530.145.7; 539.194

.. , ..
. . .. , . E-mail: byk@asd.iao.ru
, . , .
:
, , , . Key words:
Perturbation theory, summation of divergent series, Stiltjes series, anharmonic oscillator.

() . (, , ) , , . , [1, 2] [3-10]. : 20- [3].
, . , , . , , , , . , , .
, . , , . , , , -, , , .
. , [11-14] .
1.
, ()
() = e0 + 1 + e2X2 +... (1)
- , e0,eb... - . , [2].

Mn =j tnd x(t) = tnp(t )dt (2)
0 0
- ,
x(x) = p(t) dt. (3)
0
(p(t) . , (2) n.
(1) :
^ + ^+ ^2 +...^ g(X). (4)
Mo Ml M2
g(A), :

E () = p(t) g ( )dt.
0
, [2], . . ї . , - (3), p(t) - , -
, , (1) X, (0 [2].
, , .
= | ^ )
(5)
^) = 2 (/)
(11)
,
^ = (-1)4, = 0,..., N. (12)
:
* = 2 ^ = (-1)4
(13)
=[1
)

.
) = 2 V).
(6)
()
)=2
(7)
(X) = V [)
' =0 1+
(10)
=0

(14)
, , [3] , , . .
- (), =0,..., 0<<,
(13) +1 +1 . , :
N - (-1)1+., =0 ) '
(15)
S - , (14), - 8. , (4)
(X) =7"^ + ^ (X),
1 +
(
^ ()= V
- (-1)
^
.
(16)
( )

=[(() ( = V ' = V 1,, (8)
'=0 '=0
.
^ //
(X) = [+ = 2. (X). (9)
=0 ' = 0 01 + ^ '= 0

, , , :
N ( '
(X) = 2 1 = (X),
=0 1 + XI = 0

, (X)=2 (-1)
+1, +1 [ (0
[01+xt
.
- , .
(8) [2, 11, 12]. (7-10).
. (6) , +1
, , , , [12].
, (16) , . . . , (-1)" . :
=0
=0


=0
'=0
=0

() = 0 1+
+
,, + N. + 2
+ 1 2
+^ ?'1 * >(0<?{ >( 0 1 + 1 '
111 ^

+... +
1 + //2/3
+ N. +...+ +
0 0 (+1) (
(-1) "
1 (-1/
.
(18)
(18) . 1(-1/0 . , 1(-1) , , . , , , , , . -ї
(1)(2)(',)...<(+1)(/+1) , , ,
1+/2.+1 1 "+1 ' ;
()(/) - , (11) , , (13).
(17) , (13) . , (15-17) , , . , .
2.
- ї (). , (/). (0, , =0,..., (13), >. , ^ . , , , , , . . [11].
. [5, 10] . , , . V [5]:
= +1 ,
(-1)
(0 1(-1/, = 0,... (19)

(19) ї . [5, 7], (0
^()(() =
( + ) ( +
-,(/2 ^ (20)
, 0.... (14) :
1
( + ) ( +
(-!)' +-++1 | 1
( + 12 2 I + + / + 11.
1=0
- /
(13) . ,
=} ^ = /
gn g():
() = .
(21)
g(), ї , , , - (21) .
() = (),
=0 1

=0

N 0
) ^"1)'+' (^ { () 8 ) .
, , - , - 1,.
, , .
3.
. :
= -(2 + 2) + 4. (22)
=/ - . (22) , , . 0,01 ї , 1 .
, (22) [3-5]. , , , .
(1) - =0,01; 0,05; 0,1; 0,5; 1,0; 5,0 . 40.
.
0,01 0,05 0,1
0,50725620452460284095 0,53264275477185884442 0,55914632718351957675
0 0,50725620452460284095 0,53264275477185884442 0,55914632718351953369
0,50725620452460284095 0,53264275477185884443 0,55914632718351971066
1,53564827829680346290 1,65343600657645675356 1,76950264394905424513
1 1,53564827829680346290 1,65343600657645675354 1,76950264394905256115
1,53564827829680346290 1,65343600657645676872 1,76950264394921161047
2,59084579619070266576 2,87397963441678165236 3,13862430849812044456
2 2,59084579619070266576 2,87397963441678165241 3,13862430849811296841
2,59084579619070266576 2,87397963441678171887 3,13862430849859121295
3,67109494222579801861 4,17633891289287735612 4,62888280888814199031
3 3,67109494222579801861 4,17633891289287735612 4,62888280888812338445
3,67109494222579801861 4,17633891289287815223 4,62888280889243859547
4,77491311865550985124 5,54929781131652150600 6,22030090000652428174
4 4,77491311865550985124 5,54929781131652150588 6,22030090000654380106
4,77491311865550985124 5,54929781131650259833 6,22030089994815301573
5,90102667411262297563 6,98496309887139965285 7,89976722787143304229
5 5,90102667411262297563 6,98496309887139965285 7,8997672278714201454
5,90102667411262297563 6,98496309887129890796 7,89976722761205326245
\/\ 0,5 1,0 5,0
0,6961758207 0,8037706512 1,2245874494
0 0,6961758206 0,8037705972 1,2244526319
0,6961758215 0,8037707294 1,2247119169
2,3244063521 2,7378922687 4,2995080876
1 2,3244063620 2,7378916719 4,2953498495
2,3244065104 2,7378995539 4,3028206991
4,3275249789 5,1792916968 8,3179757567
2 4,3275231005 5,1791713478 8,2631338210
4,3275253343 5,1793075143 8,3248819016
6,5784019497 7,9424040035 12,9034930646
3 6,5783955979 7,9417651965 12,6223459874
6,5784035887 7,9424624079 12,9211429034
9,0287787259 10,9635844065 17,9470638785
4 9,0287847244 10,9624910111 17,6889377150
9,0287691513 10,9633166408 17,8893390176
11,6487207367 14,2031451886 23,3730829963
5 11,6487080496 14,2010837547 23,1120621444
11,6486904036 14,2023522434 23,2080076384
- (20), N (11) 10 80. () , =0-5, .
() - () 20- . , , ї.
, , (=0,01, ) 10-20. 0,01 0,1 , , , --. =0,5, 1,0 1,0, , , 1 % .

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, . , . . -. , . , , -- . . .
( 06-03-39014-_) .
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06.07.2009.