COMPUTING KLEIN-GORDON SPECTRA FRANK R OSLER AND CHRISTIANE TRETTER Abstract. We study the computational complexity of the eigenvalue problem for the Klein-Gordon

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COMPUTING KLEIN-GORDON SPECTRA

FRANK R ¨

OSLER AND CHRISTIANE TRETTER

Abstract. We study the computational complexity of the eigenvalue problem for the Klein-Gordon

equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue

of the Klein-Gordon equation with linearly decaying potential can be computed in a single limit with

guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm

that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum

of the Klein-Gordon equation and we compare our numerical results to these enclosures. Finally, we

apply both the implemented algorithm and our abstract enclosures to several physically relevant

potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.

1. Introduction and Main Results

Reliably computing eigenvalues and spectra poses major challenges across physics, spanning the wide

range from computing resonances in quantum mechanics over the wealth of extremely hard computa-

tional spectral problems in hydrodynamics and electromagnetics to the new era of spectral light tuning

in metamaterials, see e.g. [10], [4], [12], [2]. These challenges are due to inherent properties of physical

systems including lack of symmetry, coupling of diﬀerent phenomena or inﬁnite dimensional eﬀects

such as non-discrete spectrum which may cause serious failures of domain truncation methods or ﬁnite

dimensional approximations, see e.g. [16], [33]. There are two such highly undesirable failures: spectral

pollution, where ﬁnite dimensional eigenvalue approximations converge to a point that is not a true

spectral point and spectral invisibility where a true spectral point is not seen by the ﬁnite dimensional

approximations, see e.g. [13], [11], [3]. Therefore there is an urgent need for tools that allow physicists

and applied mathematicians to assess whether their computational problems are prone to these failures

and to estimate the complexity of the required computational tasks.

A key breakthrough in this direction was the introduction of the Solvability Complexity Index (SCI)

by A. Hansen in 2011 [25] and the SCI Hierarchy by J. Ben-Artzi et al. in 2015 [6] which has opened

up a completely new pathway to the analysis and numerics of spectral problems and sparked progress

for many computational problems in physics. The SCI hierarchy oﬀers a framework to compare the

complexity of computational tasks such as approximating the spectrum in Hausdorﬀ distance (or the

Attouch-Wets metric in the unbounded case) for diﬀerent classes of linear operators. This classiﬁcation

involves the number of successive limits required for the approximation, the availability of error bounds

and, in its ﬁnest form, also the type of algorithm such as arithmetic.

The enormous impact of the SCI hierarchy in computational spectral theory has several reasons.

First, it revealed why even the most common open problems such as computing the eigenvalues of

Schr¨odinger operators with bounded, even real-valued potentials using point-values of the potential

have remained open for more than 9 decades since quantum mechanics was created. Secondly, it

helped to solve such longstanding open problems not only for compact, bounded, selfadjoint and normal

operators [6, Thm. 7.5], but also for unbounded operators with certain resolvent growth and non-empty

essential spectrum. Thirdly, the corresponding results are not only theoretical, but they yield practically

implementable algorithms which have been shown to perform well for large scale problems in physics [21].

Date: October 25, 2022.

2010 Mathematics Subject Classiﬁcation. 81Q05, 47N40, 68Q25, 35P05, 81R20, 35P15.

Key words and phrases. Klein-Gordon Equation, Solvability Complexity Index, Computational complexity, Spectral

Theory, Eigenvalue approximation, Eigenvalue bounds, Mathematical Physics.

FR acknowledges support from the European Union’s Horizon 2020 Research and Innovation Programme under the

Marie Sklodowska-Curie grant agreement no. 885904. CT gratefully acknowledges the support of the Swiss National

Science Foundation (SNF), grant no. 200021 204788.

arXiv:2210.12516v1 [math.SP] 22 Oct 2022

2 FRANK R ¨

OSLER AND CHRISTIANE TRETTER

The urgent need to classify computational problems in spectral theory in the SCI Hierarchy is further

substantiated by parallel research on challenges in spectral approximation such as spectral pollution or

spectral invisibility and methods to avoid them, see e.g. [10], [11]. In recent years, since the fundamental

work [6] which contains a detailed study on computing the spectra and pseudospectra of bounded matrix

operators on `2pNqas well as results on computing spectra and pseudospectra of Schr¨odinger operators

with potentials satisfying a uniform BV bound (cf. [6, Thm. 8.3, 8.5]), the SCI theory has been further

developed in multiple directions. We mention the works [17,19,39] on matrix operators; [5,18] on

solving PDEs; [20] on the computation of spectral measures, and [7,8] on the computation of scattering

resonances.

While many of the above works obtained results on the (nonrelativistic) Schr¨odinger equation, no

SCI results seem to exist so far on relativistic models, such as the Klein-Gordon or Dirac equations. The

present article begins to ﬁll this gap: we study the spectral problem for the Klein-Gordon equation in the

framework of the SCI Hierarchy. The novelty of this contribution is computational, physically relevant

as well as mathematical. Indeed, the spectral problem for the Klein-Gordon equation is non-standard

in the sense that one is lead to the study of quadratic operator pencils rather than classical eigenvalue

equations which lead to linear monic pencils. Moreover, essential spectrum is not only present but it is

not semi-bounded; even bounded symmetric potentials may create complex eigenvalues in addition to the

two unbounded rays of real essential spectrum. Accordingly, new methods are needed to study spectral

computation which require an intimate interplay of analysis and numerics with operator theory. The

development and evaluation of these techniques, along with corresponding implementable algorithms,

are the focus of this article.

In the next two subsections we present our main results on the computational spectral problem for the

Klein-Gordon equation and we introduce the SCI hierarchy allowing us to interpret our results therein.

1.1. The Klein-Gordon equation. In quantum mechanics the Klein-Gordon equation

(1.1) ˆ´´´i~B

Bt´eϕ¯2

`c2´´i~∇´e

A¯2

`m2c4˙U“0

describes the motion of a relativistic spinless particle with mass mand charge ein an electromagnetic

ﬁeld with scalar potential ϕand vector potential ~

A; here cis the speed of light and ~is the Planck

constant. If we separate time by setting Upx, tq“: eiλ{~tupxq,xPRd,tPR, normalize cto 1, let Vbe

the multiplication operator by eϕ in L2pRdqand A0:“p´i~∇´e~

Aq2, then (1.1) leads to a quadratic

eigenvalue problem in λ, which we will cast in a more abstract framework.

To this end, let pH,x¨,¨yq be a separable Hilbert space, let A0be a nonnegative operator on Hand

H0:“A0`m2with mą0. If Vis a symmetric operator with dom H1{2

0Ădom V, the operator

V H´1{2

0is bounded in H. Then the quadratic eigenvalue problem associated with (1.1) is of the form

TVpλqu“0, λPC, where the Klein-Gordon operator polynomial (or pencil) in His given by

TVpλq:“H0´ pV´λq2,dom TVpλq “ dom H0, λ PC.

If we assume that S:“V H´1{2

0“S0`S1with a strict contraction S0and compact S1, as in [28], [29],

[30], then it is well-known that the essential spectrum of TVhas a gap around 0 and that the non-real

spectrum of TVis discrete, see [28], [29]. Moreover, one can show that if S0“0, then the essential

spectrum of TVis given by the half lines p´8,´msYrm, 8q and any other spectral points are discrete

eigenvalues. We ensure that S0“0 by making the following assumptions throughout the paper.

Hypothesis 1.1. Unless otherwise stated, assume that H“L2pRdq, and ~

A“0, i.e. A0“ ´∆,

dompA0q “ H2pRdq. Moreover, assume that

(i) VPW1,ppRdqfor some pąd,

(ii) there exists a constant Mą0 such that

}V}W1,ppRdqďM, |Vpxq| ď Mp1` |x|2q´1

2for all xPRd.(HM)

It is easy to see from the Fr´echet-Kolmogorov-Riesz theorem [34, Th. XIII.66] that Hypothesis 1.1

implies compactness of V H´1{2

0and thus S0“0. Therefore

σesspTVq “ σesspT0q“tλPC|λ2PσpH0qu “ ´aσpH0q Y aσpH0q,

COMPUTING KLEIN-GORDON SPECTRA 3

i.e. σesspTVqis independent of Vas long as Vsatisﬁes Hypothesis 1.1. Our main result concerns the

convergence of the numerical approximation of the spectrum σpTVq. Here, for subsets A, B ĂC, let

dHpA, Bqdenote their Hausdorﬀ distance.

Theorem 1.2. Let Vsatisfy Hypothesis 1.1. Then there exist computational routines Γn(depending

only on p,M), which take their input from the set of point values of Vand produce sets ΓnpVq Ă C

such that

dH`ΓnpVq, σpTVq˘Ñ0for nÑ 8.(i)

If the constant Min (HM)is known a-priori, then in addition to (i)we obtain the error bound

sup

zPσpTVq

distpz, ΓnpVqq ď 1

nfor all nPN(ii)

pfor details, see Theorem 6.9).

Remark 1.3.(i) Theorem 1.2 implies a classiﬁcation result in the Solvability Complexity Index Hier-

archy. For details, see Corollary 2.9 below.

(ii) Not only does Theorem 1.2 give an existence result, but its proof is constructive. In Deﬁnition 6.7

below we give a concrete deﬁnition of an algorithm that achieves (i), (ii) and can be implemented

on a computer. Indeed, in Section 7.2 we present numerical results obtained from a MATLAB

implementation of our algorithm.

1.2. The SCI Hierarchy. The SCI Hierarchy, introduced by Hansen [25], is a novel and rapidly

developing ﬁeld. In its initial form, the Solvability Complexity Index gave a rigorous framework to

study the “number of successive limits” needed for the numerical approximation of spectral problems.

Since then, it has evolved into a general theory of computability and error estimation for arbitrary

computational problems in mathematics. The starting point of the theory is the following rigorous

deﬁnition of a computational problem.

Deﬁnition 1.4 (Computational problem).Acomputational problem is a quadruple pΩ,Λ,M,Ξq, where

(i) Ω is a set, called the primary set,

(ii) Λ is a set of complex-valued functions on Ω, called the evaluation set,

(iii) Mis a metric space,

(iv) Ξ : Ω ÑMis a map, called the problem function.

Example 1.5.As an example, we show how a rigorous formulation of the task “compute the spectrum

of a compact operator from its matrix elements” ﬁts into Deﬁnition 1.4.

Let Ω “S8p`2pNqq be the space of compact operators on the Hilbert space `2pNq. We can formulate

the computation of the spectrum of an operator from Ω as a computational problem in the following

way. Let teiuiPNdenote the canonical basis of `2pNqand deﬁne fij : Ω ÑC;fij pAq “ xei, Aejy

and Λ :“ tfij p¨q|i, j PNu. Next, let Mbe the set of closed bounded subsets of C, endowed with the

Hausdorﬀ distance dHand let Ξ : Ω ÑM; ΞpAq “ σpAqbe the function mapping a linear operator onto

its spectrum. Then the quadruple pΩ,Λ,M,Ξqis a computational problem in the sense of Deﬁnition

1.4. The above example gives an intuitive meaning to the building blocks (i)-(iv):

(i) Ω is the set of objects that give rise to the computational problem,

(ii) Λ is the set of information an algorithm is allowed to access during the computation,

(iii) the metric on Mmeasures the approximation error,

(iv) the image of the function Ξ contains the objects to be computed.

Moving on from Deﬁnition 1.4, we would like to be able to say when a computational problem is

considered “solved” and “how good” it has been solved (see questions (1)-(4) below). To this end, we

proceed by introducing a rigorous deﬁnition of what we mean by an algorithm.

Deﬁnition 1.6 (General algorithm).Let pΩ,Λ,M,Ξqbe a computational problem. A general algorithm

is a mapping Γ : Ω ÑMsuch that for each TPΩ

(i) there exists a ﬁnite (non-empty) subset ΛΓpTq Ă Λ,

(ii) the action of Γ on Tdepends only on tfpTqufPΛΓpTq,

(iii) for every SPΩ with fpTq “ fpSqfor all fPΛΓpTqone has ΛΓpSq “ ΛΓpTq.

4 FRANK R ¨

OSLER AND CHRISTIANE TRETTER

We will sometimes write ΓpTq “ ΓptfpTqufPΛΓpTqqto emphasise point (ii) above: the output ΓpTq

depends only on the (ﬁnitely many) evaluations tfpTqufPΛΓpTq.

Deﬁnition 1.6 above is a general abstraction of what a computer algorithm does: it takes a ﬁnite

amount of input (i.e. tfpTqufPΛΓpTq) and returns some output in a metric space (usually a string of

numbers). The term general algorithm is used, because Deﬁnition 1.6 imposes no restrictions on how

the output ΓpTqis computed. We will specify more restrictive types of algorithms in the next section

by introducing a recursiveness constraint (the reader may think of the way a Turing machine produces

its output). Given the above deﬁnitions one can ask the following questions:

(1) Given a computational problem pΩ,Λ,M,Ξq, does there exist a sequence of algorithms pΓnqnPN

such that ΓnpTq Ñ ΞpTqin Mfor all TPΩ?

(2) If so, do we have explicit error bounds, i.e. dpΓnpTq,ΞpTqq ă εnfor a known sequence pεnqnPN?

(3) If Mhas certain ordering properties, do we have error bounds from above / below?

(4) If all of the above fail, does it help to “add limits”? More precisely, does there exist a family

pΓnk,...,n2,n1qn1,...,nkPNof algorithms with limnkÑ8 ¨¨¨limn1Ñ8 Γnk,...,n1pTq“ΞpTqfor all TPΩ?

These questions are nontrivial: there exist examples and counterexamples to every single one of the

above questions. In particular, there exist examples for question (4), i.e. computational problems that

inherently require more than 1 limit to solve [6, Th. 7.5]. This shows that it is not enough to merely

consider algorithms alone and motivates the following deﬁnition.

Deﬁnition 1.7 (Tower of general algorithms).Let pΩ,Λ,M,Ξqbe a computational problem. A tower of

general algorithms of height kfor pΩ,Λ,M,Ξqis a family Γnk,nk´1,...,n1: Ω ÑMof general algorithms

(where niPNfor 1 ďiďk) such that for all TPΩ

ΞpTq “ lim

nkÑ`8 ¨¨¨ lim

n1Ñ`8 Γnk,...,n1pTq.

If a computational problem requires a tower at least of height kto solve it, then we say the problem

has Solvability Complexity Index k. Notice that towers of algorithms are commonplace in applications.

Indeed, numerical approximation usually consists of a sequence (or a tower ) of algorithms, with in-

creasingly large input and increasingly precise output. As a prime example, the reader may think of

the Finite Element method, in which the input of the algorithm consists of the data of a PDE at the

mesh points and the output consists of a sequence of numbers representing the approximate values of

the solution at those mesh points.

While precise deﬁnitions will follow in the next section, we can now give an informal deﬁnition of

the SCI Hierarchy. Given a computational problem pΩ,Λ,M,Ξq, we deﬁne the classes

∆2:“ tpΩ,Λ,M,Ξq|the answer to (1) is yesu,

∆1:“ tpΩ,Λ,M,Ξq|the answer to (2) is yesu,

Π1:“ tpΩ,Λ,M,Ξq P ∆2|there exist error bounds from aboveu,

Σ1:“ tpΩ,Λ,M,Ξq P ∆2|there exist error bounds from belowu.

In addition, higher classes ∆j, Πj, Σjcan be deﬁned, based on towers of algorithms, instead of individual

sequences (see Section 2). One obtains a sequence of classes

∆1Ă pΣ1XΠ1qĂpΣ1YΠ1q Ă ∆2Ă pΣ2XΠ2q Ă ¨¨¨

into which computational problems fall. This is the SCI Hierarchy.

Within this so-called SCI Hierarchy our main results Theorems 1.2 and 2.1 mean that the compu-

tational spectral problem for the Klein-Gordon equation in Rdbelongs to the class ∆A

2if the potential

Vsatisﬁes Hypothesis 1.1 with pąd, and it belongs to the subclass ΠA

1Ă∆A

2if the constant Min

(HM) is explicitly known; here the supscript Ameans that the algorithm to approximate the spectrum

can be chosen to be arithmetic.

2. Preliminaries

2.1. Spectral theory of the Klein-Gordon equation. The spectral properties of the operator

polynomial TVare intimately related to the spectral properties of the operator polynomial

LVpλq:“I´`S˚´λH´1

0˘`S´λH´1

0˘, λ PC.

COMPUTING KLEIN-GORDON SPECTRA 5

which arises from TVby means of the quasi-similarity transformation LV“H´1{2

0TVH´1{2

0and whose

values are bounded linear operators. In particular, σppTVq “ σppLVqand σesspTVqXR“σesspLVq XR

by [30, Prop. 2.3]. Hence we can use LVto compute the point spectrum of TV. In the sequel we will

construct ﬁnite-dimensional approximations of LVin order to approximate σppLVq.

A standard calculation shows that, for λ2RσpH0q,

LVpλq“pI´λ2H´1

0qrI´Kpλqs(2.1)

where

Kpλq:“ pI´λ2H´1

0q´1´S˚S´λ`S˚H´1

0`H´1

0S˘¯;(2.2)

note that compactness of Simplies compactness of Kpλqfor all λwith λ2PρpH0q. The factorization

(2.1) implies that LVpλqis invertible if both I´λ2H´1

0and I´Kpλqare invertible. The ﬁrst factor is

invertible if and only if λ2RσpH0q, hence we obtain the characterisation

σpTVqzt˘aσpH0qu“σpLVqzt˘aσpH0qu“tλPC|I´Kpλqnot invertibleuzt˘aσpH0qu.(2.3)

The proof of Theorem 1.2 relies heavily on the following theorem, which we prove in Sections 3-5.

Theorem 2.1 (Computable approximation of Kpλq).Assume that Vsatisﬁes Hypothesis 1.1 and let

λPCzt˘aσpH0qu “ Czpp´8,´msYrm, 8qq. Then there exists a sequence of matrix approximations

pKnpλqqnPN, such that each Knpλqcan be computed in ﬁnitely many arithmetic operations from the

values tVpxq|xPQduand ››Kpλq ´ Knpλq››L2ÑL2ďC

where Cą0is given explicitly in terms of M,m,d,p,λpfor details, see Theorem 5.4 belowq.

2.2. The SCI Hierarchy in depth. In this section we dive deeper into the theory of the SCI Hierarchy.

This will allow us to fully understand the implications of Theorem 1.2 for the computational complexity

of the Klein-Gordon eigenvalue problem.

Deﬁnition 2.2 (Recursiveness).Suppose that for all fPΛ and for all TPΩ we have fpTq PRor C.

We say that Γ “Γnk,nk´1,...,n1is recursive if Γnk,nk´1,...,n1ptfpTqufPΛΓpTqqcan be executed by a Blum-

Shub-Smale (BSS) machine [9] that takes pn1, n2, . . . , nkqas input and that has an oracle that can

access fpTqfor any fPΛ.

Example 1.6 (continued).Let pΩ,Λ,M,Ξqbe as in Example 1.5. The following sequence of algorithms

has been shown to converge to σpAqfor any APΩ (with respect to dH) in [6]:

For nPNlet Ln:“n´1pZ`iZq X Bnp0qbe a ﬁnite lattice in the complex plane and, for APΩ“

S8p`2pNqq, let ΛΓnpAq:“ tfij |i, j ďnu. Note that in this case ΛΓnpAqdoes not actually depend on A,

hence condition (iii) of Deﬁnition 1.6 is automatically satisﬁed. The algorithm Γnis now deﬁned by

ΓnpAq:“ zPLnˇˇ}pAn´zq´1} ą n(,(2.4)

where Andenotes the upper left nˆnblock obtained by truncating the matrix representation of A.

Clearly, the action of Γnonly depends on those fij with i, j ďn, i.e. Deﬁnition 1.6 (ii) is satisﬁed. It

can also be shown (cf. [6, Th. 7.5 (i)]) that Γnis recursive in the sense of Deﬁnition 2.2 for any nPN

and that dHpΓnpAq, σpAqq Ñ 0 as nÑ 8 for any APΩ.

We stress that convergence of the sequence Γnis only guaranteed for compact operators APΩ as in

the example above. Indeed, for larger problem sets (e.g. the set of bounded operators Lp`2pNqq) it can be

shown that there does not exist any sequence of recursive algorithms Γnsuch that dHpΓnpAq, σpAqq Ñ 0

as nÑ 8 for all APLp`2pNqq (cf. [6, Thm. 7.5 (i)]). This motivates the next deﬁnition.

Deﬁnition 2.3 (Tower of arithmetic algorithms).Given a computational problem pΩ,Λ,M,Ξqwhere

Λ is countable, a tower of arithmetic algorithms for pΩ,Λ,M,Ξqis a general tower of algorithms

where the lowest mappings Γnk,...,n1: Ω ÑMsatisfy the following: For each TPΩ the mapping

NkQpn1, . . . , nkqÞÑΓnk,...,n1pTq“Γnk,...,n1ptfpTqufPΛpTqqis recursive, and Γnk,...,n1pTqis a ﬁnite string

of complex numbers that can be identiﬁed with an element in M.

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COMPUTINGKLEIN-GORDONSPECTRAFRANKROSLERANDCHRISTIANETRETTERAbstract.WestudythecomputationalcomplexityoftheeigenvalueproblemfortheKlein-GordonequationintheframeworkoftheSolvabilityComplexityIndexHierarchy.WeprovethattheeigenvalueoftheKlein-Gordonequationwithlinearlydecayingpotentialcanbecomputedinas...

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COMPUTING KLEIN-GORDON SPECTRA FRANK R OSLER AND CHRISTIANE TRETTER Abstract. We study the computational complexity of the eigenvalue problem for the Klein-Gordon

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