue that approaches a simple rational number , possibly hinting at the existence of an analytical solution .    in section  [ sec : dinfty ]
we consider the @xmath5 limit and describe the resulting @xmath7 expansion .
we sum up the perturbation series for the ground state energy and evaluate it at @xmath6 .
the binding energy we find agrees with variational studies to five significant figures .
we conclude in section  [ sec : conclusion ] .
in the one dimensional limit , the coulomb potential is represented by the dirac delta function @xcite .
delta function models have been used extensively also in condensed matter physics . a simple analytical wave function exists for any number of identical particles interacting via attractive potentials @xcite .
the case of all repulsive potentials with periodic boundary conditions has been treated by lieb and liniger @xcite , and yang @xcite .
more recent works have studied one dimensional systems with both attractive and repulsive delta interactions .
_ considered the dependence of the energy on the number of particles in a system of equal numbers of positively and negatively charged bosons @xcite .
li and ma studied a system of @xmath15 identical particles with an impurity with periodic boundary conditions @xcite .
the @xmath9 limit of the ps@xmath0 quantum problem is a delta function model with two attractive and one repulsive delta functions with non - periodic boundary conditions , which , to the best of our knowledge , has not yet been solved .
we present a derivation of a one dimensional integral equation for the solution to this problem , analogous to the helium case treated by rosenthal @xcite .
the time independent schrdinger equation for the relative motion of @xmath16 takes the dimensionless form @xmath17 -\frac{1}{r_1}-\frac{1}{r_2}+\frac{1}{r}\right)\psi = \varepsilon\psi , \label{eq : schrodeqexplicit}\ ] ] where @xmath18 and @xmath19 are the electron - positron distances , @xmath20 is the inter - electron distance ( in units of @xmath21 with @xmath22 ) and @xmath23 determines the energy eigenvalue , @xmath24 .
this choice of units helps compare intermediate results with rosenthal s delta function model of helium @xcite .    in the limit @xmath4 , we let @xmath25 and @xmath26 , where @xmath27 ; the gradients become partial derivatives and the coulomb potentials are replaced by dirac delta functions ( this limit is described in detail in @xcite ) .
equation ( [ eq : schrodeqexplicit ] ) is replaced by @xmath28\psi    \nonumber \\ & &   \hspace*{10mm}= \varepsilon\psi .
\label{eq : posshrodeq}\end{aligned}\ ] ] using fourier transformation , we rewrite this schrdinger equation as a one - dimensional integral equation , @xmath29 where the fourier transforms of the wave function @xmath30 are @xmath31 and @xmath32 .
we now invert the transformation ( [ eq : transg ] ) , and use the resulting @xmath33 in eqs .
( [ eq : xpsitran],[eq : xypsitran ] ) to obtain a system of two integral equations for @xmath34 and @xmath35 .
these are easily decoupled and yield      and@xmath37 once @xmath34 is found one can compute @xmath35 .
the two - dimensional eigenvalue problem is thus reduced to a one - dimensional integral equation ( [ eq : psintegralequationexplicit ] ) , which we solve numerically .
the integral equation is discretized using gauss - legendre quadrature , casting it into a system of homogeneous linear equations for @xmath38 , where @xmath39 are the abscissas .
the system has a non - trivial solution when the determinant of the discretized integral kernel vanishes .
this condition fixes the value of @xmath40 and thus the @xmath9 binding energy .
the wave function is then determined by solving the linear system for @xmath38 .
one finds that @xmath38 spans the null space of the discretized kernel and can be computed using its singular value decomposition .
we used cubic spline interpolation on the set @xmath41 to interpolate between the quadrature points and generate an approximation for @xmath34 .    once @xmath34 is known , the functions @xmath35 and @xmath42 are constructed using eq .
( [ eq : hk-1 ] ) and ( [ eq : gkk ] ) .
finally , the wave function @xmath30 is obtained by the inverse fourier transformation of @xmath42 .      .binding energy of the one - dimensional model of the positronium ion , in units @xmath43 . for @xmath44
the eigenvalue appears to converge to @xmath45 . for these large quadrature
sizes the uncertainty in the energy is in the last digit due to finite precision used in the calculation .
[ tab : the - bound - state ] [ cols="^,^ " , ]     the relative effect of round off error is shown in fig .
[ fig : accuracy ] .
we see that the error introduced by finite precision arithmetic is much smaller than the accuracy of the final ground state energy obtained by resumming the @xmath7 expansion .
higher order calculations will require better precision when the fractional error becomes of the same order as the accuracy required .
1   expansion coefficients defined as @xmath46 as a function of the order @xmath47 . the coefficients @xmath48 were obtained using the standard double precision arithmetic ( @xmath49 digits of precision ) , while @xmath50 were obtained using the arprec arbitrary precision library ( with @xmath51 digits of precision ) .
[ fig : accuracy],title="fig : " ]                                        f.  fleischer , the negative ion of positronium : decay rate measurements and prospects for future experiments , in s.  karshenboim ( ed . ) , _ precision physics of simple atoms and molecules _ ,
lecture notes in physics vol .
261281 , springer , berlin , 2008 .
we determine the binding energy of the negative positronium ion in the limits of one spatial dimension and of infinitely many dimensions . 
 the numerical result for the one - dimensional ground state energy seems to be a rational number , suggesting the existence of an analytical solution for the wave function . 
 we construct a perturbation expansion around the infinitely - dimensional limit to compute an accurate estimate for the physical three - dimensional case . 
 that result for the energy agrees to five significant figures with variational studies .