Large-order analysis of the convergent renormalized strong-coupling perturbation theory for the quartic anharmonic oscillator L. Ska ´ la, 1,2 J. C ˇ ı ´ z ˇ ek, 1,2,3 V. Kapsa, 1 and E. J. Weniger 2,4 1 Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague 2, Czech Republic 2 Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 3 Lehrstuhl fu ¨r Theoretische Chemie, Universita ¨t-Erlangen-Nu ¨rnberg, D-91058 Erlangen, Germany 4 Institut fu ¨r Physikalische und Theoretische Chemie, Universita ¨t Regensburg, D-93040 Regensburg, Germany Received 23 December 1996; revised manuscript received 22 August 1997 Two hundred coefficients of the renormalized strong-coupling perturbation expansion for the ground and first excited states of the quartic anharmonic oscillator are calculated numerically. The large-order behavior of the perturbation coefficients is analyzed, a general and comparatively simple analytic formula describing their large-order behavior is proposed, and it is shown that this formula is consistent with known results from the divergent weak-coupling expansion. The accuracy of our numerically determined coefficients is checked by summation rules. In particular, if the summation rules are supplemented by the leading terms of our large-order formula, we obtain remarkably accurate results. This independently confirms the correctness of our large-order analysis. It is shown that the renormalized strong-coupling expansion converges—in contrast to other pertur- bation expansions—for all physically relevant coupling constants. S1050-2947 97 03712-8 PACS number s : 03.65. w, 02.30.Lt, 02.70. c I. INTRODUCTION We investigate the Schro ¨ dinger equation H ( ) E ( ) for the quartic anharmonic oscillator, where H p 2 x 2 x 4 , 0. 1 This is one of the old, but nontrivial problems of quantum mechanics. As is well known, E ( ) can be expressed as a weak-coupling perturbation series in powers of , which di- verges for every 0 1–4 . Hamiltonian 1 can be transformed into an equivalent Hamiltonian H 1/3 p 2 2/3 x 2 x 4 3 . Consequently, E ( ) also possesses the strong-coupling expansion E 1/3 n 0 K n 2 n /3 . 2 This series converges if is large 3,4 . Unfortunately, the perturbative computation of the coefficients K n is very diffi- cult 5–9 . An alternative perturbative approach based upon renor- malization Wick ordering 10 or scaling 9–14 has con- siderable conceptual and technical advantages. In the quartic case, Wick ordering and scaling are closely related, and they differ by a numerical factor in the effective coupling con- stant. In the scaling approach, 0, ) is replaced by a renormalized coupling constant 0,1) according to / 3(1 ) 3/2 , and Hamiltonian 1 is transformed into a renormalized Hamiltonian H R ( ) 11,12 : H 1 1/2 H R , 3 H R p 2 x 4 /3 1 x 2 x 4 /3 , 4 E 1 1/2 E R . 5 In contrast to Eq. 1 , the perturbation in H R ( ) is a differ- ence of two terms, which partly compensate for each other 11,12 . The renormalized energy E R ( ) can either be ex- pressed as a divergent weak-coupling expansion in 12 , or as a strong-coupling expansion in 1 14 , E R n 0 c n n n 0 n 1 n . 6 The advantage of the renormalized approach is due to the fact that E R ( ) is finite for 0,1 E R (0) 1 and E R (1) 0 , since the troublesome pole (1 ) 1/2 is ex- plicitly factorized out in Eq. 5. The weak-coupling expansion for E R ( ) diverges almost as strongly as the corresponding weak-coupling expansion for E ( ) 10,12 . In contrast, it was shown in theorems 1 and 2 of Ref. 14 that the strong-coupling expansion for E R ( ) is analytic at 1, which implies that it converges if is close to 1. Moreover, Table V in Ref. 14 indicates that this strong-coupling expansion actually converges for all physically relevant 0,1). The main purpose of this paper is to study the large-order behavior of the perturbation series coefficients in the strong- coupling case. We show that this large-order behavior is ex- ceptionally simple in the renormalized case. This provides us with an interesting insight which can be used even for the study of the strong-coupling expansion 2 . In contrast to a large-order analysis of divergent expansions, our large-order analysis can be used directly for numerical purposes. II. NUMERICAL CALCULATIONS In this paper, we compute numerically 200 coefficients n for the ground and first excited states of the quartic anhar- monic oscillator, perform their large-order analysis, and pro- pose an analytic large-order formula for n . With the help of this formula, we show that the strong-coupling expansion for PHYSICAL REVIEW A DECEMBER 1997 VOLUME 56, NUMBER 6 56 1050-2947/97/56 6 /4471 6 /$10.00 4471 © 1997 The American Physical Society