An indeterminacy criterion is proven for the moment problem associated with the coefficients of a Borel summable power series of Stieltjes type which diverge faster than (2n) !. As an application we show that the Stieltjes type continued fraction corresponding to the Rayleigh–Schrödinger perturbation expansions for the energy eigenvalues of the anharmonic oscillators (x2(m+1) and in any finite number of dimensions) does not converge to the eigenvalues if m≳2. In particular, this implies the nonconvergence of the Padé approximants to the eigenvalues of p2+x2x2(m+1) if m≳2.

1.
S.
Graffi
,
V.
Grecchi
, and
B.
Simon
,
Phys. Lett. B
32
,
631
(
1970
).
2.
B.
Simon
,
Phys. Rev. Lett.
25
,
1583
(
1970
).
3.
J. P.
Eckmann
,
J.
Magnen
, and
R.
Sénéor
,
Commun. Math. Phys.
39
,
251
(
1975
).
4.
J. J. Loeffel and A. Martin, Proceedings Program 25th Conference, Strasbourg, May, 1970.
5.
J. J.
Loeffel
,
A.
Martin
,
B.
Simon
, and
A. S.
Wightman
,
Phys. Lett. B
30
,
656
(
1969
).
6.
S.
Graffi
,
V.
Grecchi
, and
G.
Turchetti
,
Nuovo Cimento B
4
,
313
(
1971
).
7.
See e.g., G. Doetsch, Theorie und Amvendangen der Laplace Transformation (Springer‐Verlag, Berlin, 1937), Chap. 14.1.
8.
N. I. Akhiezer, The Classical Moment Problem (Oliver and Boyd, Edinburgh, 1965).
9.
R. P. Boas, Entire Functions (Academic, New York, 1954).
10.
B.
Simon
,
Ann. Phys.
58
,
76
(
1970
).
11.
H. S. Wall, Continued Fractions (Van Nostrand, New York, 1947).
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