After the connection formulas have been obtained, we can focus back the attention on the eigenvalue problem (7.6). We assume that the potential wellq has the structure
q(y, λ) =V(y) +λ,
whereV(y) is an analytic function, possessing a critical point at y= 0,
If the value of λ is close enough to V(0), and the local behavior of V around y= 0 is parabolic, two simple turning-points, yL and yR, exist close to the y-origin. Three Stokes lines emanate from each ofyLandyR, and the same number of anti-Stokes lines. We remind that Stokes lines are defined by Re[Ryy0q(ζ, λ)1/2dζ] = 0, while anti-Stokes
lines are defined by the specular condition Im[Ryy0q(ζ, λ)1/2dζ] = 0 withy0 being either yL oryR. In figure 7.3 is reported a plot of such curves in the complex y-plane, which depicts the typical configuration corresponding to turning-points localized near at the critical point of a locally parabolic potential well. (Precisely, this plot is based on the case that will be discussed in § 8.2.2.)
On the Stokes and anti-Stokes lines WKBJ solutions exhibit limiting behaviors, on the first the exponential is purely oscillatory while on the second it is purely grow- ing/decaying without oscillations.
We introduce the four WKBJ solutions φ1L, φ2L,φ1R and φ2R, where the subscript indicates the specific choice y0 = yL or y0 = yR. The sector of definition of both φ1L and φ2L is the one contained in between µL2 and µL3 (unshaded left region in Fig.7.3).
Similarly φ1R and φ2R are defined in between µR2 and µR3 (unshaded right region in
Fig.7.3). Also, the branch choice implies that φ1L is the exponentially small solution for y moving left along the negative real axis, while φ2R is small for y moving right along the positive real axis.
As we work under free-space conditions, only the vanishing components of the so- lution (i.e. φ1L and φ2R) are present in the far field, the white regions extending indefinitely in figure 7.3. To determine eigenvalues one has to impose matching in the middle region S (shaded region in Fig.7.3) between left- and right-handed solutions, that correspond toφ1Land φ2Rin the lateral far-field sectors. Moving from the far field intoS the two asymptotic solutions φ1L and φ2R must be continued inside S by using the connection formulas. In other words, φ1L and φ2R must be replaced by different expressions in order to be asymptotic to the same exact solution insideS. By following the convention that the four functionsφare extended insideS by analytic continuation looping counter-clockwise around the turning-points, the connection formulas, as found
Π Π 2 -Π -Π2 Π Π 2 -Π -Π2 yL yR µL1 µL2 µL3 µR1 µR2 µR3
Figure 7.3: Stokes (dashed) an anti-Stokes (continuous) lines for turning-points close to the origin. The shaded region is the (open) set S.
in the previous section, are
φ1L99Kφ1L+iφ2L, (7.11a)
φ2R99Kφ2R+iφ1R. (7.11b)
We remark again that these are analogs of Jefferey’s connection formulas [52, 13], generalized to connect asymptotic solutions valid in different sectors of the complex plane around a turning-point, rather than the two parts of the real line divided by a turning-point for real self-adjoint problem.
After using the connection formulas, we can enforce matching inside S and obtain an eigenvalue conditon. This can be performed in either a symmetric or anti-symmetric manner
For compactness of notation let QL = ǫ−1/2 Z y yL q(ζ, λ)1/2dζ, (7.13) QR = ǫ−1/2 Z y yR q(ζ, λ)1/2dζ, (7.14)
equation (7.12) (dropping the prefactor q−1/4) becomes
eQL+ieQL =±e−QR ±ieQR,
where all the integrals are now path-independent in S. Introducing also QLR =
ǫ−1/2RyR
yL q
1/2dζ, the above equation is equivalent to
eQLR+QR +ie−QLR−QR =±e−QR ±ieQR,
which can be recombined as
eQReQLR∓i=e−QR−ie−QLR ±1. (7.15)
The last relation can be satisfied if the terms in brackets are equal to zero. After a few straightforward manipulations, the above eigenvalue condition for complex turning- points yields the following implicit relation in the eigenvalues λ
exp ǫ−1/2 Z γ q(ζ;λ)1/2dζ =±i, (7.16)
whereγ is an arbitrary path in the complex plane connecting yLtoyRwithout looping around one of them (because of the multivalueness ofq(ζ;λ)1/2). Notice how the above
integral condition appears as a natural extension of the well-known result for self- adjoint problems, where the turning-points lie on the real line [13]. Equation (7.16)
can be rewritten as
Z
γ
q(ζ;λ)1/2dζ =iǫ1/2π n+12, n = 0,1,2... (7.17)
which constitutes an implicit relation for a set of eigenvalues λ, corresponding to even (odd) eigenmodes forn even (odd). The left-hand side is a function ofλ that involves an integral of the elliptic kind, of which the limits of integration contain themselves a dependence fromλ. Such relation can be inverted only numerically.