We discuss a close connection between the formula of Banks, Bender, and Wu for the asymptotics of the Rayleigh-Schrödinger coefficients of the two-dimensional rotationally symmetric anharmonic oscillator and the behavior of resonances of the hydrogen Stark problem in two regimes: small field (Oppenheimer's formula) and large field (where we obtain the new results arg E → −π/3, ∣E∣ ∼α[F(lnF)^(2/3) for F, the electric field strength, going to infinity). We also announce a rigorous proof of Bender-Wu-type formulas

Benassi, L.

Grecchi, V.

Harrell, E.

Simon, B.

Caltech Authors - Main

VOLUME 42, NUMBER 11 PHYSICAL REVIEW LETTERS 12 M~RcH 1979
C. C. Chang, J. R. %'u, T. W. Koontz, and H. D.
Holmgren, Bull. Am. Phys. Soc. 28, 929 (1978).
S. Ferroni, G. Ricco, G. A. Rottigini, M. Sanzone,
and G. LoBianco, Nuovo Cimento 85, 15 (1976).
F. Ajzenberg-Selove, Nucl. Phys. A268, 1 (1976},
and references contained therein.
Bender-Wu Formula and the Stark Effect in Hydrogen
L. Benassi and V. Grecchi
Istituto Matematico and Istituto di Ei'sica, Unieexsita di j/Iodena, Modena, Italy
and
E. Harrell
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 0' 39
B. Simon
DePartment of Mathematics and DePartment of Physics, Pnnceton University, Pnnceton, New Jersey 08540
{Received 23 October 1978)
We discuss a close connection bebveen the formula of Banks, Bender, and Wu for the as-
ymptotics of the Rayleigh-Schrodinger coefficients of the two-dimensional rotationally sym-
metric anharmonic oscillator and the behavior of resonances of the hydrogen %ark problem
in teo regimes: small field {Oppenheimer's formula) and large field {where we obtain the
new results argg —~/3, I F I -o.'[&OnF) ]'I' for F, the electric field strength, going to infini-
ty). We also Announce a rigorous proof of Bender-Wu-type formulas.
u(0) =0 (m =1,2, . . .), u(x)-x"' (m =0), (4)
which is the Hamiltonian of the two-dimensional
oscillator —a+ a r'+ pr4 reduced to the subspace(f (r)e' ]'. We will let p, „)(a',P) denote the (n
+ l)th eigenvalue of (3). Using the Bender-Wu
methods, h was studied by Banks, Bender, and
Wu, ' who found, in particular, that the ground-
In this note, we wish to discuss various as-
pects of the hydrogen Stark Hamiltonian (in atom-
ic units)
H = —25 —g~ ~+I'g3,
and, in particular, the celebrated formula of Op-
penheimer' for the width, I', of the ground state
in small positive field (Z = 1),
I'= —2 ImE(E) =4F 'exp(- —,F ')[I+O(F)] (2).
We will link this closely to certain ideas devel-
oped by Bender and Wu during the past ten years'
involving the large-n behavior of the Rayleigh-
Schrodinger coefficients of the ground state of an
anharmonic oscillator. Equation (1) is connected'
to the Hamiltonian
It =-d'/dx'+ (m' ——,')x '+ nx'+Px
with the boundary conditions~
state Rayleigh-Schrodinger coefficients a„, de-
fined by
u, "&(1,P) -pa„P",
obey (n-~)
Here we will discuss three results whose techni-
cal details will appear elsewhere":
(A) The equivalence of Eqs. (2) and (6) modulo
technical conditions";
(B) a rigorous proof of formulas of Bender-Wu-
type including the one-dimensional anharmonic
oscillator, formula (6) and formula (2) (Ref. 6);
(C) an analysis of the relation between (6) and
the large Fbehavior o-f E(F) (Ref. 7); in particu-
lar, for one state that we consider here (which
is probably' the continuation of the ground state)
for E-~,
argE- —m/3+O((lnF) '),
~E~ -aE' (lnF) '+PE' '(lnlnF)
+ O(E"'(lnF) "'), (8)
704 1979 The American Physicai Society
VOLUME 42, NUMBER 11 PHYSICAL REVIEW LETTERS 12 MARcH 1979
where'
0. =2 ' '=0.31498,
P = ~o = 0.83995.
As a preliminary to our discussion of these
imp. "'(I,s "0) - 8P 'exp(--'P ')[ItO(P)],
Bio
and the result of Simon" (n ~ 2),
a„=(- 1)"m 'f x " 'Imp, (l, e "x)dx.
Roughly speaking, '4 (9) is equivalent to (6). We
remark that the relation between (6) and (9) can
be played also with E so that by a result of Herbst
and Simon, "Eq. (2) implies that E(E)-QA„E'"
with
(10)
As a final preliminary, we recall the Symanzik
scaling relation for y, „(Ref. 13) which results
floDl /~A. X:
p„(X'a, X'P) = Xp, „(o,P),
and the resulting fact" that if p„(l,P) is continued
along circles! P! =const, then
The key to the connection between (1) and (3) i.s
the discovery of Titchmarsh" that by use of
squared parabolic coordinates, the equation B(
!
=E( separates into two copies of (2) so that the
points, we make several remarks. For Z, E
real, the Hamiltonian H of Eq. (1) has a purely
continuous spectrum', ' resonance energies are
defined by the method of complex scaling (dila-
tion analyticity") which has recently been extend-
ed to the hydrogen Stark problem. '" Secondly,
Eq. (6) is actually deduced' from the fact that
Fl/3/( 2E)1/2 ~ P 4Z/F&/3
Noting that a/f has a limit as E- 0, we can
identify the ground state" of hydrogen F-(Z, 0)
= —
—,
'Z' as m = n, = n, = 0. Thus,
Re(~) = O(E"'),
and therefore
Im(w') = O(Re(~)') im(~ ) = O(E"') Im(~)
= O(E) Im(wf), (16)
since r„'=O(E"') Multip. lying Eq. (14) by m and
taking imaginary parts, we find that
resonance eigenvalues F.(Z, E) are given by solu-
tions of the implicit equation"
( )(-2EF)+~ (m)(-2E, Fe ' )=4Z (13)
Using the scaling relation (11), (13) is equivalent
to
f(N/) =&,
where
f(M/)=n '[p ' '(1 ~')+p ' '(1 e " ')]
Im(mg) = Imp 0 ' (1,e "Re(E/[-2E(2, Z)]"'))+O(im(m')),
since p, ,/' (1,Re(--)) is real and the derivatives of p, ,~') with respect to P are uniformly bounded as P
-0, !argP! &-, 7/ —5." Thus, using (16) we see that
1m[42/(- 2E)"']= 1m'.")(I,s ™E/[—2~ (Z, o)]"')[1+o(F)],
so long as
Imp, ')(l, e '"[aE+O(E')])=Imp, , ' (1,e"aE)[1+O(E)] (18)
Thus, (2) is equivalent to (9).
As for point (B), we point out that in Ref. 6 a rigorous proof of (9) and also the original Bender-Wu'
formula is given. In this proof, a key role is played by a further transformation of Titchmarsh" which
relates (3) to the Schrodinger operator" obtained by replacing Fx, in (1) by Fx Aformu-la fo.r the
width is obtained in the model by a method whose first-order version is similar to the procedure of
Bender and Wu' of taking a trial wave function which is of WEB type in certain regions and "a,symptot-
ically exact"" in others. We go further than they do by obtaining a convergent expansion" allowing
control on the possible errors. Sufficient control is obtained on derivatives to justify all formal ma-
VOLUME 42, NUMBER 11 PHYSICAL REVIEW LETTERS 12 MARCH 1979
f(e '"'u)=e'"'u '[p,„(i,e "u')+y.„(i,e '"u')]=e'"'u '[g (I., e "u') —p, „,(l, s"u')],
using the second etluality in Eq. (12). Thus if u &0 and n, =n„f will go to zero M. ore generally
analysis" of the asymptotics of p, „shows that f can only go to zero if n, =n, and u stays in the region
iargui - m/6; thus as F-~ along the real axis, iEi - ~ with
(19)
nipulations mentioned above and thus to prove (2). We note that our proof does not seem to shed any
light on the recent functional integration demonstrations"" of the Bender-Wu formula.
Finally, we turn to a study of the limit E-+~, Z fixed. As a preliminary, we note that Herbst" has
shown that for E fixed nonzero and Z =0, there are no resonances and no spectrum for the complex
scaledH. It follows that for F fixed, as Z-0, iEi —~. Thus the only solutions of (14) with f-0 have
u —0 also. At first sight it does not appear that f —0 is possible as w - 0 but if we change variables
ton=8'"'se, we see that
3 7T argE ( 0.2 (20)
This agrees with the result ~ that there are no resonances in a region (Ei —n & argE & —,'n —e; —~Ei &M )
for Z, F fixed. If we now suppose that the asymptotic formula (9), which can be written
u 'p. ,(l, e '"u') -u 'p, (l, 8"u') ™- 16iu 4 exp(- 3 u ')[1+0(u')],
extends to i argui small, then (19) implies that (14) is equivalent to
exp(-,'~ ') = &tc'[C+0(w')],
with v/6 & argse&q/2 and a nonzero constant C. Taking logarithms of (21) we find that
w
' = & log) + 6 Inw + 0(1)= 2 log) —2 ln in& + 0 (1) = —~ logF —2 ln lnF + 0(1),
(21)
which implies (7) and (8).
We close by mentioning four important ques-
tions raised by part (C) of our analysis: (1) We
have found a solution of (14) for each m, n, =n,
for P-0 (F large) and f ~ (F small). If one
takes the small-E solution and tries to increase
Il, does it go over to the large-&' solution" ?
(2) Is it possible to obtain (7),(8) without using
the Titchmarsh reduction? This is relevant to
the large-J" behavior of resonances in complex
atoms. (3) The results of Herbst" imply that for
Z, E fixed there are only finitely many resonanc-
es in any region of the E plane. Since, for I" =0,
there are infinitely many bound states one ex-
pects infinitely many resonances E„(F), which by
the above result of Herbst must have iE„(F)i-~
at n- ~. Since tv„ thus goes to zero while f (tc„)
stays finite, one would expect" that argu„- 0
which would imply that argE„(F) ——s/3 as n- ~.
Is this true'? (4) We expect that a resonance
which solves (14) for F small, n, en, will cross
enough Bender-Wu cuts" as I' is increased" so
that eventually ~, =n, . Thus, for I" large, n, =n»
there would be many solutions of (14) all with E
having the same large-E asymptotic behavior.
Can one prove this'P
It is a pleasure to thank S. Graffi and I. Herbst
for their interest and valuable comments. This
research was supported in part by the National
Science Foundation, Grant No. MCS 78-01885 and
706
in part by the Istituto Nazionale di Fisica Nu-
cleare, Sezione di Bologna One .of us (E.H. ) ac-
knowledges receipt of a National Science Founda-
tion National Needs Fellowship.
~J. R. Oppenheimer, Phys. Hev. 31, 66 (1928); see
also L. D. Landau and E. M. Lifshitz, Quantum Mechan-
ics (Pergamon, New York, 1958).
2C. Bender and T. T. Wu, Phys. Rev. 184, 1231
(1969), and Phys. Rev. Lett. 16, 461 (1971), and Phys.
Rev. D 7, 1620 (1973).
38. Graffi and V. Grecchi, Lett. Math. Phys. 2, 335
(1978), and to be published.
4For m =0, all solutions of (3) go to zero, namely as
x'i2 or x'i~lnx, so we must specify boundary conditions
carefully in that case.
5T. Banks, C. Bender, and T. T. Wu, Phys. Rev. D
8, 3346 (1973).
6E. Harrell and B. Simon, "The Mathematical Theory
of Resonances Whose Widths are Exponentially Small "
(to be published).
VL. Benassi and V. Grecchi, "Resonances in the
Stark Effect and Strongly Asymptotic Approximants"
(to be published); this paper will also discuss numeri-
cal calculation of resonances.
Bof course we expect that the answer is yes; but we
cannot rule out the bizarre possibility that as E is in-
VOLUME 42, NUMBER 11 PHYSICAL REVIEW LETTERS 12 M~RcH 1979
creased from 0, E{I )- at some point E, and then at
E
&
)E
&
a new solution comes in from infinity which
agrees with the E -~ as I' is increased.
The exact values of + and P are probably mainly of
theoretical interest. For numerical purposes it is
probably better to use the implicit Eq. (19) ~ the situa-
tion reminds one of that in the large-field Zeeman ef-
fect; see J. Avron, I. Herbst, and B. Simon, (to be
published) .
~ J. Avron and I. Herbst, Commun. Math. Phys. 52,
239 (1977); I. Herbst, Math. Zeit. 155, 55-70 (1977);
B. Simon, to be published.
An extensive review of both xnathematica1 and com-
putational aspects of complex scaling is to be published.
See W. Reinhardt, Int. J. Quantum Chem. 10, 359
(1976), for calculations on the Stark problem-.
~~I. Herbst, to be published.
B. Simon, Ann. Phys. (N.Y.) 58, 76 (1970).
'4(9) implies (6) but, rigorously speaking (6) only im-
plies (9) in an averaged sense; but if one supposes
that ImBp/BP and Imp have the same small-P depen-
dence, then {6) implies (9).
"I.Herbst and B. Simon, Phys. Bev. Lett. 41, 67
(1978).
~8K. C. Titchmarsh, Eigenfunction ExPansions Asso
ciated with Second Order Differential Equations (Ox-
ford Univ. Press, Oxford, 1958).
YThe occurrence of e '" rather than e'" in (13) is con-
nected (Refs. 3 and 12) with the demand that ImF ~ 0.
With this choice, E is an actual eigenvalue of - 2 e
x ~ -e ezra ~+ e Fx3 for 0 & Im& & 37t as required by
the general theory. The m in {13)is the eigenvalue of
Lz for the problem {1).
It is merely for notational convenience that we con-
sider m = n
&
=n
2
= 0. In Refs. 6 and 7 the general case
is considered.
'BThis operator is further discussed in S. Graffi,
V. Grecchi, S. Lavoni, and M. Maioli, {to be published).
2 In the Bender-Wu case, the trial function is a para-
bolic cylinder function in the near region; in our case,
a Whittaker function —both are exact in the limit P,
respectively I, goes to zero. Near the distant turning
points Airy functions are used in both cases.
'The expansion is obtained by the method of variation
of parameters from the theory of ordinary differential
equations. This idea with first order being purely of
WEB form appears already in N. Fromin and P. O.
Fromin, ~KB Approximations, Contributions to the
Theory (North- Holland, Amsterdam, 1965). Recently
as expansion based on the asymptotic exact form in the
non-WEB region has been introduced independently by
W. Crutchfield, thesis, Princeton University (unpub-
lished), and E. Harrell, to be published. It is the lat-
ter idea that we use; see also E. Harrell, Commun.
Math. Phys. 60, 73 (1978).
L. N. Lipatov, Pis'ma Zh. Eksp. Teor. Fiz. 25, 116
(1971) [JETP Lett. 25, 104 (1977)]; E. Brezin et al. ,
Phys. Rev. D 15, 1544, 1558 {1977).
One can go part way towards a rigorous justification
of the path space argument, namely one can justify the
method on T 'Tg exp(—Tiioi) 1) for each fixed T. The
situation is described in B. Simon, Functiona/ Integra-
tion and Quantum Physics (Academic, New York, to be
published) .
S. Graffi and p. Grecchi, to be published.
"See Ref. 8.
26The only barrier to proving this is that we are deal-
ing with w„'[p„,(l,w„)+p„(l,w„3e")j wit hnq, n 2 —~
as ~n 0
27In addition to the cubic "branch point" structure
that p(1,P) has at P =0, there are infinitely many singu-
larities (believed to be square-root branch points)
coalescing at P = 0, argP =+ 2~, and our convention
above is to draw circular arcs as branch cuts between
these points. These are the Bender-Wu cuts. When
one is crossed, two values of n are interchanged.
A calculation forn&=m=0, n2=1 suggests that this
is precisely what happens; see Ref. 7.
707


Bender-Wu Formula and the Stark Effect in Hydrogen

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Bender-Wu Formula and the Stark Effect in Hydrogen

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