Strong-coupling expansion for the ground-state energy in the|x|αpotential

Abstract

Using lattice techniques we examine the strong-coupling expansion for the ground-state energy of a $g{|x|}^{\ensuremath{\alpha}}(\ensuremath{\alpha}>0)$ potential in quantum mechanics. We are particularly interested in studying the effectiveness of various Pad\'e-type methods for extrapolating the lattice series back to the continuum. We have computed the lattice series out to 12th order for all $\ensuremath{\alpha}$ and we identify three regions. When $\ensuremath{\alpha}<\frac{2}{3}$ the lattice series diverges faster than ($2n$)! and no extrapolation technique appears to work. When $\frac{2}{3}\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}2$ the lattice series diverges roughly like $\ensuremath{\Gamma}((\frac{2}{\ensuremath{\alpha}}\ensuremath{-}1)n)$; here, diagonal Pad\'e extrapolation schemes give excellent results. When $\ensuremath{\alpha}\ensuremath{\ge}2$ the lattice series has a finite radius of convergence; here, completely-off-diagonal Pad\'e extrapolants work best. As $\ensuremath{\alpha}$ increases beyond 2 it becomes more difficult to obtain good continuum results, apparently because the sign pattern of the lattice series seems to fluctuate randomly. The onset of randomness occurs earlier in the lattice series as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$.