I N T E R N R r I O N A L J O U R N A I . OF Q U A N T U M C H E M I S T R Y . VOI,. 46. 375-381 (lY'l3)

A Multiple-Transition-Point WKB Investigation of Complex Energy Resonances

NILS ANDERSSON* Department of Theoretical Physics, Uppsala Universify, Thunbergsvagen 3, S-752 38 Uppsala, Sweden

Abstract

Approximate conditions determining the complex energy resonances of a one-dimensional model potential, originally discussed by Moiseyev et al., are derived within a multiple-transition-point WKB

treatment. It is shown that an erroneous energy spectrum is generated if too few transition points are considered. The numerical results of a correct semiclassical treatment of the problem, where several transition points are included, agree well with the, presumably exact, results of Rittby, Elander, and Brandas. 0 1993 John Wiley & Sons, Inc.

Introduction

In general, determining complex resonance energies by means of direct integration leads to a numerically unstable problem. This is because the boundary conditions determining a scattering resonance require wave functions that behave as purely outgoing waves. Such solutions to the Schrodinger equation, usually referred to as Siegert states, are asymptotically divergent on the real coordinate axis. To avoid this difficulty, and also to make the wave function vanish asymptotically, the considered coordinates can be allowed to assume complex values. This approach has the great advantage that resonance phenomena can be investigated by familiar bound-state techniques. However, in the standard treatment, significant errors may enter in a very subtle way and easily pass undetected.

In this paper, we study the resonance states of the one-dimensional model potential, initially suggested by Moiseyev et al. [l]:

In a series of papers by Rittby, Elander, and Brandas [2-41, this potential was investigated using a method that combines complex scaling with Weyl's theory. It was found that the resonances can be divided into two classes: those preceding and those following a certain threshold energy. At the threshold, the real part of the resonance

*Present address: Dept. of Physics and Astronomy, University of Wales, College of Cardiff, Cardiff CF2 3YB, U.K.

0 1993 John Wiley & Sons, Inc. CCC 0020-7608/93/030375-07

376 ANDERSSON

energy attains a maximum value, while the imaginary part of the resonance energy is decreasing throughout the spectrum. As discussed by Rittby et al. [2-41, it seems as if the real part of the resonance energy is oscillatory beyond the threshold. This conclusion was contradicted by Korsch et al. [5]. Using the complex-rotated Milne method [6], they found the real part of the resonance energy to be monotonically decreasing in the region following the threshold. The numerical results of Korsch et al. were confirmed by a WKB analysis, using a uniform three-turning-point formula originally derived by Connor [7]. The similarities between the potential (1) and the radial potential Vor2 exp(-r), for which a similar threshold energy had been found [8], were also pointed out.

Multiple spectra, such as the ones discussed above, associated with an exponential wall, has been discussed in detail numerically and analytically by Atabek and Lefebvre 191. Conclusively, in some situations, the resonance spectrum may change completely if the complex-rotation angle exceeds a critical value. Consequently, as demonstrated by Rittby et al. [3], in the investigation of the potential (l), a large value of the complex-rotation angle generates the spectrum obtained by Korsch et al. [5 ] . Hence, there exist several different angular sectors for the complex coordinate plane where the obtained wave function vanishes asymptotically. The natural question is whether these different wave functions all fulfil the purely outgoing-wave condition determining the resonances.

In the present investigation, we will, by considering several transition points situated in the complex coordinate plane, show that a straightforward wKB analysis of the potential (1) yields the resonance spectrum described by Rittby et al. We will also see that the occurrence of a qualitatively different spectrum, as discussed by Korsch et al., can be explained using simple semiclassical arguments.

The WKB Method in the Complex Plane

In the WKB method, the solution of the differential equation

d2$ 2P dz2 dz2 f i2 d2@ + R(z)$ = - + - [E - V(Z)l$ = 0,

is, locally, given by a linear combination of the two functions:

where q(z) is an approximate function explicitly given by

In a rigorous semiclassical investigation of the differential equation (2), it is important to examine the positions of the transition points, i.e., the zeros and poles of the analytic function R ( z ) in the complex plane. In the vicinity of these points, the WKB approximation is no longer valid. However, in the complex coordinate plane, we can always trace the desired solution of (2) at a safe distance from the points where the

COMPLEX ENERGY RESONANCES 377

approximation breaks down. Nevertheless, the same linear combination of the two approximate functions (3 ) cannot be used to represent the desired solution of ( 2 ) in different regions of the complex plane surrounding a zero of R(z) . This is due to the so-called Stokes phenomenon.

A useful concept in an analysis of the differential equation ( 2 ) is the pattern of anti- Stokes lines. These are defined as contours along which the quantity q(z) dz is purely real. In the WKB approximation, three such anti-Stokes lines emerge from each zero of R ( z ) . As follows directly from (3) , the approximate solutions of ( 2 ) behave as traveling waves along an anti-Stokes line. Consequently, it is natural to introduce the boundary condition of outgoing waves, defining a resonance state, on such lines in the complex coordinate plane. As long as the solution of (2) is continued along an anti-Stokes line, a given linear combination of the two approximate functions (3) is preserved. This is, of course, provided that the solution is not traced through the vicinity of a transition point. Hence, to solve (2) using the WKB method, the approximate solutions have to be continued through the regions of the complex coordinate plane containing transition points.

Multiple-transition-point WKB Conditions

To make the approximate function q ( z ) single-valued, we have to introduce cuts in the complex coordinate plane. In the present investigation, we will assume that cuts are introduced according to Figure 1 and that the phase of R’”(z) is chosen in such a way that the solution fl(z) has the outgoing-wave behavior required of the desired solution of ( 2 ) as z - +m. Hence, at the point z1 of Figure l(a) (on the appropriate anti-Stokes line), we have

To continue this solution through the region close to the transition point tl, we use the F-matrix method developed by Froman and Froman [ 101. Provided that the transition point t1 lies well separated from all other transition points, we obtain the solution corresponding to (5) at the point z2 (cf. Chap. 7 in [lo]):

where C is a constant. Since the function R ( z ) and also the boundary conditions determining the resonances

are symmetric with respect to an exchange z - - z , it is possible to restrict an analysis of the differential equation ( 2 ) to the right half-plane.* Because of the symmetry of the problem, there are two different sets of solutions. These are either of odd or even parity. For a solution of odd parity, it is obvious that the wave function q ( z )

*If the symmetry of the problem is not considered (cf. the analysis of Korsch et al. [ S ] ) , twice as many transition points have to be considered. This means that the one-transition-point WKB condition of the present paper is, in fact, equivalent to a two-transition-point formula should symmetry not be considered.

378 ANDERSSON

Figure 1. In (a), the pattern of anti-Stokes lines corresponding to the potential (1) and E = 3.25 - l . l l i , approximating the resonance state n = 5, is drawn. (b) corresponds to the situation for E = 4.64 - 10.83i, i.e., n = 15. The transition points considered as relevant in the derivation of the WKB conditions (7) and (11) are denoted by t i . Other specific points used in the analysis are given by z i . Cuts introduced to make q(z ) a single-valued function are represented by wavy lines. (The scale of the figure is such that each tick

corresponds to one unit in the coordinate plane.)

must vanish at the origin. This implies a WKB condition determining the odd parity resonance states. On the other hand, a solution of even parity is characterized by a vanishing derivative of $(z) at the origin. Since the derivative of q ( z ) vanishes at z = 0, the WKB condition determining the even parity solutions immediately follows. Hence, we have

(7) n = 1,3,5,. . . odd parity

2J," q ( z ) d z = (n + $)- { n = 0,2,4,. . . even parity,

where n is an integer. The condition (7) is valid, provided that the transition point tl lies well separated

and no other transition point is of relevance for the solution. If, however, we have the situation depicted in Figure l(b), the transition point t2 must also be considered. According to (6), the desired boundary condition of outgoing waves at z1 of Figure l(b) implies that the solution at the point z2 can be written

We have changed the lower limit of integration by introducing the convenient definition

COMPLEX ENERGY RESONANCES 379

If we again apply the F-matrix used in deriving (6), we find that the corresponding solution, at the point z 3 of Figure l(b), is given by

where D is a constant. It follows that the WKB condition, considering both tl and t 2 , can be written

The formula is valid as long as the two transition points considered are not close-lying. By repeated use of the same procedure, the appropriate WKB condition considering

also t 3 can be derived. Hence, we obtain the three-transition-point formula:

and if four transition points are considered, we have

(1 + e2ia34)}] = (n + - T . (13) 2 7 2 1 " q ( z ) d z - i log[l + e2ia12{1 + e 2 i ~ 2 3

It is important to note that the integer n, the quantum number labeling the resonance states, is unambiguously introduced in the one-transition-point condition (7). However, due to the multivaluedness of the complex logarithm in the conditions (11-13), there is a possible ambiguity in the quantum number when these are used. To avoid difficulties, the necessary cuts in the complex logarithm must be properly considered in a numerical evaluation of the WKB conditions (11-13).

Numerical Results and Conclusions

Numerical calculations using the WKB conditions (7,ll-13) have been performed with parameters J = 0.8 and A = 0.1, in atomic units. With these parameters, the potential has a single bound state and, as it seems, infinitely many resonance states in the continuous part of the energy spectrum. The results of the numerical calculations are presented graphically in Figures 2(a)-(d). Comparison has been made with the numerical results of Rittby et al. [2-41. It can easily be verified that, if the one- transition-point condition (7) is used, the results of the present investigation correspond to the WKB results obtained by Korsch et al. [ 5 ] . Since the nonuniform WKB analysis of the present paper gives results as accurate, the uniform WKB formulae used by Korsch et al. are not needed for determining the resonance energies. Nevertheless, it must not be forgotten that, when used in the appropriate situation, the uniform WKB formulae are powerful (cf., e.g., [7]).

Furthermore, it follows from Figure 2(a) that the results obtained using the one- transition-point WKE3 condition (7) agree well with the results of Rittby et al. in the

380 ANDERSON

4.5

4.0

3.5

Re En 4.5

4.0

3.5

. . m . . . .

1 1 1 1 1 1 1 1 1

0 10 20 30 40

. m .

I . .".. -5 D I W . .. . . .-.. .. ..

m 1 . 1 . .

m . . I I I I I I I I I

0 10 20 30 40

-1m h, Figure 2. The numerical results obtained using the WKB conditions are compared with the results of Rittby et al. The WKB resonances are denoted by squares, whereas the presumably exact results are represented by dots. (a) corresponds to results obtained using (7), whereas (b)-(d) correspond to conditions (11)-(13), respectively. Re E is plotted as a function of

-Im E .

region preceding the threshold energy. To reveal the expected oscillations in the real part of the resonance energy after the threshold (cf. [2-4]), further transition points have to be included in the WKB analysis. It follows that already when two transition points (in the right half-plane) are considered, i.e., in calculations using (ll), the first oscillations appear. The four-transition-point results agree almost completely with those obtained by Rittby et al., at least to the accuracy of Figure 2(d). Hence, the WKB method can be used to determine the complex energy resonances corresponding to the model potential (1). It must be pointed out, however, that the wm conditions discussed in the present paper are not the appropriate formulae for determining bound states of the potential.

The failure of the complex-rotated Milne method proposed by Korsch et al. [5,6] can be explained from semiclassical theory. Let us assume that the complex-rotation angle is chosen larger than a critical value, the existence of which is obvious from Figures l(a) and (b). The boundary condition of outgoing waves is then introduced in an asymptotic region of the complex coordinate plane where the anti-Stokes lines are parallel to the imaginary coordinate axis. Somewhat simplified, one may conclude that an outgoing wave in this region never escapes to spatial infinity, i.e., z - +m.

It is, however, impossible to solve the connection problem from this region of the

COMPLEX ENERGY RESONANCES 38 1

complex coordinate plane to the proper asymptotic region, since a large number of transition points must be considered in doing so.

Acknowledgment

The author would like to thank Dr. Karl-Erik Thylwe for immensely helpful discussions on complex energy resonances and related topics.

Bibliography

[ I ] N. Moiseyev, P.C. Certain, and F. Weinhold, Mol. Phys. 36, 1613 (1978). 121 M. Rittby, N. Elander, and E. Brandas, Phys. Rev. A 24, 1636 (1981). 131 M. Rittby, N. Elander, and E. Brandas, Phys. Rev. A 26, 1804 (1982). [4] M. Rittby, N. Elander, and E. Brandas, Mol. Phys. 45, 553 (1982). (51 H. J . Korsch, H. Laurent, and R. Mohlenkamp, Phys. Rev. A 26, 1802 (1982). [6] H. J. Korsch, H. Laurent, and R. Mohlenkamp, J. Phys. B 15, 1 (1982). [7] J. N. L. Connor, in Semi-Classical Methods in Molecular Scattering and Spectroscopy, M. S. Child,

[8] H.J. Korsch, H. Laurent, and R. Mohlenkamp, Mol. Phys. 43, 1441 (1981). [9] 0. Atabek and R. Lefebvre, Nuovo Cim. B 76, 176 (1983).

Ed. (Riedel, Dordrecht, 1980), p. 45.

1101 N. Froman and P. 0. Froman, JWKB Approximation, Contributions fo the Theory (North-Holland, Amsterdam, 1965).

Received January 22, 1992 Accepted for publication October 29, 1992