Volume 69B, number 3 PHYSICS LETTERS 15 August 1977 THE PERTURBATIVE EXPANSION AND THE INFINITE COUPLING LIMIT G. PARISI + LH.E.S., Bures-sur- Yvette, France Received 26 May 1977 Using as numerical input the first orders of the perturbative expansion, we compute the ground state energy of the anharmonic oscillator in the infinite coupling limit. In field theory the perturbative expansion in the coupling constant is not convergent and it is often re- garded as useless for numerical purposes unless the coupling constant is small. However, in these last years it has been shown that it is possible to resume the per- turbative expansion and to obtain reasonable results for coupling constants of order 1 provided that we have a sufficiently good knowledge of the analytic properties of the function we want to compute. The aim of this note is to show that in a particular case (the one dimensional anharmonic oscillator) we can resume the perturbative expansion for arbitrary values of the coupling constant and to construct uni- form approximants in the whole interval 0 - oo. The Lagrangian is L2 ~dx ] + ½¢2 . (1) The sum of all the connected vacuum-vacuum graphs is one half of the ground state energy (E(g)) of the Hamiltonian: H = p2 + x 2 + 2gx 4. (2) E(g) has an asymptotic expansion in powers of g E(g) ~ F-,n=oAng n which is not convergent. Our goal is to compute the function E(g) in the in- finite coupling limit using the numerical knowledge of the first few A n . Without additional information the situation would be hopeless. However we know: I) The asymptotic expansion for E(g) can be Borel summed: co oo E(g)=l / expl-/]B(fl)d/3; B('fl)= S An/n' /3n * On leave of absence from INFN (Frascati). the function B03) being analytic in the positive half plane [ 1]. IIa) When g goes to infinity E(g)~Eoog1/3 + O(g 1/3) [2, 31. IIb) For large g the following expansion is conver- gent: oo E(g) =Eoogl/3 (l t- ~ aKg-2K/3 1 IIIa) The nearest singularity to the origin of the Borel transform B03) is at the point/3 = -1/3 [4-6]. IIIb) The behaviour of the Borel transform near the singularity is known. IIIc) The Borel transform is believed to be analytic in the cut complex plane (the cut goes from -1/3 to _co). IVa) The function E(g) and B(/3) have the following representations [7] : oo ~(g) = K(o) + g f 1/(g + g')D(g') dg'; 0 B(/3) = B(0) + f [ 1 - exp(/3/g')l D(g') dg' (5) 0 1Vb) D(g') I> O. This impressive list of results gives us the possibility of finding convergent sequence of approximants to the function E(g). It is well known that IV a and b imply that the diagonal Pad~ approximants converge to E(g), however in constructing the Pad~ approximants we do not use all the information we have at our disposal. It would be quite useful to find approximants which sat- isfy all the properties I-IV at once. In this note we construct a sequence of approximants using the prop- erties I, IIa, IIIa and IVa which converge uniformly to 329