## Abstract

In this paper, we confirm, with absolute certainty, a conjecture on a certain oscillatory behaviour of higher auto-ionizing resonances of atoms and molecules beyond a threshold. These results not only definitely settle a more than 30 year old controversy in Rittby *et al.* (1981 *Phys. Rev. A* **24**, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) and Korsch *et al.* (1982 *Phys. Rev. A* **26**, 1802–1803 (doi:10.1103/PhysRevA.26.1802)), but also provide new and reliable information on the threshold. Our interval-arithmetic-based method allows one, for the first time, to *en*close and to *ex*clude resonances with guaranteed certainty. The efficiency of our approach is demonstrated by the fact that we are able to show that the approximations in Rittby *et al.* (1981 *Phys. Rev. A* **24**, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) *do* lie near true resonances, whereas the approximations of higher resonances in Korsch *et al.* (1982 *Phys. Rev. A* **26**, 1802–1803 (doi:10.1103/PhysRevA.26.1802)) do *not*, and further that there exist two new pairs of resonances as suggested in Abramov *et al.* (2001 *J. Phys. A* **34**, 57–72 (doi:10.1088/0305-4470/34/1/304)).

## 1. Introduction

Reliable and precise information on the location of resonances is very hard to obtain. While numerical approximations are widely used in physics, so far there has been no way to show that they produce results *near*, or *not near*, true resonances. The reason is that computations of complex eigenvalues in the presence of continuous spectrum are not backed up by any convergence results. This paper presents a new method that, for the first time, permits one to locate resonances with absolute certainty and high accuracy and, at the same time, to show that numerical approximations *fail* to lie near true resonances. We provide new and reliable information on the oscillatory behaviour of the real parts of certain resonance strings and on the threshold beyond which it occurs.

The key ingredient in our method is interval arithmetic. It allows us to carry out every computational step with absolute accuracy by operating on intervals rather than on numbers. Remarkably, this theoretical idea has had convincing impact in different practical physical applications recently: to control the stability of difficult nonlinear systems in robotics to navigate a sailboat autonomously over a distance of 100 km (see [1]); to perform rigorous global optimization of impulsive planet-to-planet transfer (see [2]) or to rigorously govern the long-term stability in particle accelerators (see [3]).

In this paper, we demonstrate the efficacy of interval approaches for the computation of resonance enclosures and exclosures with absolute certainty. The power of our method is substantiated by the fact that it can be applied to definitely settle a more than 30 year old controversy in [4,5] which could not be resolved by any other method before.

In connection with auto-ionizing resonances of atoms and molecules lying above a ionization threshold, Moiseyev *et al.* [6] studied resonances of the Sturm–Liouville problem
*i* (with

In the two subsequent papers [4] and the more detailed version [7], Rittby *et al.* combined complex scaling with some Weyl-type analysis and numerical integration methods to compute 44 resonance approximations, including approximations for the first resonance and bound state suggested in [6]; the second resonance therein was further studied by Engdahl & Brändas in [8] by computing lower bounds for norms of Riesz projections. The main conclusion of [4,7] is that there exists a complex threshold *ϵ*_{thresh} with Re(*ϵ*_{thresh})>0, Im(*ϵ*_{thresh})<0 such that all resonance approximations of (1.1) satisfy Re(*ϵ*)≤Re(*ϵ*_{thresh})∼4.68 and beyond this threshold, i.e. for Im(*ϵ*)<Im(*ϵ*_{thresh})<0, their real parts exhibit a certain oscillatory behaviour.

Shortly after the publication of [4] and the submission of [7], Korsch *et al.* announced in [5], comment that they had computed a different set of resonance approximations beyond the threshold which did not exhibit any oscillatory behaviour, whereas their earlier computations of lower resonances in [9] had not shown such a disagreement. They used a complex-rotated Milne method and they believed to have backed up their computations by some WKB approximations. Korsch *et al.* concluded that the results of Rittby *et al.* for higher resonances were incorrect; they conjectured this might be due to numerical instabilities or to the too limited range 0<*θ*<*π*/4 of angles in the complex scaling method in [4,7].

In an immediate reply (see [10], reply to comment), Rittby *et al.* [5] defended their results and attributed the discrepancies of the results to wrongly chosen outgoing boundary conditions. They argued that the asymptotic solutions of the complex Riccati equation associated with (1.1) undergo a dramatic change when *θ* passes the critical value *θ*_{crit}=*π*/4 of the potential in (1.1) and hence the rotation angle *θ*=50° used by Korsch *et al.* was too large. Because of this and the stability of the computations in [4,7] against variations of the rotation angle *θ*, Rittby *et al.* [4,7] believed to have found approximations to true resonances. About 10 years later, Andersson corroborated the arguments and conclusions of Rittby *et al.* by a careful multiple-transition point WKB analysis and explained the failure of the complex-rotated Milne method of Korsch *et al.* by semi-classical theory in [11].

Almost 20 years after the 1982 dispute, the resonance problem (1.1) was studied as an example in two papers in the mathematical literature. In [12], for more general classes of exponentially decaying potentials, Brown *et al.* developed a resonance-finding procedure for resonances close to points of spectral concentration on the real axis. This method relies on analytic continuation of the Weyl–Titchmarsh function rather than on complex scaling and was first used by Hehenberger *et al.* [13] in numerical computations for the Stark effect. As an example, Brown *et al.* computed approximations to the first three resonances of (1.1) which were very close to the ones found in [7]; note that *q* and spectral parameter λ in [12] are related to the potential *V* and spectral parameter *ϵ* in (1.1) by

Not long after, Abramov *et al.* [14] proved some global analytical bounds for resonances for various classes of potentials. They combined complex scaling with operator theoretic techniques such as numerical ranges and Birman–Schwinger type arguments. Moreover, for the particular case of (1.1), they also performed numerical computations. The analytical results in [14] supported the conjecture of Rittby *et al.* that a wrong asymptotic boundary condition was used by Korsch *et al.* [5]. The numerical results of [14] reproduced the resonances found in [4,7] and they suggested three pairs of additional resonances. Each pair consists of an even and an odd resonance so close to each other that they could not be computed accurately. These new resonance pairs may be related to the oscillatory behaviour of the real parts; because two of these pairs satisfy −9.57∼Im(*ϵ*_{thresh})<Im(*ϵ*)<0.

As it was rightly put in [14], none of the above methods for finding resonances can be used *to locate them accurately, but there is clear evidence that they exist*. Moreover, none of these methods allows for a proof that a numerically computed candidate for a resonance is *not* near any true resonance.

The method presented here permits us to settle both questions definitely and adds new information on the threshold beyond which oscillatory behaviour of the real parts of resonances occurs. We prove that the 44 numerical approximations of resonances from [4,7] do lie near true resonances and that the numerical approximations labelled 16–28 in [5] do *not* lie near true resonances. Moreover, we prove that two of the additional pairs of resonances conjectured in [14] do exist. Our provably correct computations are based on a combination of two key tools, the argument principle on the analytic side and interval arithmetic on the computational side.

Briefly, our approach is as follows. By means of complex scaling *x*→e^{iθ}*x* with *θ*∈[0,*π*/4), the resonances *ϵ* of (1.1) are given in terms of the eigenvalues *z*=*e*^{2iθ}(2*ϵ*−1.6) of a Sturm–Liouville problem on *et al*. [15]. Roughly speaking, this means that all computations, from adding numbers up to integration, amount to working with two-sided estimates; e.g. the sum of two real numbers *a*∈[*a*_{1},*a*_{2}] and *b*∈[*b*_{1},*b*_{2}] is the interval [*a*_{1}+*b*_{1},*a*_{2}+*b*_{2}] which is guaranteed to contain *a*+*b* (see [16], §2 for a more detailed description). If we obtain that
*n*_{0} eigenvalues of the complex-scaled Hamiltonian in the rectangle *n*_{0} resonances in the rotated rectangle

Our method is the first, in both physical and mathematical literature, that accomplishes the following three different tasks:

Enclose resonances with prescribed accuracy, by choosing the size of the rectangle accordingly small and achieving

*n*_{0}=1.Exclude resonances in certain rectangles by achieving

*n*_{0}=0.Check if the number of resonances in a rectangle of arbitrary size computed with non-reliable methods is correct by checking if it coincides with

*n*_{0}.

## 2. Complex scaling and lack of analytic information

There are various mathematical definitions of resonances and different methods to study them; for details, we refer to the comprehensive review articles by Simon [17], Siedentop [18] and Harrell [19]. Here, we use the method of complex scaling where resonances are characterized as eigenvalues of certain non-self-adjoint Schrödinger operators.

As an example, we consider the spectral problem (1.1), with *ϵ*−1.6, it is easy to see that (1.1) is equivalent to the spectral problem
*L* in the Hilbert space *y*′, *y*′′ denote the weak derivatives and ∥⋅∥_{2} denotes the norm of

According to the method of complex scaling ([21,22], [23], §5 and also [24]), for every *θ*∈[0,*π*/4), the spectral problem (2.1) is equivalent to the spectral problem for the operator *H*_{θ} in *z* is an eigenvalue of (2.2) if and only if λ=*e*^{−2iθ}*z* is a resonance of (2.1) or, equivalently, if *ϵ*=(λ+1.6)/2=(*e*^{−2iθ}*z*+1.6)/2 is a resonance of (1.1).

Because *q*_{θ} is even, the spectral problem (2.2) for the operator *H*_{θ} is equivalent to the two spectral problems
*τ*_{θ}*y*:=−*y*′′+*q*_{θ}*y* in

If *y*_{0}, *γ*=±1. Because *y*_{0} and *y*_{0}′ in 0 yields that *y*_{0}′(0)=0 if *γ*=1 and *y*_{0}(0)=0 if *γ*=−1. Hence, _{0} by setting *y*_{0}(*x*):=−*y*_{0}(−*x*), *y*_{0}(*x*):=*y*_{0}(−*x*),

Because the potential *q*_{θ} is complex-valued and hence all the above operators *H*_{θ} along with

Analytic bounds for resonances are commonly based on numerical range estimates for each complex-scaled problem (2.2) with *θ*∈[0,*π*/4) (comp. [14]). For the set of resonances of (1.1), we obtain the following result.

### Theorem 2.1

*The resonances of (*1.1*) in the sector* *are contained in the closed convex set
**where* *for θ∈[0,π/4) with
*

### Proof.

The set *a*_{+}(*θ*)≥0 and hence *ϵ*)≤0.8 and Im(*ϵ*)≤0.

Thus, it is sufficient to show that every resonance *ϵ*_{0})>0.8, Im(*ϵ*_{0})≤0 belongs to *θ*∈[0,*π*/4], the point λ_{0}:=2*ϵ*_{0}−1.6 lies in the sector *a*_{+}(*θ*), we define
*θ*∈[0,*π*/4),
_{0} of *L* with

Figure 1 shows that the only available analytic information is much too coarse to judge the validity or non-validity of resonance approximations. Therefore, it is necessary to employ a method yielding both guaranteed and much more accurate enclosures and exclosures for eigenvalues of non-self-adjoint eigenvalue problems.

## 3. Eigenvalue enclosures for complex-valued potentials

The algorithm we use to establish guaranteed eigenvalue enclosures was developed and described in detail in [25,16]. Briefly, it consists of the following two steps. For the sake of simplicity, we consider the Dirichlet problem (2.4); the approach to the Neumann problem (2.5) is completely analogous.

*Step* *A*. *Solving a truncated problem with guaranteed error bounds*. In order to truncate problem (2.4), we restrict the potential *q*_{θ} to an interval [0,*X*] and set it equal to 0 on *y*′′=*zy* in *X*], we have to solve is

The algorithms for the calculation of the analytic function Δ and for the contour integral over a chosen starting box *n*_{0} of eigenvalues of the truncated problem (3.1). Repeating this procedure by suitably subdividing the box *ε*_{Z} that contains exactly one eigenvalue *z*_{trunc}.

*Step* *B*. *Use Levinson asymptotics to enclose the eigenvalues of problem* (2.4). If *y*_{2}(⋅,*z*) is the unique (suitably normalized) solution of the differential equation in (2.4) belonging to *z*_{true} is an eigenvalue of (2.4) if and only if *y*_{2}(0,*z*_{true})=0. Levinson's theorem (see e.g. [25], theorem 3.3) shows that
*X*≥0 such that *α*_{X,θ}<1. Hence, if *y*_{2}(⋅,*z*)] is an interval-valued solution of the truncated problem on [0,*X*] satisfying the interval-valued initial conditions
*y*_{2}(0,*z*)∈[*y*_{2}(0,*z*)]. By means of the interval arithmetic argument principle already used in step A, we now obtain enclosures for the zeros of *y*_{2}(0,*z*), and hence for the eigenvalues *z*_{true} of (2.4) of desired precision.

For the above-described method, several parameters have to be provided; in particular, the length *X* of the truncated interval has to be determined such that *α*_{X,θ}<1. To this end, we note that
*a*≥0,
*θ*∈[0,*π*/4) with *A*_{X,θ} to obtain a rigorous computable upper bound *A*_{X,θ} and hence for *α*_{X,θ},
*x*∈[0,2*θ*]⊂[0,*π*/2]. If *θ* is a decimal fraction whose fractional part has three digits, the sum *T*_{X,θ}(*m*) is rational and can be evaluated exactly. We choose a rigorous computable lower bound *T*_{X,θ}(*m*) as the unique decimal number whose fractional part has six digits and *f*(*t*):=(10/*t*) *e*^{−tX2/10}(*X*+5/*tX*), *t*∈(0,1), is decreasing, hence, again by Taylor's theorem with remainder in Lagrange form, we obtain that, for all *m*, *T*_{X,θ}(*m*) to obtain a rigorous computable upper bound *A*_{X,θ}(*m*,*n*) with *θ*↦*A*_{X,θ}(*m*,*n*), *θ*∈[0,*π*/4), is increasing, the rigorous computable upper bound *θ*_{0}:=0.75<*π*/4 is also an upper bound of *A*_{X,θ}(*m*,*n*) for *θ*∈(0,*θ*_{0}). Only in two of our computations (for the resonances numbered 37^{−} and 44^{+}), we needed parameter values *θ* that are larger than *θ*_{0}=0.75; their upper bound *X*=50, *m*=2, *n*=32 and obtain the rigorous computable lower bounds *X*=50 the condition *θ*≤0.785238 very close to *π*/4∼0.7853981635.

## 4. Guaranteed resonance enclosures and exclosures

In [10], reply to comment, Rittby *et al.* listed a set of 44 approximate resonances *et al.* in [5], comment; here, the superscript + occurs for even *k*, whereas − occurs for odd *k*. The differences in modulus between these two approximate resonance strings are smaller than 2⋅10^{−3} up to ^{−2} from

We computed guaranteed enclosures for all 44 approximate resonances by Rittby *et al.* from [4] as well as exclosures for the approximate resonances *et al.* from [5], comment. In addition, we enclosed the two pairs of resonances discovered numerically in [14] that are visible by the complex scaling method.

All computed enclosures for resonances, except for one of these pairs, were performed with interval length *X*=50, varying scaling angle *θ* as displayed in the tables, and corresponding guaranteed upper bound *α*_{X,θ} as in table 1 at the end of §3. The enclosure for one of the additional resonance pairs in [14] turned out to be by far more challenging than all other computations.

We employ the interval arithmetic-based software library VNODE developed by Nedialkov *et al.* (see [15]) where all operations are performed with complex ‘intervals’, i.e. rectangles [*z*]=[*x*]+[*y*]*i*, where [*x*], *k* and parity ± in the list of approximate resonances of Rittby *et al.* in [10, reply to comment, table I].

Note that the resonances coming from the boundary condition *y*(0)=0 have parity ‘−’, because the eigenfunctions of the corresponding eigenvalues of (2.4) are odd, whereas those coming from the boundary condition *y*′(0)=0 have parity ‘+’, because the eigenfunctions of the corresponding eigenvalues of (2.5) are even (see §2).

### (a) Guaranteed enclosures for resonance approximations by Rittby *et al.*

First, we present the computed enclosures for the 44 resonances

Table 2 contains the enclosures for resonances λ=*e*^{−2iθ}*z* via enclosures for eigenvalues *z* of (2.2) restricted to *y*(0)=0, i.e. eigenvalues of problem (2.4); table 3 contains the corresponding enclosures using eigenvalues *z* of (2.2) restricted to *y*′(0)=0, i.e. for eigenvalues of problem (2.5). Table 4 contains the enclosures for the 44 resonances

The enclosing boxes for the resonances *ϵ* of (1.1) are obtained from the enclosing boxes for the eigenvalues *z* of (2.2) as follows. If [*u*_{1},*u*_{2}]+[*v*_{1},*v*_{2}]*i* is an enclosing box in the *z*-plane, then the enclosing box [*x*_{1},*x*_{2}]+[*y*_{1},*y*_{2}]*i* for a resonance λ=e^{−2iθ}*z* of (2.1) is the smallest axis-parallel box that contains the rotated box e^{−2iθ}([*u*_{1},*u*_{2}]+[*v*_{1},*v*_{2}]*i*). The corresponding enclosing box for a resonance *ϵ*=(λ+1.6)/2 of (1.1) is obtained from

The values of the 44 approximate resonances of (1.1) listed in [10], reply to comment, table I, which were computed by Rittby *et al.* in floating point arithmetic without error bounds, are displayed in the right column in table 4; they agree with our enclosures at least up to order 10^{−4}. Thus, our guaranteed enclosures prove that all values computed by Rittby *et al.* do indeed lie near true resonances.

### (b) Guaranteed exclosures for resonance approximations by Korsch *et al.*

On the other hand, we applied our method to the numerical values of the resonance approximations of Korsch *et al.* numbered 16^{+},17^{−},…,27^{−}, 28^{+} in [10], reply to comment, table II; note that the resonance approximations 29^{−},…,40^{+} therein can not be seen by the complex scaling method.

Using larger boxes around these numerical values, we found that in each case the interval-valued argument principle yields an interval [*c*_{1},*c*_{2}] with *l*_{k}∈[0.1,2] are listed in table 5. For every resonance approximation *z*-plane is denoted by *l*_{k} in the *z*-plane is chosen such that
*ϵ*-plane is scaled and rotated owing to the relation *l*_{k}/2 and is rotated clockwise by the angle 2*θ* around the midpoint *m*_{k}:=(e^{−2iθ}*M*_{k}+1.6)/2. The minimal distance *d*_{k} of *d*_{k}>(*l*_{k}/4)−0.025≥0 (figure 3).

Hence, our guaranteed exclosures prove that none of the numerical values of Korsch *et al.* numbered 16^{+},17^{−},…,27^{−},28^{+} lies near a true resonance of (1.1).

### (c) Enclosures of resonance approximations by Abramov *et al.*

Finally, we considered the three pairs of additional resonances found in [14, p. 72], one pair near each of the points
*ϵ*−1.6 are

We computed guaranteed enclosures for the two pairs of resonances near *z* of (2.4) with boundary condition *y*(0)=0 with odd eigenfunction (denoted by superscript ‘−’) and one eigenvalue *z* of (2.5) with boundary condition *y*′(0)=0 with even eigenfunction (denoted by superscript ‘+’). The guaranteed enclosures we obtained for the four resonances

The computation of the resonance pair *a*. Choosing *θ*=0.735, our provably correct computations showed that for each of the two boundary conditions there is only one resonance ^{−2iθ}([23,24]+[0.05,1]*i*) containing these two boxes as well as the numerical value *et al.* there is only one resonance for each of the two boundary conditions. Altogether, we thus proved that there is precisely one pair of disjoint resonances *et al.* and that this approximation has distance approximately 1⋅10^{−1} to the true resonance pair

The computation of the resonance pair *R*>0 in addition to rotation of the variable by an angle *θ*∈[0,*π*/4). The potential *q*_{θ,R} and the eigenvalue parameter *z* in the spectral problem for the corresponding operator *H*_{θ,R} (compare (2.2), (2.3)) then become
*R*=1.

In order to apply Levinson's theorem, we needed to find suitable *X*≥0, *θ*∈[0,*π*/4) and *R*>0 such that *α*_{X,θ,R}<1. Proceeding as for usual complex scaling, instead of (3.3), we used

The main benefit of the additional stretching is that the upper bound *A*_{X,θ,R} decays exponentially fast in *R*. As for usual complex scaling, we then applied Taylor's theorem with remainder in Lagrange form to obtain the rigorous computable upper bound *X*=10, *θ*=0.76 and *R*=10.

With these parameters, we succeeded to enclose the resonances *y*(0)=0 and *y*′(0)=0, respectively. The corresponding values in the *z*-plane are both in the box *R*^{−2} e^{−2iθ}(1702.54+5.35*i*)≈0.918−17.001*i* and side length *R*^{−2}10^{−1}=1⋅10^{−3}, rotated clockwise by the angle 2*θ*=1.52; the second set, which is the one displayed in table 6, is the smallest axis-parallel box containing this rotated box. Note that these enclosures for ^{−2} from the value *et al.* [14].

Hence, our guaranteed enclosures prove that not far from each of the two numerically computed values *et al.* there is indeed a pair of true resonances of (2.1); the distance is approximately 2⋅10^{−2} for ^{−1} for

## 5. Conclusion

In this paper, we have presented a method which, for the first time, permits one to compute resonances in atomic physics with absolute certainty. At the same time, it allows one to detect with absolute certainty wrongly computed resonance approximations. The absolute reliability of our approach is based on a combination of interval arithmetic and the argument principle. To prove the efficiency of our method, we have established guaranteed *en*closures for all numerical resonance approximations of Rittby *et al.* in [4,7] for problem (1.1) and guaranteed *ex*closures for the numerically computed values of Korsch *et al.* in [5] that are visible to complex scaling, thus definitely settling a dispute between these two groups of authors. The greatest challenge was to provably enclose two additional pairs of approximate resonances computed by Abramov *et al.* in [14] that were found neither by Rittby *et al.* nor by Korsch *et al.* Thus, we have proved the conjecture in [4,7] that the real parts of auto-ionizing resonances of certain atoms and molecules exhibit an oscillatory behaviour beyond a threshold and we have added new information on this threshold originating in the two new confirmed pairs of resonances.

Figure 4*a* displays all our results in the rectangle 0≤Re(λ)≤15, −70≤Im(λ)≤0: in the top right corner of the λ-plane, the analytic exclusion from theorem 2.1 (grey-shaded), the enclosed approximate resonances 1^{−},…,38^{+} of Rittby *et al.* surrounded by circles, the additional ones by Abramov *et al.* as star and square, and the claimed approximate resonances 1^{−},…,29^{−} of Korsch *et al.* as asterisks; note that the resonances 0^{+}, 29^{−} and ^{+},…,28^{+} of Korsch *et al.,* our excluding box is shown (grey-shaded). Figure 4*b* illustrates that for resonance 16^{+} it was especially difficult to find a box that simultaneously *ex*cludes the computed value of Korsch *et al.* and does *not* contain the value computed by Rittby *et al.*

## Acknowledgements

The authors gratefully acknowledge the support of the following funding bodies: Swiss National Science Foundation, SNF, grant no. 200020_146477 (S. Bögli, C. Tretter); EPSRC grant no. GR/R46410/01 (B.M. Brown); Leverhulme Trust grant no. RPG 167 (M. Marletta). C. Tretter thanks the Isaac Newton Institute of Mathematical Sciences, Cambridge, UK, where a first draft of this paper was written. All authors thank H. Siedentop for drawing their attention to this problem.

- Received June 20, 2014.
- Accepted September 1, 2014.

© 2014 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.