Royal Society Publishing

Analysis of Regge poles for the Schrödinger equation

A. Hiscox, B. M. Brown, M. Marletta


We study the question addressed by Barut and Dilley (Barut & Dilley 1963J. Math. Phys. 4, 1401–1408) of counting the number of Regge poles for a radial Schrödinger equation. Using the asymptotics of Rudolph Langer, we acquire estimates for the free solutions at infinity for large generalized complex angular momentum |λ|. These estimates allow us to calculate the Wronskian of two particular solutions, which is the function whose zeros are the Regge poles, for large |λ| in the right-half λ-plane. These angular momentum asymptotics are rigorously related to the large-radius asymptotics by generalizing Marianna Shubova’s idea of formulating an integral equation for the solution at infinity. This leads to the proof that for integrable potentials there are only finitely many Regge poles. This should be compared with the ideas of Barut and Dilley, who require that the potential be analytic in the right-half plane with r2V (r) remaining bounded.

1. Introduction

Complex angular momentum (CAM) was conceived by Watson (1918) to assist in the study of diffraction and scattering of short-length electromagnetic waves (Connor 1990). This CAM approach was resurrected with renewed vigour when the theory of Regge poles was introduced in 1959 by Tullio Regge. Regge showed that, when considering solutions of the Schrödinger equation for non-relativistic scattering by a screened Coulomb potential (Eden 1971), it is useful to regard the angular momentum, ℓ, as a complex variable (Collins 1977). Regge proved that for a variety of potentials, the only singularities of the scattering amplitude in the complex ℓ-plane are poles (Regge 1959), or, as they are now called, Regge poles.

In recent times, there has been renewed interest in Regge theory, no doubt owing to the rise in the development of disciplines such as reactive molecular collisions (Connor et al. 1981; Connor 1990; Sokolovski et al. 1995) and particle physics, and in particular, the theory of resonance angular scattering (Sokolovski et al. 2007). The theoretical aspects of low-energy electron elastic scattering are best studied using Regge poles, particularly in gaining an understanding of the formation of temporary anion states during electron attachment, which is fundamental to the physical mechanism by which the scattering process deposits energy (Felfli et al. 2008). Furthermore, Im(ℓ) can be used to distinguish between the shape resonances and the stable bound states of the anions formed as Regge resonances in the electron–atom scattering, whereby Im(ℓ) is vastly smaller for the stable bound states (Msezane et al. 2009).

In this paper, we discuss the number of Regge poles when V is an integrable potential. More precisely, we find that outside a certain sector in the complex λ-plane there are no Regge poles, from which we can deduce that, in total, there are only finitely many Regge poles. The Regge poles are the zeros of an entire function of λ, which is the Wronskian determinant constructed out of two solutions of a radial Schrödinger equation; one solution that is regular at the origin and another at infinity, satisfying a given asymptotic boundary condition. The idea of the proof is to compute this Wronskian for sufficiently large |λ|. In particular, we consider the large |λ| asymptotics of the solution near the origin, with the Coulomb potential, using the Frobenius method, generalizing this to an integrable potential by perturbing the problem by a bounded potential. For the solution at infinity, we initially consider the free problem as a base equation, which can be solved explicitly using Hankel functions, and we acquire large |λ| asymptotics for these using the work of Langer (1932). We can thus obtain the asymptotic form of the Wronskian and, indeed, show that it has no zeros. From the solution at infinity for the free problem, we can formulate an exact integral equation for the general problem. This allows us to demonstrate that, as long as the potential is integrable, we will not, for sufficiently large |λ|, change the Wronskian that we had obtained previously. The basic approach is similar to that used by Shubova (1988) in a rather different context of resonances.

2. Background and notation

The radial Schrödinger equation for an electron in a potential field V is Embedded Image 2.1 where λ=ℓ+1/2 is the generalized CAM (Regge 1959) and Embedded Image. We look for the scattering solution ψ(r)∼rα as r→0, which means that according to equation (2.1), we have α=λ+1/2 or α=−λ+1/2. For values of λ in the right-half plane, the boundary condition Embedded Image 2.2 ensures that ψ(r) is regular at the origin. Also, we make sure that for large r the asymptotic boundary condition, Embedded Image 2.3 is satisfied (Thylwe & Sokolovski 2005).

Through multiplication on the left of equation (2.1) by ψ(r) and integration over Embedded Image, we get an exact expression for μ:=λ2−1/4. Given a trial function ψtr(r), an estimate of μ can be obtained, namely μtr. It can then be shown that (Sukumar & Bardsley 1975) Embedded Image 2.4

If ψtr(r) satisfies equation (2.2), then the first term on the right-hand side of equation (2.4) vanishes, provided Re(λ+λtr)>0. Therefore, the convergence of λtr to λ is only possible if Re(λ)>0.

The question we wish to pose is: are there finitely or infinitely many Regge poles for problem (2.1)? It is known that for the Coulomb potential, Vcoul(r):=−2Z/r, where Z is the nuclear charge, there are no Regge poles in the right-half plane (Thylwe & Connor 1985). The motivation for such a question was the rational approximation to the Thomas–Fermi potential (Belov et al. 2004); however, the work of Barut & Dilley (1963), which requires that the potential has finite second moment and can be analytically continued into the right-half plane, provides the answer that there are finitely many Regge poles in the Thomas–Fermi case. We shall consider analogous questions for the class of integrable potentials. More precisely, let us denote by V a potential with the following two properties: V (r)∼−2Z/r as r→0 and Embedded Image for all a>0.

Let us define our notation. Let ψL,0(r,λ) be the solution of equation (2.1), regular at the origin with VVcoul, and let us denote by ψL(r,λ) the solution of (2.1), regular at the origin with our general V identified earlier. Also, let ψR,0(r,λ) be the solution at infinity with V ≡0 and denote by ψR(r,λ) the solution at infinity with our general V . We shall find fixed r>0 large |λ| asymptotics for ψL and ψR,0 and, from these asymptotics, compute their Wronskian determinant WR,0, which is defined by Embedded Image

We repeat that the Regge poles are the zeros of the Wronskian. Moreover, we sometimes write λR for Re(λ) and λI for Im(λ).

3. The series solution at the origin

Consider equation (2.1) where VVcoul. We have, using the Frobenius method, a series solution at the origin given by the formula Embedded Image 3.1

A standard calculation of inserting equation (3.1) into equation (2.1) shows that Embedded Image

Choosing α=λ+1/2 or α=−λ+1/2, we have Embedded Image which results in the following system of equations Embedded Image and hence Embedded Image with a 0:=1 and a1=−Z/α. Recall that choosing α=λ+1/2 ensures the regularity of ψL,0(r,λ) at the origin. Therefore, Embedded Image

We have |aj|≤(const.)/|λ+1/2|, ( j=1,2,3,…) as Embedded Image in the right-half plane, and thus Embedded Image uniformly with respect to λ. That is, for fixed r and large |λ| Embedded Image 3.2

To generalize this to an integrable potential, we introduce a perturbation into equation (2.1) by a bounded potential U, i.e. Embedded Image 3.3

Making the change of variable Embedded Image 3.4 which means d/dr=−exd/dx, and with Embedded Image, we have Embedded Image

Writing this as a first-order system, we have Embedded Image 3.5 where Embedded Image

The matrix A0(λ) has two eigenvalues μ1(λ)=−(λ+1/2) and μ2(λ)=λ−1/2, with corresponding eigenvectors v1(λ)=(1,−(λ+1/2)) and v2(λ)=(1,λ−1/2). Denote by P(λ) the 2×2 matrix with columns v1 and v2. As in Eastham (1989), making the transformation Z=P(λ)Y , system (3.5) becomes Embedded Image 3.6 where Λ(λ)=diag(μ1(λ),μ2(λ)) and R(x,λ)=P−1(λ)A1(x)P(λ). We now define Embedded Image and Embedded Image

It is easily verifiable that Φ(x,λ) is a fundamental matrix for the system Embedded Image

Define Embedded Image 3.7

A calculation shows that any Y that satisfies the integral equation Embedded Image 3.8 is, for any a≥0, a solution of equation (3.6). By the Levinson theorem (Eastham 1989), the linear differential system (3.6) has solutions with the asymptotic form Embedded Image 3.9 where ej denotes the coordinate vector whose jth component is unity and other components are zero. We need an explicit estimate for the o(1) term in equation (3.9), and it can indeed be obtained from equation (3.8). Suppose o(1) has components o(1)i. Then, by Eastham (1989), we have Embedded Image

Hence |o(1)i|≤(const.) for Re(λ)>0. Recall that we look for solutions that behave like rλ+1/2 as r→0. Therefore, in view of the transformation (3.4), we must choose j=1 in equation (3.9) Embedded Image and since Z=P(λ)Y , Embedded Image 3.10

Hence, from equation (3.10), we have Embedded Image, Embedded Image and therefore by equation (3.4) Embedded Image 3.11

Also from equation (3.10), we have Embedded Image and so by equation (3.4) again Embedded Image 3.12

4. The Hankel solution at infinity

For a fixed r and large |λ|, the centrifugal term dominates V . Thus, it is instructive to consider the free problem Embedded Image 4.1

It is well known (Abramowitz & Stegan 1965) that solutions of equation (4.1) are Embedded Image 4.2

We want, for a fixed r>0, large |λ| asymptotics for ψR,0(r,λ). These can be determined from a seminal paper by Langer (1932). Using the notation of Langer (1932), we have Embedded Image 4.3

Consider first the case whereby k>0, then for λ in the first quadrant such that |λ|>kr, we have Embedded Image 4.4

We need Langer’s variable ξ, which is defined as (Langer 1932) Embedded Image

On evaluating this integral, we get, in our case, Embedded Image 4.5 where Embedded Image 4.6

Next, the function g (Langer 1932, p. 470) is defined by Embedded Image

The asymptotic forms of Embedded Image and Embedded Image are given in Langer (1932; p. 471, table 50) according as to which region Ξ(h)z resides. By equation (4.4), z lies in region IV in Langer (1932; p. 466, fig. 1) and consequently h=1 or h=2 (Langer 1932; p. 469). Therefore, the dominant terms of Embedded Image and Embedded Image are given by Embedded Image respectively. Hence, for large |λ| Embedded Image 4.7 and Embedded Image 4.8

Therefore, by equation (4.2), we have for large |λ| (taking j=1 in equation (4.2)) Embedded Image 4.9 where Embedded Image 4.10 from equations (4.3), (4.5) and (4.6); subject to the following identification: Embedded Image

To calculate the asymptotic form of the Wronskian WR,0, we also require the large |λ| asymptotics for Embedded Image. Note that from equation (4.2), we look for large |λ| asymptotics for Embedded Image 4.11

We have the well-known formula (Arfken & Weber 2005) Embedded Image 4.12 and so Embedded Image 4.13

For λ−1 such that |λ−1|>kr, we have Re(z)<0 and −π/2<Im(z)<0. However, for λ+1 such that |λ+1|>kr, we need to ensure that Embedded Image which implies that −π/2<Im(z)<0 and Embedded Image. Thus, in this case, z lies in region III and hence h=0 or h=1. Hence, the dominant term of Embedded Image is gei(ξπ/4) with λ being replaced by λ+1 in g and in equation (4.10). Therefore, it is readily seen that Embedded Image 4.14

Up to this moment, we have assumed that k>0. When k<0, we have Embedded Image, where Embedded Image and Embedded Image. This implies that Re(z)<0 and π/2<Im(z)<π, which means that z lies in region IV leaving equation (4.9) unaltered. Regarding equation (4.14), we only need to consider ρ=λ−1. But then we must have Embedded Image and so π/2<Im(z)<π and Embedded Image. We now find that z is in region II and hence h=0 or h=1. The dominant term of Embedded Image is gei(ξπ/4), with λ being replaced by λ−1 in g and in equation (4.10). The asymptotic form (4.14) for large |λ| is thus also unchanged. We mention here the well-known fact that there are no poles in the fourth quadrant (Delos & Carlson 1975).

(a) Forming the Wronskian

We are now in a position to compute the Wronskian of ψL(r,λ) and ψR,0(r,λ) using equations (3.2), (3.11), (3.12), (4.9) and (4.14). So, for large |λ|, Embedded Image

For large |λ|, it is clear that the (λ+1/2)ψR,0(r,λ)/r term is the dominant term, thus we only need to consider Embedded Image 4.15 of which the only possible zeros occur when Embedded Image which is never the case.

5. The solution with integrable potential

We have clearly demonstrated that there are no Regge poles in the right-half plane for the free problem when |λ| is sufficiently large. We will now show that this is still the case under the influence of a potential V ∈L1. As alluded to earlier, we use an idea of Shubova (1988). The idea is to formulate an integral equation for the solution ψR, behaving like Embedded Image for large r, in terms of the Hankel-type functions of §4 whose behaviour for large |λ| we appreciate. Thus, if large |λ| asymptotics of the integral equation can be calculated, then we can use these asymptotics, along with those of the Hankel-type functions, to acquire large |λ| asymptotics of the solution ψR and consequently compute the Wronskian W(ψL,ψR).

We already know that Embedded Image are two linearly independent V ≡0 solutions at infinity. Consider the integral equation Embedded Image 5.1 where WR is the Wronskian of ψR,0 and Embedded Image, and Embedded Image

It can be easily shown that any solution of equation (5.1) satisfies the ODE (2.1). We now consider the problem of finding large |λ| asymptotics for ψR(r,λ) by first computing large |λ| asymptotics of WR, which will be relatively straightforward as we already have the large |λ| behaviour of ψR,0 and Embedded Image. Therefore, it follows from equations (4.2), (4.7), (4.12) and (4.13), and Langer (1932; p. 471, table 50) that Embedded Image

Noting that Θ(r,s) is bounded and Embedded Image for all a>0, we find from equations (4.10) and (5.1) that ψR(r,λ)∼ψR,0(r,λ) for large |λ|. Thus, by equation (4.15), there are no Regge poles of (2.1) for sufficiently large |λ|. Therefore, we have proved the following.

Theorem 5.1

Suppose Embedded Image then equation (2.1) has no Regge poles outside some sector Sλ:={λD(0;d):Re(λ)>0}, where D(0;d) is a disc centred at the origin of sufficiently large radius d>0.

It has been established that there exists no Regge poles outside the sector Sλ. However, the situation inside Sλ is unclear. With regard to how all solutions of equation (2.1) behave as functions of λ, we recall small r and large r asymptotics (2.2) and (2.3), respectively. In formulating their Wronskian and factoring out the rλ−1/2, we get a function that is entire in λ. The Regge poles are the zeros of this function, and from a standard result (Holland 1973) in complex analysis, which says that an entire function on a finite region has finitely many zeros, we have the following.

Theorem 5.2

Suppose Embedded Image then equation (2.1) has finitely many Regge poles in the right-half plane.

6. Conclusion

In this article, we have shown that if the potential V (r) is such that |V (r)| is integrable over the positive semi-axis, then the associated Regge pole problem has only finitely many poles in the right-half plane. This weakens the assumptions in Barut & Dilley (1963), who required that r2V (r) be bounded and that V (r) has an analytic continuation into the right-half plane. The new results allow treatment of potentials such as V (r)=C/[(1+ar)2(1+b(r−1)2)], which is similar to a Thomas–Fermi potential but has singularities at Embedded Image, or V (r)=1/(r+1)3/2, for which r2V (r) is unbounded. What we have not yet developed, and what we believe could be of considerable interest, would be an estimate of the number of Regge poles in terms of a norm of the potential, in the spirit of the Cwikel–Lieb–Rosenblum estimates for the number of bound states.


The authors would like to thank two anonymous referees for many helpful suggestions that substantially improved this manuscript. M. Marletta also thanks Professor Sergey Naboko of St Petersburg University for helpful discussions during a visit to St Petersburg funded by INTAS.


    • Received February 18, 2009.
    • Accepted June 2, 2009.


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