An approximate series solution of the semiclassical Wigner equation K.S. Giannopoulou ∗†& G.N. Makrakis
‡
May 22, 2017
arXiv:1705.06754v1 [math-ph] 18 May 2017
Abstract We propose a new approximate series solution of the semiclassical Wigner equation by uniformization of WKB approximations of the Schr¨ odinger eigenfunctions. Keywords Schr¨ odinger equation, Wigner equation, semiclassical limit, geometric optics, caustics, Weyl quantization, Weyl symbols, uniform stationary phase method AMS (MOS) subject classification: 78A05, 81Q20, 53D55, 81S30, 34E05, 58K55
Contents 1 Introduction
2
2 Moyal star product and the Wigner equation
5
3 Eigenfunction-series solution of the 3.1 Eigenvalues of L and M . . . . . 3.2 Eigenfunctions of L and M . . . 3.3 The eigenfunction series expansion
Wigner equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 7 8 8
4 Airy approximation of the Wigner eigenfunctions 4.1 The WKB-Wigner eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Airy approximation of W`,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Airy approximation of W`,n,m . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 10 11 14
5 Approximate series expansion of the Wigner function 18 5.1 Approximation of the expansion coefficients . . . . . . . . . . . . . . . . . . . 19 5.2 Approximation of the energy density . . . . . . . . . . . . . . . . . . . . . . . 21 A QM in configuration space: Schr¨ odinger equation 22 A.1 Eigenfunction series expansion of the wave function . . . . . . . . . . . . . . . 22 A.2 WKB asymptotic expansion of eigenfunctions . . . . . . . . . . . . . . . . . . 23 B Stationary phase formulae 25 B.1 The case of a simple stationary point . . . . . . . . . . . . . . . . . . . . . . . 25 B.2 The case of two coalescing stationary points (uniform formula) . . . . . . . . 25 C Berry’s semiclassical Wigner function
26
∗ Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway. Email:
[email protected] † Acknowledgement This work has been partially supported by the “Maria Michail Manasaki” Bequest Fellowship & the ERCIM “Alain Bensoussan Fellowship Programme” ‡ Department of Mathematics & Applied Mathematics, University of Crete, Heraklion, Crete, Greece & Institute of Applied and Computational Mathematics, FORTH, Heraklion, Crete, Greece. Email:
[email protected]
1
1
Introduction
We consider the Cauchy problem for the 1-d Schr¨odinger equation with oscillatory initial data 2 iut (x, t) = − uxx (x, t) + V (x)u (x, t) , x ∈ Rx , t ∈ [0, T ) , (1.1) 2 u (x, t = 0) = u0 (x) = A0 (x) exp iS0 (x)/ , (1.2) where the potential V (x) ∈ C ∞ (Rx ) is real valued, T is some positive constant, A0 (x) ∈ C0∞ (Rx ), S0 (x) ∈ C ∞ (Rx ), and is a semiclassical parameter (0 < 0 when (x, p) lies in the upper meniscus Σ+ n , and p < 0 when (x, p) lies in the lower meniscus Σ− . There are not exist any stationary points in the interior of Λ∗n . Obviously the n stationary points (4.26) coalesce to σ(x, p) = 0 when (x, p) approaches Λn . Therefore, in this case we can use Berry’s semiclassical formula (C.11) in Appendix C. By inspection√of Berry’s chord construction we identify the uniformity parameter α in (C.8) by α = p − 2En − x2 √ 2 when (x, p) lies in Σ− . when (x, p) lies in Σ+ , and α = p + 2E − x n n n Then, by (C.11) we get the approximation formula ! p 2−2/3 −1/3 −1 −2/3 p − 2En − x2 W1,n (x, p) ≈ π (2En ) Ai , 1/3 −1/2 (2En ) (2En − x2 ) which, is further approximated by W1,n (x, p)
≈ π
−1 −2/3
−1/3
(2En )
Ai
p2 + x2 − 2En 1/3
2/3 (2En )
!
,
(4.27)
√ near the upper branch p = 2En − x2 of Λn , according to the uniformization procedure which we introduced in [22]. Similarly we get the approximation ! p2 + x2 − 2En −1/3 −1 −2/3 W2,n (x, p) ≈ π (2En ) Ai , (4.28) 1/3 2/3 (2En ) 12
√ near the lower branch p = − 2En −√x2 of Λn . Away from the branches p = ± 2En − x2 of Λn , and inside Σ± n , the approximations (4.27), (4.28) are still valid, and, as it should be, they coincide with the standard stationary phase approximations of W`,n . In the interior of Λ∗n , W`,n , ` = 1 , 2, are asymptotically small to any order of . We must note that the formulae (4.27), (4.28) are formally the same, but they are valid in different regions, since they have been derived by applying Berry’s semiclassical formula (C.11) near two different branches of the Lagrangian curve Λn .Thus, for later use and notational convenience, we introduce the function ! p2 + x2 − 2En −1/3 −1 −2/3 g . (4.29) Wn (x, p) := π (2En ) Ai 1/3 2/3 (2En )
Outside Λn , where the stationary points of F`,n , ` = 1, 2 are imaginary (a pair) and again coalesce to a double point on Λn , we formally apply the uniform stationary phase method (see eq. (B.3) in Appendix B). Then, we get the approximation (4.29). This procedure somehow corresponds to the analytic continuation of the asymptotic formulae in the complex space. This is not rigorous since the Airy function becomes exponentially small and the algebraic remainders cease to carry any approximation information. Construction of the asymptotics of W`,n , ` = 3, 4
The stationary points of the Wigner phases F`,n , ` = 3, 4, are simple, when (x, p) lies in the interior of Λ∗n , and there not any stationary points in Σ± n . Therefore, in this case we use the standard stationary phase formula (B.2), and we derive the following approximations √ • For 0 6 x < 2En and p > 0 , W3,n (x, p) ≈ W4,n (x, p) ≈
• For 0 6 x
0 , 1
W3,n (x, p) ≈
2π 3/2
W4,n (x, p) ≈
2π 3/2
1
i
√ e F3,n (+σ0 ) ei3π/4 (p2 + x2 )−1/4 (2En − p2 − x2 )−1/4
,
i
√ e− F3,n (+σ0 ) e−i3π/4 (p2 + x2 )−1/4 (2En − p2 − x2 )−1/4
√ • For − 2En < x 6 0 and p < 0 , W3,n (x, p) ≈ W4,n (x, p) ≈
i 1 √ e F3,n (−σ0 ) ei3π/4 (p2 + x2 )−1/4 (2En − p2 − x2 )−1/4 , 2π 3/2 i 1 √ e− F3,n (−σ0 ) e−i3π/4 (p2 + x2 )−1/4 (2En − p2 − x2 )−1/4 3/2 2π
where F3,n (±σ0 ) = F3,n (σ = ±σ0 , x, p) with σ0 (x, p) := p
|p|
p2 + x2
p
13
2En − p2 − x2 .
We observe that the approximation of W4,n (x, p) is the complex conjugate of the approxima tion of W3,n (x, p) . We must also note that in Σ± n , W`,n (x, p) , ` = 3 , 4, are asymptotically small to any order of . By rather complicated transformations we can relate the phases F3,n (±σ0 ) to the phase of the Airy approximation (4.29) of W`,n , ` = 1, 2. These transformations have been motivated by the asymptotics of Laguerre polynomials (see e.g. [44]), which are the exact Wigner eigenfunctions for the harmonic oscillator. Then, by standard asymptotics of the Airy function, we can see that in the interior of Λ∗n the sum (W3,n + W4,n ) matches with the g . asymptotics of (4.29), and therefore we get the approximation (W3,n + W4,n )≈W n Furthermore, by applying the so called complex stationary phase formula [41], (Ch. X, Sec. 3) we show that W3,n (x, p) = 0 ,
W4,n (x, p) = 0
out of Λn , since the imaginary stationary points contribute opposite terms in the amplitudes due to phase jumps. These results conform with the fact that, the approximation (4.30) is exponentially small in the exterior of Λn , by the asymptotics of the Airy function. The uniform approximation of Wn (x, p) In Table 1 we summarize the contribution of different regions in the strip {|x| < g . R} to Wn . The main contribution comes from the leading Airy term W n region √ p > 2En − x2 √ p ≈ 2En − x2 inside Λ∗n √ p ≈ − 2En − x2 √ p > − 2En − x2
p
2En , p ∈
main contribution to Wn (x, p) g W1,n ≈W n g W1,n ≈ Wn g W3,n + W4,n ≈W n g W2,n ≈ W n g W ≈W n 2,n
Table 1: The main contribution to Wn g Recall that Λ∗n is the dual curve of Λn (see eq. (4.23) and Figure 1), and W n given by (4.29). Thus, by the uniformization procedure [22], the Airy approximation of Wn is given by
Wn (x, p)
≈
g W n (x, p)
−1/3
:= π −1 −2/3 (2En )
Ai
p2 + x2 − 2En 1/3
2/3 (2En )
!
.
(4.30)
for any (x, p) in the strip.
4.3
Airy approximation of W`,n,m
The construction of the asymptotics of off-diagonal Wigner integrals W`,nm , ` = 1, 2 (eq. (4.13)), is much more complicated than that of the diagonal terms, because the double stationary points of F`,nm (σ; x, p) ` = 1, 2, do not coalesce to zero on certain Lagrangian manifolds (contrary to what happens when n = m). For this reason we cannot use Berry’s semiclassical Wigner function, but we must work with uniform stationary phase approximation (B.3). Without loss of generality, suppose that n > m and then En > Em . In the construction of the approximation we assume that n, m are large and small, so that n, m = constant, and n − m = constant > 0. Then, the eigenvalues En = (n + 1/2) ≈ n =: En and Em = (m + 1/2) ≈ m =: Em can be treated as constants.
14
Construction of the asymptotics of W`,n,m , ` = 1, 2
By the stationarity condition ∂σ F`,nm (σ, x, p) = 0 we find that the stationary points are given by σ1,2 (x, p) = where
|p| p 2 xenm ± (p + x2 )(2Enm − p2 − x2 ) − e2nm , p2 + x2 p2 + x2
Enm :=
1 (En + Em ) 2
and enm :=
1 (En − Em ) . 2
(4.31)
(4.32)
It turns out that these points are real when (x, p) lies in the meniscus − Σnm = Σ+ nm ∪ Σnm . 2 This meniscus is defined as the intersection of the ring ρ2nm ≤ x2 + p2 ≤ Rnm lying between the Lagrangian curves
Λ1,nm Λ2,nm
=
2 {(x, p) ∈ R2 : p2 + x2 = Rnm } 2
2
2
= {(x, p) ∈ R : p + x =
ρ2nm }
,
(4.33)
,
(4.34)
where Rnm
:=
ρnm
:=
p 1 p ( 2En + 2Em ) , 2 p 1 p ( 2En − 2Em ) 2
(4.35) (4.36)
and the exterior of their dual curve Λ∗nm ,
Λ∗nm := Λ∗1,nm ∪ Λ∗2,nm ,
(4.37)
where • if σ = σ1,2 ≥ 0 (see Figure 2) Λ∗1,nm := {(x, p) ∈ R2 : p2 + (x −
p
En /2) = Em /2} .
(4.38)
Λ∗2,nm := {(x, p) ∈ R2 : p2 + (x +
p 2 Em /2) = En /2} ,
(4.39)
Λ∗1,nm := {(x, p) ∈ R2 : p2 + (x −
p
Em /2) = En /2} ,
(4.40)
Λ∗2,nm := {(x, p) ∈ R2 : p2 + (x +
p
En /2) = Em /2} .
and
• if σ = σ1,2 ≤ 0 (see Figure 3)
and
15
2
2
2
(4.41)
p
Λ1,nm
Λn : p2 + x2 = 2En
Λ2,nm
p − Em /2
p
En /2
x
Λ∗1,nm Λ∗2,nm
Figure 2: Area of existence of stationary points of F1,nm (σ = σ1,2 > 0) p
Λ1,nm
Λn : p2 + x2 = 2En
Λ2,nm
p − En /2
p
Em /2
x
Λ∗2,nm Λ∗1,nm
Figure 3: Area of existence of stationary points of F1,nm (σ = σ1,2 < 0) Also, it turns out that there are no real stationary point for (x, p) lying in the interior of Λ2,nm . The stationary points (4.31) coalesce to the double point xenm σ ¯0 (x, p) := 2 , (4.42) p + x2 when (x, p) ∈ Λ1,nm ∪ Λ2,nm . Thus, the curves Λ1,nm , Λ2,nm may be considered as the analogue of Λn (eq. (4.22)),which arouse in the diagonal case, since, formally, ρnm = 0 and Rnm = Rn := 2En for n = m. Then, by applying the uniform stationary phase formula (B.3) to W1,nm , we derive the p 2 following approximation, for small and (x, p) near the branch p = Rnm − x2 > 0, 2 2 2 p + x − Rnm −4/3 2 , W1,nm (x, p) ≈ π −1 e−i(n−m)φ −2/3 Rnm (Rnm − ρ2nm )1/3 Ai −1/3 4/3 2 2/3 Rnm (Rnm − ρ2nm )
where the angle φ is defined by φ ≡ φ(x, p) := arctan(p/x), and Rnm , rnm given by (4.35) and (4.36), respectively. In a similar way, for the second integral W2,nm , we derive the approximation 2 2 2 p + x − Rnm −4/3 2 , W2,nm (x, p) ≈ π −1 e−i(n−m)φ −2/3 Rnm (Rnm − ρ2nm )1/3 Ai −1/3 4/3 2 2/3 Rnm (Rnm − ρ2nm ) 16
p 2 for small and near the branch p = − Rnm − x2 < 0 . It must be emphasized that although the above two formulas are formally the same, they are valid for different values of p. Thus we are lead to define 2 2 2 p + x − Rnm −1 −i(n−m)φ −2/3 −4/3 2 ] . W e Rnm (Rnm − ρ2nm )1/3 Ai nm (x, p) := π −1/3 4/3 2 2/3 Rnm (Rnm − ρ2nm ) (4.43) Construction of the asymptotics of W`,n,m ,` = 3 ,4
The phases F3,nm (σ, x, p) and F4,nm (σ, x, p) (eqs. (4.18) and (4.19), respectively) have simple real stationary points in the shaded region of Figures 4 , 5, and a pair of complex stationary points out of Λn . p
Λ1,nm
Λn : p2 + x2 = 2En
Λ2,nm
p − Em /2
p
x
En /2
Λ∗1,nm Λ∗2,nm
Figure 4: Area of existence of stationary points of F`,nm , ` = 3, 4 (σ = σ1,2 > 0) By standard stationary phase method, and using involved transformations which are quite analogous to those applied to the approximation of W`,n , ` = 3, 4, we obtain that W`,nm , ` = 3, 4, are approximated by the Airy approximation (eq. (4.43)) . The uniform approximation of Wnm
for any fixed (x, p) in the strip In Table 2 we present the main contribution to Wnm p {|x| < 2En , p ∈ R}.
region p 2 p > Rnm − x2 p 2 p ≈ Rnm − x2 ∗ inside Λnm p 2 p ≈ − Rnm − x2 p 2 p > − Rnm − x2
main contribution to Wnm (x, p) ] W1,nm ≈ Wnm ] W1,nm ≈W nm ] W3,nm + W4,nm ≈ W nm ] W2,nm ≈W nm ] W2,nm ≈ W nm
Table 2: The main contribution to Wnm (x, p) Therefore, the leading approximation of Wnm (x, p) is Wnm (x, p)
≈
] W nm (x, p)
−4/3 2 := π −1 e−i(n−m)φ −2/3 Rnm (Rnm − ρ2nm )1/3 Ai
17
2
2
p +x −
4/3 2 2/3 Rnm (Rnm
2 Rnm −1/3
− ρ2nm )
,
(4.44)
p
Λ1,nm
Λn : p2 + x2 = 2En
Λ2,nm
p − En /2
p
Em /2
x
Λ∗1,nm Λ∗2,nm
Figure 5: Area of existence of stationary points of F`,nm , ` = 3, 4 (σ = σ1,2 < 0) for any (x, p) in the interior of the strip {|x|
0 by their definition and the construction of the approximation. On the other hand, ηincoh (x, t) oscillates and changes i sign as time evolves. Since as the time t increases, the exponential terms e− (En −Em )t tend weakly to zero, we expect that the incoherent part is weakly negligible for large time. For the incoherent component of the density we have ηincoh (x, t) ≈ ∞ ∞ X X
n=0 m=0 ,m6=n
×
Z
∞
e
i
2 1/3 (0) e− (En −Em )t π −1 −2/3 R−4/3 (R2 cg nm nm − ρnm ) nm
−Em ) arctan(p/x) − i (En
−∞
Ai
2 x2 + p2 − Rnm
4/3 2 2/3 Rnm (Rnm
−
−1/3 ρ2nm )
dp .
(5.23)
The integrals in (5.23) cannot calculated analytically at a general space-time point (x , t). Nevertheless, it can be shown, by using the Riemann-Lebesgue lemma and the exponential decay of the Airy function for large positive argument, that they are convergent. At the special position x = 0, the exponential term disappears. Then we can calculate the integrals by using the formula (5.10), and we get
ηincoh (x = 0, t) ≈
×2
2/3 −1/3
∞ X
∞ X
i
n=0 m=0 ,m6=n
−2/3 2 Rnm (Rnm
−
(0) e− (En −Em )t cg nm
ρ2nm )1/6 Ai2
2 Rnm
4/3
2 − ρ2nm )−1/3 2/3 Rnm (Rnm
!
.
(5.24)
At this point we can make the following important remark. In geometrical optics, focal points and caustic formation appear along space-time curves. Therefore, from the fact that the coherent part of the amplitude is time-independent, we expect that generation of effects is associated with the incoherent part. This implies, in turn, that possible amplification of the wave intensity, as diminishes, should be encoded in the off-diagonal coefficients. The detailed investigation of the formation of the singularities as the time evolves, is an open important problem which requires the calculation or the approximation of the integrals in (5.23). A preparatory step would be the systematic study of the trigonometric series (5.24) which is still very complicated due to the Airy functions appearing in its coefficients.
Appendices A A.1
QM in configuration space: Schr¨ odinger equation Eigenfunction series expansion of the wave function
We assume that the potential V (x) ∈ C ∞ (Rx ) is positive, real valued, and lim|x|→∞ V (x) = ∞. For each fixed ∈ (0, 0 ), with arbitrary 0 > 0, the spectral problem 2 2 b v (x) = − d + V (x) v (x) = E v (x) . H (A.1) 2 dx2 22
has purely discrete spectrum with eigenvalues arranged in an increasing sequence, 0 < E0 < E1 ≤ . . . ≤ En ≤ . . . ,
lim En = +∞
n→∞
.
(A.2)
The corresponding eigenfunctions vn (x) ∈ L2 (Rx ) form an orthonormal set with respect to the L2 − inner product (see e.g. [5, 27]). Then, by separation of variables (Fourier method), the solution u (x, t) of the Cauchy problem 2 (A.3) i∂t u (x, t) = − ∂xx + V (x) u (x, t) , x ∈ Rx , t ∈ [0, T ) , 2 u (x, t = 0)
i
= u0 (x) = A0 (x) e S0 (x) .
(A.4)
with initial data A0 (x) ∈ C0∞ (Rx ) and S0 (x) ∈ C ∞ (Rx ), and for some positive constant T < ∞, is given by the eigenfunction series u (x, t) =
∞ X
i
cn vn (x)e− En t ,
(A.5)
n=0
where coefficients cn are the L2 −projections of the initial data (A.4) onto the eigenfunctions cn (0) = (u0 , vn )L2 (R)
for all n = 0, 1, 2, . . . .
(A.6)
The solution u (x, t) belongs to L2 (Rx ) and it conserves the quantum energy, that is ku (x, t)kL2 (Rx ) = ku0 (x)kL2 (Rx ) , for any fixed t ∈ [0, T ). Although the eigenfunction expansion of the wavefunction is an exact solution of the Cauchy problem (A.3)-(A.4), it is, in general, very slowly convergent for small values of the semiclassical parameter . In such cases asymptotic approximations of the eigenvalues and eigenfunctions are constructed by the WKB method, but these approximate eigenfunctions have singularities on the turning points where their amplitude blows up.
A.2
WKB asymptotic expansion of eigenfunctions
According to the WKB method, the eigenvalues En are approximated for small , by the Bohr-Sommerfeld quantization rule (see, e.g., Fedoriuk [18],Yosida [45]). Assuming that the potential has the shape of a single well, and x1 , x2 are the turning points satisfying V (x1 (En )) = V (x1 (En )) = En , with x1 (En ) < x2 (En ), this rule implies that the eigenvalues En are approximate solutions of the equation f (En )
≡
Z
x2 (En )
) x1 (En
p 1 2(En − V (x)) dx ≈ π n + , 2
(A.7)
for large n, so that n being constant. The asymptotic approximations of the corresponding eigenfunctions, are given by • x > x2 (En ) vn (x)
≈
ψn (x)
= [2(V (x) −
−1/4 En )]
1 exp −
Z
x
) x2 (En
! p 2(V (t) − En ) dt
, (A.8)
• x1 (En ) < x < x2 (En ) vn (x)
≈
ψn (x)
=
[2(En
−1/4
− V (x))]
cos
1
• x < x1 (En ) 23
Z
x
) x2 (En
p
π 2(En − V (t)) dt + 4
!
, (A.9)
V (x)
E
x1
x2
x
Figure 6: Single well potential
vn (x)
≈
ψn (x)
n
= (−1) [2(V (x) −
−1/4 En )]
1
exp
Z
x
) x2 (En
! p 2(V (t) − En ) dt
. (A.10)
Clearly, these approximations break down at the turning points, where V (x) = En , since the amplitudes [2(V (x) − En )]−1/4 diverge at these points. In the classically allowed region x1 (En ) < x < x2 (En ) the eigenfunction is rapidly oscillatory and in the classically forbidden regions x > x2 (En ) and x < x1 (En ) is exponentially decaying. In the case of the harmonic oscillator V (x) = x2 /2, the integral in (A.7) is computed analytically, and we get the approximation 1 En ≈ n + . (A.11) 2 In the oscillatory region En > x2 /2, the approximation (A.9) is written in the form ! 1/2 1/2 Z p π 2 2 1 x 2 −1/4 2 2En − t dt + . vn (x) ≈ ψn (x) = 2En − x cos π π √2En 4 (A.12) In order to emphasize the two-phase structure of the WKB approximations near the turning points, we rewrite (A.12) in terms of complex exponentials vn (x)
1/2 i + i − 2 ≈ ψn (x) = An + (x) e Sn (x) + An − (x) e Sn (x) . π
(A.13)
where the amplitudes An ± and the phases Sn ± are given by the formulae 1/2 −1/4 ±iπ/4 2 2En − x2 e , π Z x p + − 2En − t2 dt . Sn (x) = −Sn (x) = √
An ± (x) :=
1 2
(A.14) (A.15)
2En
p The amplitudes A± n (x) diverge at the turning points x = ± 2En (caustics) and the expo±iπ/4 nentials e take care of the phase shift there. The approximation (A.11) of the eigenvalues coincide with the exact eigenvalues En = (n + 1/2) , n = 0, 1, . . ., while the exact eigenfunctions are given by (see, e.g. [20, 39]) 2
vn (x)
e−x /2 √ = Hn (π)1/4 2n n!
x √
24
,
for n = 0, 1, . . . ,
(A.16)
where Hn (x) := (−1)n ex
2
dn dxn
e−x
2
are the Hermite polynomials.
By exploiting appropriate asymptotic expansions of the Hermite polynomials [15], we can check that the asymptotic expansion of (A.16) coincides, in the leading order, with the WKB approximation (A.12) in the oscillatory region.
B
Stationary phase formulae
B.1
The case of a simple stationary point
We consider the integral I(λ) =
Z
b
f (x) eiλφ(x) dx
(B.1)
a
where f ∈ C[a, b], and φ ∈ C 2 [a, b] is a real-valued function with a simple stationary point x = c ∈ (a, b) such that φ0 (c) = 0 and φ00 (c) 6= 0. Then, the following approximation formula holds 1/2 2π + O(λ−3/2 ) , λ → ∞ , (B.2) I(λ) = eiλφ(c)+iδπ/4 f (c) λ|φ00 (c)|
with δ = sgnφ00 (c) (see e.g. [7], Ch. 6, or [9], Ch. 2).
B.2
The case of two coalescing stationary points (uniform formula)
We consider the integral I(λ, α) =
Z
∞
f (x) eiλφ(x,α) dx,
−∞
where the phase depends on the parameter α > 0, and look for the asymptotic behavior of I, as λ → ∞. We assume again that f ∈ C[a, b], and that the phase function φ ∈ C ∞ has two stationary points, x1 (α) and x2 (α), which approach the same limit x0 when α → 0. Let φxx (x1 , α) < 0 and φxx (x2 , α) > 0. Then, the approximation h i I(λ, α) = eiλφ0 (α) 2πA0 (α)λ−1/3 Ai(−λ2/3 ξ) − 2πiB0 (α)λ−2/3 Ai0 (−λ2/3 ξ) + C(λ, ξ) ,(B.3) with
φ0 (α) =
ξ(α) =
1 (φ(x1 (α), α) + φ(x2 (α), α)) , 2
3 (φ(x1 (α), α) − φ(x2 (α), α)) 4
and
"
A0 = 2−1/2 ξ 1/4 p
f (x2 ) φxx (x2 , α)
"
B0 = −2−1/2 ξ −1/4 p
+p
f (x1 )
2/3
f (x1 ) |φxx (x1 , α)| f (x2 )
|φxx (x1 , α)|
(B.4)
,
#
−p φxx (x2 , α)
(B.5)
, #
(B.6)
,
(B.7)
holds uniformly for any α > 0. As α → 0+ we use the following approximations, φ0 (α) ≈ φ(0, 0) ,
ξ
1/4
"
≈ −∂xα φ
∂xxx φ 2 25
−1/3 #1/4 α ,
(B.8)
(B.9)
and | ∂xx φ(x1 (α), α) | ≈ (−2∂xxx φ ∂xα φ α)1/2 ,
(B.10)
| ∂xx φ(x2 (α), α) | ≈ (−2∂xxx φ ∂xα φ α)1/2 ,
(B.11)
where the derivatives are calculated at the point (x, α = 0). The approximation (B.3) has been constructed by Chester, Friedman and Ursell in [13] for the case of analytic phases, and a concise derivation is presented in [9].
C
Berry’s semiclassical Wigner function We consider the Wigner transform Z i 1 ψ (x + σ) ψ (x − σ) e− 2pσ dσ W (x, p) = π R
(C.1)
of the WKB wave function ψ (x) = A(x) eiS(x)/ ,
(C.2) 0
where the amplitude A and the phase S are smooth, real-valued functions, and that S (x) is globally convex. We write (C.1) in the form of Fourier integral Z 1 1 D(σ, x) ei F (σ,x,p) dσ , (C.3) W (x, p) = π R where D(σ, x) = A(x + σ)A(x − σ)
(C.4)
F (σ, x, p) = S(x + σ) − S(x − σ) − 2pσ
(C.5)
is the Wigner amplitude, and
is the Wigner phase. For any fixed (x, p), the critical points of the phase F (σ, x, p) are the roots of 0
0
∂σ F (σ, x, p) = S (x + σ) + S (x − σ) − 2p = 0 .
(C.6)
000
Assuming that S (x) 6= 0, we have ∂σσ F (σ = 0, x, p) = 0 ,
000
∂σσσ F (σ = 0, x, p) = 2S (x) 6= 0 ,
(C.7)
and therefore σ = 0 is a double stationary point of F . Berry [6] has introduced an invariant geometrical interpretation of this equation (Figure 7), by observing that (C.6) has a pair of symmetric roots ±σ0 (x, p) such that the point 0 P = (x, p) be the middle of a chord QR (Berry’s chord) with endpoints Q(x−σ0 , S (x−σ0 )) 0 0 and R(x − σ0 , S (x + σ0 )) on the Lagrangian “manifold” (curve) Λ = {p = S (x)} of the WKB function. As P approaches toward Λ, the chord QR becomes to the tangent of Λ and σ0 (x, p) → 0 . It is clear that the two stationary points of (C.6) coalesce to the double point σ = 0 as (x, p) moves towards Λ. Since the ordinary stationary-phase formula (B.2) fails for the integral (C.3) since ∂σσ F (σ = 0, x, p) = 0, we must to apply the uniform stationary formula (B.3). For applying this formula, we need first to identify the small parameter α, which controls the distance between the stationary points ±σ0 (x, p) of the Wigner phase. In order to do this, we expand F in Taylor series about σ = 0 , F (σ, x, p)
0 σ 3 000 σ 2 00 S (x) + S (x) + . . . = S(x) + σS (x) + 2 6 0 σ 2 00 σ 3 000 − S(x) − σS (x) + S (x) − S (x) + . . . − 2pσ 2 6 0 1 000 = −2(p − S (x))σ + S (x)σ 3 + O(σ 5 ) . 3
26
T Q R
P Λ
x + σ0
x
x − σ0
Figure 7: Lagrangian curve & Berry’s chord It becomes evident that for P lying close enough to Λ, the parameter α has to be identified as 0 α = α(x, p) := p − S (x) , (C.8) since by 0
000
∂σσ F (σ, x, p) = −2(p − S (x)) + S (x)σ 2 + O(σ 4 ) , 0
we easily see that σ = 0 is a double stationary point for p = S (x) . Then, for any fixed x, we rewrite the Wigner phase F in the form F (σ, α, x)
and we have
0
= S(x + σ) − S(x − σ) − 2σ(α + S (x)) 0 = S(x + σ) − S(x − σ) − 2σS (x) − 2σα
,
(C.9)
000
Fσα (σ = 0, α, x) = −2 6= 0 . (C.10) These are exactly the conditions on the phase which are required for applying the uniform asymptotic formula (B.3), and we get the approximation ∂σσ F (σ = 0, x, p) = 0 ,
∂σσσ F (σ = 0, x, p) = 2S (x) ,
f (x, p) W (x, p) ≈ W
:=
1/3 2 A2 (x) | S 000 (x) | ! 1/3 22/3 2 0 ×Ai − 2/3 (p − S (x)) , S 000 (x) 22/3 2/3
(C.11) 0
which holds simultaneously for small , and (x, p) near the Lagrangian curve Λ = {p = S (x)} of the WKB function. f (x, p) as the semiclassical Wigner function, We refer to the phase-space function W corresponding to the WKB function (C.2).
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