RIMS Kôkyûroku Bessatsu
B5 (2008), 75–88
Adiabatic Transition Probability for a Small
Eigenvalue Gap at Two Points
By
Takuya Watanabe∗
Abstract
Let us consider a 2-level adiabatic transition problem. We study by using an exact WKB
method the asymptotic behavior of the transition probability as the eigenvalue gap parameter,
as well as the adiabatic parameter, tends to 0.
§ 1.
Introduction
In these notes we consider the 2 × 2 system of first order differential equations:
(
)
d
V (t) ε
(1.1)
ih ψ(t) = H(t, ε)ψ(t), H(t, ε) =
,
dt
ε −V (t)
where h and ε are small positive parameters. h is called an adiabatic parameter. V (t)
satisfies the following assumptions:
(A) V (t) is real-valued on R and there exist two real numbers 0 < θ0 < π/2 and µ > 0
such that V (t) is analytic in the complex domain:
S = {t ∈ C; |Im t| < |Re t| tan θ0 } ∪ {|Im t| < µ}.
(B) There exist two real non-zero constants Er , El and σ > 1 such that
{
V (t) =
5Er + O(|t|−σ )
as Re t → +∞ in S,
El + O(|t|−σ )
as Re t → −∞ in S.
Received April 10, 2007, Accepted June 23, 2007.
2000 Mathematics Subject Classification(s): 34E20. 34E25. 81Q20.
Key Words: Adiabatic limit, Transition probability, Exact WKB method.
∗ University of Hyogo, Graduate School of Material Science, Himeji, Shosha, Japan.
c 2008 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
°
76
Takuya Watanabe
The 2 × 2 matrix H(t, ε) is trace-free real-symmetric matrix. H(t, ε) has positive and
√
negative eigenvalues E± (t, ε) = ± V (t)2 + ε2 .
r
r
l
l
Under the conditions (A) and (B), there exist four solutions ψ+
, ψ−
, ψ+
, and ψ−
to (1.1) uniquely defined by the following asymptotic conditions:
)
(
] − sin θ
[ i√
r
r
, as Re t → +∞ in S,
Er2 + ε2 t
ψ+
(t) ∼ exp +
h
cos θr
)
(
[ i√
] cos θ
r
r
ψ−
(t) ∼ exp −
Er2 + ε2 t
,
as Re t → +∞ in S,
h
sin θr
)
(
(1.2)
[ i√
] − sin θ
l
l
, as Re t → −∞ in S,
ψ+
(t) ∼ exp +
El2 + ε2 t
h
cos θl
)
(
[ i√
] cos θ
l
l
,
as Re t → −∞ in S,
ψ− (t) ∼ exp −
El2 + ε2 t
h
sin θl
where tan 2θr = ε/Er and tan 2θl = ε/El (0 < θr , θl < π/2). These solutions are
r
r
l
l
called the Jost solutions to (1.1). The pairs of Jost solutions (ψ+
, ψ−
) and (ψ+
, ψ−
)
2
are orthonormal bases on C for any fixed t.
(
)
Definition 1.1.
The scattering matrix S(ε, h) = skl (ε, h) 1≤k,l≤2 is defined as
the change of bases of Jost solutions:
l
l
r
r
(ψ+
ψ−
) = (ψ+
ψ−
)S(ε, h).
Note that the scattering matrix S(ε, h) is unitary. The transition probability P (ε, h) is
defined by
P (ε, h) = |s21 (ε, h)|2 = |s12 (ε, h)|2 .
In this paper we assume that
(C) V (t) vanishes at two points t = x, y (x > y) on R.
Then the difference of eigenvalues (eigenvalue gap) attains the minimum 2ε at t = x, y.
It is expected that the transition probability is governed by the zeros of V (t).
In the special case where V (t) = at (a > 0) called Landau-Zener model, it was
shown by Landau and Zener in 1932 ([L], [Z]) that the transition probability is given
by the so-called Landau-Zener formula:
[ πε2 ]
(1.3)
P (ε, h) = exp −
ah
for any ε > 0, h > 0. Note that small parameters ε and h play opposite roles for P (ε, h)
from (1.3). Our problem is studying what makes a contribution to the asymptotic
behavior of P (ε, h) as the adiabatic parameter h and the eigenvalue gap ε tend to 0.
77
Adiabatic Transition Probability
§ 2.
Results
We first introduce so-called turning points which play an important role in the
exact WKB method. They are by definition the zeros of V (t)2 + ε2 . Let n (resp. m) be
the order of the zero t = x (resp. y) and suppose V (x) = V 0 (x) = · · · = V (n−1) (x) = 0
and V (n) (x) 6= 0 (resp. V (y) = V 0 (y) = · · · = V (m−1) (y) = 0 and V (m) (y) 6= 0). We can
assume V (n) (x) > 0 without loss of generality. Then there are 2n simple turning points
around t = x, which are denoted by xj (ε) and xj (ε) (j = 1, . . . , n), and they behave
like
(
)1/n
[ (2j − 1)πi ]
n!
xj (ε) ∼ x +
ε1/n
as ε → 0.
exp
2n
|V (n) (x)|
There are also 2m simple turning points around t = y, which are denoted by yj (ε) and
y j (ε) (j = 1, . . . , m), and they have similar asymptotic behaviors.
We define the action integrals Aj (ε) and Bj (ε) by
∫
(2.1)
Aj (ε) = 2
xj (ε)√
∫
V
(t)2
+
ε2
dt,
x
Bj (ε) = 2
yj (ε)√
V (t)2 + ε2 dt,
y
where the integration path of Aj (ε) (resp. Bj (ε)) is the complex segment from x to
xj (ε) (resp. from y to yj (ε)) and the branch of the square root is ε at t = x. Moreover
we introduce the real-valued action integral:
∫ x√
R(ε) = 2
V (t)2 + ε2 dt.
y
We shall observe the asymptotic behaviors of the action integrals for sufficiently small
ε in § 3.2.
§ 2.1.
Related Results
There exist many preceding studies on adiabatic transition problems (see [HJ],
[T]). We here refer to the results by Joye-Mileti-Pfister and Joye in the case where the
eigenvalue gap is fixed. They indicated in [JMP], [J1] that the asymptotic behavior of
the transition probability as h → 0 is determined by the geometrical structure generated
by the Stokes lines closest to the real axis among those passing through turning points
(see § 4.1).
On the other hand when the eigenvalue gap, in particular the off-diagonal part ε in
our Hamiltonian H(t, ε), tends to 0, the number of real zeros of V (t) and their vanishing
order are relevant. Here is a result in the case where V (t) vanishes only at one point
t = x.
Theorem 2.1 ([W]).
Assume V (t) vanishes only at one point t = x.
78
Takuya Watanabe
(i) If n = 1, then there exists ε0 > 0 such that
[ 2
]
P (ε, h) = exp − Im A1 (ε) (1 + O(h))
h
(2.2)
h → 0,
as
uniformly for any ε ∈ (0, ε0 ).
(ii) If n ≥ 2, then there exists ε0 > 0 such that for any ε ∈ (0, ε0 )
(2.3)
¯
(
))
[i
]
[i
]¯¯2 (
¯
h
n+1
¯
¯
P (ε, h) = ¯exp A1 (ε) + (−1)
exp An (ε) ¯ 1 + O (n+1)/n
h
h
ε
as h/ε(n+1)/n → 0.
Remark. The asymptotic formula (2.2) is a natural extension of the LandauZener formula (1.3) (see [H], [J2]). In the case where n ≥ 2, the asymptotic behavior of
P (ε, h) is not determined only by the geometrical structures of Stokes lines but by the
relations between ε and h in addition to the asymptotic condition: h/ε(n+1)/n → 0 (see
[W, Proposition 2.1]).
§ 2.2.
Main Results
We state our results according to the vanishing order n, m.
Theorem 2.2.
ε ∈ (0, ε0 )
(2.4)
If n = m = 1, there exists ε0 > 0 such that we have for any
¯
[i
]¯¯2 (
( h ))
[i(
¯
)]
¯
A1 (ε) + R(ε) − exp B1 (ε) ¯¯ 1 + O 2
P (ε, h) = ¯exp
h
h
ε
as h/ε2 → 0.
Theorem 2.3.
any ε ∈ (0, ε0 )
(2.5)
If n = 1 and m ≥ 2, there exists ε0 > 0 such that we have for
(
))
[ 2
](
h
P (ε, h) = exp − Im A1 (ε) 1 + O (m+1)/m
h
ε
as
h
ε(m+1)/m
→ 0.
Theorem 2.4.
If m ≥ n ≥ 2, there exists ε0 > 0 such that we have for any
ε ∈ (0, ε0 )
¯
[i(
[i(
¯
)]
)]
n+1
¯
exp
A (ε) + R(ε) + (−1)
A (ε) + R(ε)
P (ε, h) = ¯exp
h 1
h n
(
))
[i
]
[i
]¯¯2 (
h
n
n+m+1
+ (−1) exp B1 (ε) + (−1)
exp Bm (ε) ¯¯ 1 + O (n+1)/n
h
h
ε
Adiabatic Transition Probability
79
as h/ε(n+1)/n → 0. In particular, if m > n we have
¯
(
[
))
¯
]
[i
]¯¯2 (
h
i
¯
n+1
¯
(2.6)
P (ε, h) = ¯exp A1 (ε) + (−1)
exp An (ε) ¯ 1 + O (n+1)/n
¯
h
h
ε
as h/ε(n+1)/n → 0.
Remark. The principal terms of the asymptotic formulae (2.4) and (2.3) are the
same in the sense that the action integrals which appear in each principal term are
defined by the two turning points closest to the real axis. In fact they have essentially
the same Stokes geometry. Similarly the principal term of (2.5) is the same as that
of (2.2) and the asymptotic formula (2.6) is exactly the same as (2.3). From above
considerations, it is expected that turning points around the lowest order zero make a
major contribution to the asymptotic behavior of P (ε, h).
If V (t) vanishes at more than two points, we can express the scattering matrix by
means of the product of the transfer matrices (see § 4).
§ 3.
§ 3.1.
Preliminary
Exact WKB Method for 2 × 2 System
We review the exact WKB method for 2 × 2 systems introduced in [FLN], which is
a natural extension of the method in [GG] for Schrödinger equations.
Let us consider the following 2×2 system of first order differential equations:
(
)
h d
0 α(t)
(3.1)
φ(t) =
φ(t).
i dt
−β(t) 0
The functions α(t) and β(t) are assumed to be holomorphic in a simply connected
domain Ω ⊂ C and they do not vanish in a domain Ω1 ⊂ Ω. These zeros of α(t)β(t) are
called turning points and coincide with those mentioned in §2. Note(that )
the equation
1 1 i
φ(t).
(1.1) can be reduced to this anti-diagonal system (3.1) by ψ(t) 7→
2 i 1
The exact WKB solutions to (3.1) are the following form:
(3.2)
φ± (t, h; t0 , t1 ) = e±z(t;t0 )/h M± (z(t))w± (z(t), h; z(t1 ))
for two base points t0 ∈ Ω, t1 ∈ Ω1 , where
∫ t√
z(t; t0 ) =
(3.3)
α(τ )β(τ ) dτ,
t0
80
Takuya Watanabe
)
K(z(t))−1 K(z(t))−1
,
∓iK(z(t)) ±iK(z(t))
√
(
(3.4)
(3.5)
M± (z(t)) =
K(z(t)) =
4
β(t)
,
α(t)
( (
(
(
) )
))
e
∑ w
z(t),
h;
z(t
)
w±
z(t), h; z(t1 )
1 )
±,2k (
(
) =
.
w± (z(t), h; z(t1 )) =
o
z(t),
h;
z(t
)
w
w±
z(t), h; z(t1 )
1
±,2k−1
k≥0
Here the sequences of functions {w±,n (z; z1 )}∞
n=0 , where z1 = z(t1 ) are defined by the
integral recurrent relations:
(3.6)


w±,0 (z; z1 ) = 1,
w±,−1 (z; z1 ) = 0





∫ z

0

w
± 2h (ζ−z) K (ζ)
e
(z;
z
)
=
w
(ζ, ζ1 ) dζ,
±,2k+1
1
K(ζ) ±,2k
z1



∫ z 0


K (ζ)



w±,2k−1 (ζ; ζ1 ) dζ.
 w±,2k (z; z1 ) =
z1 K(ζ)
The phase function (3.3) is a solution to an eikonal equation and has singularities of
turning points at branch points of the integrand. On the other hand, the symbol function
(3.5) are determined by the integral recurrent relations. Hence we see that the elements
of the function w± (z; z1 ) converge absolutely and uniformly in a neighborhood of z = z1
(see [FLN], [W]). Such solutions (3.2) constructed by the way above are exact solutions
to (3.1).
The exact WKB solutions (3.2) are holomorphic in a neighborhood of t = t1 , and
extended analytically to Ω because (3.2) satisfy the equation (3.1) with holomorphic
coefficients in Ω. We call t0 the base point of the phase and t1 the base point of the
symbol. We remark that the pair of exact WKB solutions φ+ (t, h; t0 , t1 ), φ− (t, h; t0 , t1 )
are linearly independent.
We state some properties of the exact WKB solutions. The Wronskian between
(
)
e
e
e
two exact WKB solutions [φ(t), φ(t)]
= det φ(t) φ(t)
is given by w+
:
Proposition 3.1.
Any exact WKB solutions φ+ (t; t0 , t1 ) and φ− (t; t0 , t2 ) with
the same base point t0 of the phase satisfy the Wronskian formula:
e
[φ+ (t; t0 , t1 ), φ− (t; t0 , t2 )] = 2i w+
(z(t2 ); z(t1 )).
We note that the Wronskian is independent of the variable t because the matrix of
right side of (3.1) is trace-free.
The convergent series (3.5) of the function w± (z(t), h; z(t1 )) constructed by (3.6)
are also asymptotic expansions on h in the domains Ω± ⊂ Ω1 in which there exists a
curve from t1 to t along which ± Re z(t) increases strictly.
81
Adiabatic Transition Probability
Proposition 3.2.
There exist a positive integer N and a positive constant h0
such that, for all h ∈ (0, h0 ) we have
e
w±
(z(t), h; z(t1 ))
(3.7)
−
N
−1
∑
w±,2k (z(t), h; z(t1 )) = O(hN ),
k=0
(3.8)
o
w±
(z(t), h; z(t1 )) −
N
−1
∑
w±,2k−1 (z(t), h; z(t1 )) = O(hN ),
k=0
uniformly in Ω± .
We introduce the so-called Stokes line, which characterizes the asymptotic behavior
of the exact WKB solution as h tends to 0.
Definition 3.3.
The Stokes lines emanating from t = t0 in Ω are defined as the
set:
∫ t√
{
}
t ∈ Ω; Re
α(τ )β(τ ) dτ = 0 .
t0
A Stokes line is a level set of the real part of the WKB phase function z(t; t0 ).
If Re z(t) strictly increases along an oriented curve, such a curve is called a canonical
curve. In fact a canonical curve is transversal to Stokes lines. We can characterize the
asymptotic behavior of the Wronskian between the linearly independent exact WKB
solutions in terms of canonical curve.
Proposition 3.4.
If there exists a canonical curve from t1 to t2 ,
[φ+ (t, h; t0 , t1 ), φ− (t, h; t0 , t2 )] = 2i (1 + O(h))
§ 3.2.
as
h → 0.
Action Integral
In this subsection we give the asymptotic behaviors of the action integrals (2.1).
These are important to study the decay rate of P (ε, h) and the geometrical structures
of the Stokes lines.
We see that there exist the relations between the action integral defined in §2
and the phase function (3.3): Aj (ε) = iz(xj (ε); x)/2, Bj (ε) = iz(yj (ε); y)/2. Put
V (n) (x)
V (t) =
(t − x)n vx (t − x), where vx (t) are holomorphic near t = 0 and vx (0) = 1
n!
V (m) (y)
(resp. V (t) =
(t − y)m vy (t − y)) respectively.
m!
Lemma 3.5.
expansion:
(3.9)
Aj (ε) =
Aj (ε) is an analytic function of ε1/n at t = x and has the following
∞
∑
q=1
Xq exp
[ (2j − 1)qπi ]
ε(n+q)/n
2n
(j = 1, . . . , n),
82
Takuya Watanabe
(
)q/n [ q−1
]
π Γ(q/(2n))
n!
d
−q/n
where Xq =
(vx (z)
)
.
(n + q) Γ(q) Γ((n + q)/(2n)) |V (n) (x)|
dz q−1
z=0
√
Remark.
expansion.
Bj (ε) is also an analytic function of ε1/m at t = y and has a similar
To consider the scattering matrix, we define the action integrals R∞ (ε) and R−∞ (ε),
which have ±∞ as the end points of the integration paths, by
∫
R∞ (ε) = 2
∞
x
∫
R−∞ (ε) = 2
√
( V (t)2 + ε2 − λr ) dt,
−∞
√
( V (t)2 + ε2 − λl ) dt,
y
√
√
where λr = Er2 + ε2 and λl = El2 + ε2 . Note that R∞ (ε) and R−∞ (ε) are realvalued as well as R(ε).
§ 4.
Outline of Proofs
The proof of each theorem is reduced to studying the asymptotic expansions of the
elements of the scattering matrix as h → 0 by means of the exact WKB method in
§ 3.1. In short the scattering matrix can be expressed by the products of the change of
bases (transfer matrix ) between the exact WKB solutions which has a valid asymptotic
expansion on h in a complex domain separated by Stokes lines near the real axis. We
first study the geometrical structures of Stokes lines in some cases and define such exact
WKB solutions. In § 4.2 we give the expression of the scattering matrix with the transfer
matrix near each zero of V (t).
§ 4.1.
Stokes Geometry
In this paper we call Stokes geometry the geometrical structures of the Stokes lines
emanating from turning points. We remark that the Stokes geometry near the real axis
differs according to the orders of zeros of V (t). See Figures 1, 2, and 3 for different
Stokes geometries.
Let r, r̄, l, and ¯l be four base points of the symbol as in each figure. We make
the branch cuts dashed lines as in each figure. Recalling that we take the branch of
√
V (t)2 + ε2 which is ε at t = x, we see that around the real axis Re z(t) is increases as
Im t decreases and Im z(t) increases as Re t increases.
83
Adiabatic Transition Probability
For any n, m ∈ N we consider four exact WKB solutions:
[ z(t; x ) ]
1
φ+ (t; x1 , r) = exp +
M+ (z(t))w+ (z(t); z(r)),
h
[ z(t; x ) ]
1
M− (z(t))w− (z(t); z(r)),
φ− (t; x1 , r) = exp −
h
(4.1)
[ z(t; y ) ]
m
φ+ (t; ym , l) = exp +
M+ (z(t))w+ (z(t); z(l)),
h
[ z(t; y ) ]
m
φ− (t; y m , l) = exp −
M− (z(t))w− (z(t); z(l)).
h
Note that each exact WKB solution has a valid asymptotic expansion on h in the
direction toward its phase base point from its symbol base point.
Im t
l
y1
δc
x1
r
y
x
¯l
Re t
r̄
ȳ1
δ̄c
x̄1
Figure 1. n = m = 1, Im A1 (ε) < Im B1 (ε)
Let δc and δ c as in Figures 1, 2 and 3 be the intermediate symbol base points in S
satisfying
(4.2)
max{|Re z(xn )|, |Re z(y1 )|} < |Re z(δc )| < min{|Re z(xn−1 )|, |Re z(y2 )|}
for sufficiently small ε. In fact we note, from Lemma 3.5, that the inequality Re z(xn ) >
Re z(y1 ) holds for sufficiently small ε if n > m. Then we consider the intermediate exact
WKB solutions:
[ z(t; x ) ]
n
φ+ (t; xn , δc ) = exp +
M+ (z(t))w+ (z(t); z(δc )),
h
[ z(t; x ) ]
n
φ− (t; xn , δ c ) = exp −
M− (z(t))w− (z(t); z(δ c )),
h
[ z(t; y ) ]
1
φ+ (t; y1 , δc ) = exp +
M+ (z(t))w+ (z(t); z(δc )),
h
84
Takuya Watanabe
[ z(t; y ) ]
1
φ− (t; y 1 , δ c ) = exp −
M− (z(t))w− (z(t); z(δ c )),
h
whose asymptotic expansions in h are valid in the direction toward four turning points
from the symbol base points δc , δ c .
Im t
δy
l
y1
ym
δc
x1
r
y
x
¯l
ȳm
r̄
ȳ1
δ̄c
δ̄y
Re t
x̄1
Figure 2. n = 1, m ≥ 2, Im B1 (ε) > Im Bm (ε)
§ 4.2.
Scattering Matrix
We first give the relations between the Jost solutions and the exact WKB solutions
(4.1) (see [Ra], [FR], [W]).
Lemma 4.1.
The following relations hold:
[ z r (x ) ]
1
= QC1 (ε, h) exp +
φ+ (t; x1 , r),
h
[ z r (x ) ]
1
r
φ− (t; x1 , r),
ψ−
(t) = −iQC2 (ε, h) exp −
h
[ z l (y ) ]
m
l
ψ+
(t) = QC3 (ε, h) exp +
φ+ (t; ym , l),
h
[ z l (y ) ]
m
l
ψ− (t) = −iQC4 (ε, h) exp −
φ− (t; y m , l),
h
)
r
ψ+
(t)
1
where Q =
2
(
1 i
, and
i 1
∫
t
r
z (t) = i
∞
√
( V (τ )2 + ε2 − λr ) dτ + iλr t,
(
λr =
√
Er2 + ε2
)
85
Adiabatic Transition Probability
∫
t
l
z (t) = i
−∞
√
(
√
( V (τ )2 + ε2 − λl ) dτ + iλl t,
λl =
)
El2 + ε2 .
The coefficients Ck (ε, h) are some constants depending only on ε and h, and Ck (ε, h) =
1 + O(h) as h tends to 0 uniformly with respect to small ε.
We will denote such constants simply by 1 + O(h).
Im t
δy
l
δc
y1
ym
δx
x1
xn
r
y
x
¯l
ȳm
δ̄y
x̄n
ȳ1
δ̄c
Re t
x̄1
δ̄x
r̄
Figure 3. m ≥ n ≥ 2, Im A1 (ε) > Im An (ε), Im B1 (ε) > Im Bm (ε)
We next define the transfer matrices Tr (ε, h) and Tl (ε, h) by
r
r
(ψ+
(t) ψ−
(t)) = (φ+ (t; x1 , r) φ− (t; x1 , r))Tr (ε, h),
l
l
(t)) = (φ+ (t; ym , l) φ− (t; y m , l))Tl (ε, h),
(t) ψ−
(ψ+
and the transfer matrices Tx (ε, h), Ty (ε, h) around t = x, y and Tc (ε, h):
(φ+ (t; xn , δc ) φ− (t; xn , δ c )) = (φ+ (t; x1 , δr ) φ− (t; x1 , r))Tx (ε, h),
(φ+ (t; ym , l) φ− (t; y m , l)) = (φ+ (t; y1 , δc ) φ− (t; y 1 , δ c ))Ty (ε, h),
(φ+ (t; y1 , δc ) φ− (t; y 1 , δ c )) = (φ+ (t; xn , δc ) φ− (t; xn , δ c ))Tc (ε, h).
We see that the transfer matrices Tr (ε, h), Tl (ε, h), and Tc (ε, h) are diagonal matrices
given by


[ i
]
0
exp 2h (A1 − R∞ + 2λr x)

[ i
] (1 + O(h)),
(4.3) Tr = 
(R − A1 − 2λr x)
0
exp
2h ∞
86
Takuya Watanabe


[ i
]
0
exp 2h (Bm − R−∞ + 2λl y)

] (1 + O(h)),
[ i
(4.4) Tl = 
(R
− B m − 2λl y)
0
exp
2h −∞
where O(h) is uniform with respect to small ε, and


[ i
]
0
exp 2h (An − B1 + R)

[ i
] .
(4.5) Tc = 
0
exp − (An − B 1 + R)
2h
Then the asymptotic formula of S(ε, h) as h tends to 0 is given by
Proposition 4.2.
The scattering matrix S(ε, h) is the product of the 2 × 2 matrices Tr (ε, h), Tl (ε, h), Tx (ε, h), Ty (ε, h), and Tc (ε, h):
(4.6)
S(ε, h) = Tr−1 (ε, h) Tx (ε, h) Tc (ε, h) Ty (ε, h) Tl (ε, h).
We finally state the asymptotic formulae of the transfer matrices Tx (ε, h) and
Ty (ε, h). Put




ξ11 (ε, h) ξ12 (ε, h)
η11 (ε, h) η12 (ε, h)
,
.
Ty (ε, h) = 
Tx (ε, h) = 
ξ21 (ε, h) ξ22 (ε, h)
η21 (ε, h) η22 (ε, h)
Proposition 4.3.
If n = 1, then
Tx (ε, h) satisfies the following asymptotic formulae:
(h)
ξ11 (ε, h) = 1 + O 2
ε
]
[ 1
ξ12 (ε, h) = i exp − Im A1 (ε) (1 + O(h))
h
[ 1
]
ξ21 (ε, h) = i exp − Im A1 (ε) (1 + O(h))
h
(h)
ξ22 (ε, h) = 1 + O 2
ε
as
h
→ 0,
ε2
as
h → 0,
as
h → 0,
as
h
→ 0.
ε2
If n ≥ 2, then
(
[ i
])
]
[ i
ξ11 (ε, h) = exp
(A1 − An ) + (−1)n exp
(A1 − 2A1 + An )
2h
2h
(
(
))
h
h
× 1 + O (n+1)/n
as
→ 0,
ε
ε(n+1)/n
(
[ i
]
[ i
])
n+1
(A − An ) + exp
(A − 2A1 + An )
ξ12 (ε, h) = i (−1)
exp
2h 1
2h 1
(
(
))
h
h
× 1 + O (n+1)/n
as
→ 0,
(n+1)/n
ε
ε
87
Adiabatic Transition Probability
(
[ i
]
[ i
])
ξ21 (ε, h) = i (−1)n+1 exp
(An − A1 ) + exp
(2A1 − A1 − An )
2h
2h
(
(
))
h
h
× 1 + O (n+1)/n
as
→ 0,
(n+1)/n
ε
ε
]
[ i
])
(
[ i
(An − A1 ) + (−1)n exp
(2A1 − A1 − An )
ξ22 (ε, h) = exp
2h
2h
(
(
))
h
h
× 1 + O (n+1)/n
as
→ 0.
ε
ε(n+1)/n
This proposition can be proved by means of the idea in [W]. The elements of Tx (ε, h)
are expressed by Wronskians between the exact WKB solutions defined in § 4.1. We see
that there exist a canonical curve for each Wronskian in case n = 1. In the case where
n ≥ 2, taking account of intermediate base points of the symbol δx and δ x , we also
make sure of the existence of canonical curves. Therefore we obtain this proposition by
Proposition 3.4. Similarly we get
Proposition 4.4.
If m = 1, then
Ty (ε, h) satisfies the following asymptotic formulae:
η11 (ε, h) = 1 + O
(h)
ε2
]
[ 1
η12 (ε, h) = (−1)n i exp − Im B1 (ε) (1 + O(h))
h
]
[ 1
η21 (ε, h) = (−1)n i exp − Im B1 (ε) (1 + O(h))
h
(h)
η22 (ε, h) = 1 + O 2
ε
as
h
→ 0,
ε2
as
h → 0,
as
h → 0,
as
h
→ 0.
ε2
If m ≥ 2, then
])
(
[ i
]
[ i
η11 (ε, h) = exp
(B1 − Bm ) + (−1)m exp
(B1 − 2B 1 + Bm )
2h
2h
(
(
))
h
h
× 1 + O (m+1)/m
as
→ 0,
ε
ε(m+1)/m
]
[ i
])
(
[ i
n
m+1
(B − B m ) + exp
(B − 2B 1 + B m )
η12 (ε, h) = (−1) i (−1)
exp
2h 1
2h 1
(
(
))
h
h
× 1 + O (m+1)/m
as
→ 0,
(m+1)/m
ε
ε
]
[ i
])
(
[ i
(Bm − B 1 ) + exp
(2B1 − B 1 − Bm )
η21 (ε, h) = (−1)n i (−1)m+1 exp
2h
2h
(
(
))
h
h
× 1 + O (m+1)/m
as
→ 0,
(m+1)/m
ε
ε
]
[ i
])
(
[ i
(B m − B 1 ) + (−1)m exp
(2B1 − B 1 − B m )
η22 (ε, h) = exp
2h
2h
(
(
))
h
h
× 1 + O (m+1)/m
as
→ 0.
(m+1)/m
ε
ε
88
Takuya Watanabe
From (4.3), (4.4), (4.5), Propositions 4.2, 4.3, and 4.4, we obtain the asymptotic
expansions of the scattering matrix.
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