Nuclear Physics B145 (1978) 445-458 © North-Holland Publishing Company CONVERGENCE PROPERTIES OF THE PADE APPROXIMANTS ON A LATTICE J. JURKIEWICZ Institute of Physics, Jagellonlan University, Cracow *, Poland J. WOSIEK Institute of Computer Science, Jagellonian University, Cracow, Poland Received 19 July 1978 The system described by the one-dimensional linear potential is solved analytically on a lattice. A new method of calculating the continuous limit of the energy eigenvalues is proposed. This method makes use of the known analytic structure of the approximated function and has very good convergence properties. 1. Introduction The lattice formulation of QCD [ 1-3 ] provides a possibility of calculating the particle spectrum of the theory. In the zeroth-order approximation one considers the strong-coupling limit, i.e., the lattice with infinite spacing d between the lattice sites. Using the Rayleigh-Schr6dinger perturbation theory the bound-state masses can be expanded in a perturbative power series in the inverse of the coupling con- stant. The continuous limit of the theory is the weak-coupling limit (d ~ 0), where the perturbation parameter X becomes infinite. The important mathematical problem is how to extract the information about the asymptotic behaviour of the bound-state masses' eigenvalues from the few known coefficients of the power series in X. This information is usually obtained using the Padd method, where the diagonal Pad6 approximants are calculated for the perturba- tive series of the ratio of two eigenvalues [10]. The quality of approximations used in these calculations can be tested on simple low-dimensional models (cf. e.g. [4,5]). Particularly interesting are systems exactly solvable both on the lattice and in the continuous limit. One such system (harmonic oscillator in one dimension) was discussed in our previous work [6]. The main result was that the sequence P2vN(X) converges relatively slowly and one can not use the * Address: Institute of Physics, Jagellonian University, Reymonta 4, 304359 Krak6w, Poland. 445