Resummation of anisotropic quartic oscillator: Crossover from anisotropic to isotropic large-order behavior

We present an approximative calculation of the ground-state energy for the ani V(x,y)=omega(2)/2(x(2)+y(2))+g/4[x(4)+2(1-delta)x(2)y(2)+y(4)]. Using an instanton solution for the isotropic limit delta=0, we obtain the imaginary part of the ground-state energy for small negative g as a series expansion in the anisotropy parameter delta. From this, the large-order behavior of the g expansions accompanying each power of delta are obtained by means of a dispersion relation in g. The g expansions are summed by a Borel transformation, yielding an approximation to the ground-state energy for the region near the isotropic limit. This approximation is found to be excellent in a rather wide region of delta around delta=0. Special attention is devoted to the immediate vicinity of the isotropy point. Using a simple model integral we show that the large-order behavior of a delta-dependent series expansion in g undergoes. a crossover from an isotropic to an anisotropic regime as the order k of the expansion coefficients passes the value k(cross)similar to 1/\delta\.