Quasi-Normal Modes from Bound States The Numerical Approach Sebastian H. Völkel 1 2 3

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Quasi-Normal Modes from Bound States: The Numerical Approach

Sebastian H. Völkel 1, 2, 3

1Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam, Germany

2SISSA - Scuola Internazionale Superiore di Studi Avanzati,

via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste

3IFPU - Institute for Fundamental Physics of the Universe, via Beirut 2, 34014 Trieste, Italy∗

(Dated: May 15, 2023)

It is known that the spectrum of quasi-normal modes of potential barriers is related to the spec-

trum of bound states of the corresponding potential wells. This property has been widely used to

compute black hole quasi-normal modes, but it is limited to a few “approximate” potentials with cer-

tain transformation properties for which the spectrum of bound states must be known analytically.

In this work we circumvent this limitation by proposing an approach that allows one to make use

of potentials with similar transformation properties, but where the spectrum of bound states can

also be computed numerically. Because the numerical calculation of bound states is usually more

stable than the direct computation of the corresponding quasi-normal modes, the new approach

is also interesting from a technical point of view. We apply the method to diﬀerent potentials,

including the Pöschl-Teller potential for which all steps can be understood analytically, as well as

potentials for which we are not aware of analytic results but provide independent numerical results

for comparison. As a canonical test, all potentials are chosen to match the Regge-Wheeler potential

of axial perturbations of the Schwarzschild black hole. We ﬁnd that the new approximate potentials

are more suitable to approximate the exact quasi-normal modes than the Pöschl-Teller potential,

particularly for the ﬁrst overtone. We hope this work opens new perspectives to the computation

of quasi-normal modes and ﬁnds further improvements and generalizations in the future.

I. INTRODUCTION

Although black hole perturbation theory is in general

a non-trivial ﬁeld of research, it has provided some sur-

prisingly simple results. In general relativity, as well as

in many modiﬁed theories of gravity, it is possible to

derive so-called master equations that break down the

full scale of the problem into ﬁnding complex frequency

eigenvalues of eﬀective potentials in a one-dimensional

Schrödinger equation or modiﬁcations of it [1–4]. Non-

rotating black holes are signiﬁcantly easier to treat and

it is quite generic to ﬁnd single or coupled wave equa-

tions also in modiﬁed gravity. However, similar results

for rotating black holes are limited to general relativity

and very few speciﬁc theories for which the complicated

calculations could be carried out or are limited to small

spins, see Refs. [5–9] for recent developments.

The close relation to quantum mechanics immediately

calls for similar methods to solve the ﬁnal eigenvalue

problem and the literature on adopted as well as new

methods has grown immensely, see e.g. Refs. [10–14] for

reviews. Methods to compute black hole quasi-normal

modes range from purely analytic, semi-analytic and fully

numerical ones. The question of which method to choose

from ultimately depends on the speciﬁcs of the problem

and the desired insights that should be gained. Methods

that provide quasi-normal mode frequencies with pristine

precision, such as the Leaver method [15], do not pro-

vide analytic results and might be diﬃcult to adjust for

∗sebastian.voelkel@aei.mpg.de

new potentials. Analytic approaches, such as the appli-

cation of the Wentzel-Kramers-Brillouin (WKB) method,

can provide analytic results, but those are approximate

and may not apply to all parts of the quasi-normal mode

spectrum [16–24]. For example, the higher order WKB

method can be very precise for the ﬁrst few overtones,

but it qualitatively fails for large overtones. See Ref. [25]

for a standard textbook on the topic or Ref. [26] for an

early application to related problems in quantum me-

chanics. A recent review that is more speciﬁc to the

application of the WKB method to black holes can be

found in Ref. [27].

One method that is particularly insightful with respect

to standard quantum mechanics is the inverted potential

method proposed by Mashhoon [28], which is based on

earlier work of Heisenberg [29] and was further explored

by him and collaborators in Refs. [30–32]. It establishes

a mapping between the bound states of a potential well

and the quasi-normal modes of the corresponding poten-

tial barrier if certain transformation properties exist and

the bound states are known analytically, by e.g. the fac-

torization method Ref. [33]. While the complicated po-

tentials of black holes do not allow for the analytic com-

putation of bound states, the method has been applied

to approximate potentials for which analytic results are

known, e.g. the Eckart potential [34], the Pöschl-Teller

(PT) potential [35], and more recently a modiﬁed PT po-

tential [36]. The simplicity and ease of use of this method

made it very popular in the literature, but as with other

analytic methods, it has certain limitations. The main

limitation is that the method cannot be readily applied

to potentials for which the spectrum of bound states can

only be computed numerically. Therefore the number of

arXiv:2210.02069v2 [gr-qc] 12 May 2023

suitable potentials in the standard approach is limited.

Since then extensions of the original idea have been

studied. In Ref. [37] it has been suggested to use an

anharmonic oscillator potential to compute quasi-normal

modes from bound states in a perturbative way and in

Ref. [38] a perturbative, complex valued WKB matrix

approach has been presented. Based on the Bender-Wu

method [39] and subsequent work [40], it was shown in

Ref. [41] that a similar perturbative treatment, based

on using an anharmonic oscillator potential, Padé ap-

proximants and the Borel summation, allows to compute

quasi-normal modes from bound states with very high

precision. This idea was further improved in Ref. [42],

which demonstrates that it also allows for more precise

calculation of overtones.

The purpose of this work is to extend the method to

more general potentials for which bound states are ob-

tained numerically, which does not require one to limit

oneself to potentials that can be locally represented by a

Taylor expansion around their minimum. The key idea

behind the numerical extension is to compute an ana-

lytic representation of the spectrum of bound states by

using the numerical results and then apply the neces-

sary transformations to it. We use a Taylor expansion

of the bound state spectrum around a given parameter

choice, but other representations can in principle be used

as well. Limiting numerical calculations to the bound

state problem is rewarding because it is generically more

stable and in practice much easier to perform than those

of quasi-normal modes. We demonstrate the performance

of the new method by applying it ﬁrst to the PT poten-

tial, for which all steps can be veriﬁed analytically. Then

we study two potentials for which we are not aware of

analytic results in the literature. In all cases we choose

the parameters of the potential such that they corre-

spond to an approximation of the Regge-Wheeler (RW)

potential that describes the axial perturbations of the

Schwarzschild black hole [1], which we regard as the de-

fault benchmark test of the method. Our results demon-

strate that the method is simple to use and provides more

precise results than using the PT potential as known an-

alytic approximation.

The paper is structured as follows. In Sec. II we ﬁrst

review the analytic approach and then outline the new

method. The application of the method to diﬀerent po-

tentials is demonstrated in Sec. III. We discuss our ﬁnd-

ings and provide remarks for further extensions of the

method in Sec. IV. Finally, our conclusions can be found

in Sec. V. In Appendix Vwe provide additional material

for the shooting method. Throughout this work we use

units in which G=c= 1.

II. METHOD

A. Bound States and Quasi-Normal Modes

The starting point of this work is to consider the stan-

dard Schrödinger equation

dx2Ψ(x) + En−V(x, P )Ψ(x)=0,(1)

where V(x, P )is a potential with some parameter(s) P

and Enis the corresponding spectrum of eigenvalues for

a given choice of boundary conditions. Very qualita-

tively, if V(x, P )describes a potential well, the physi-

cal boundary conditions for bound states are those for

which Ψ(x)→0for x→ ±∞. Depending on the proper-

ties of V(x, P ), this can give rise to a ﬁnite or inﬁnite set

of eigenvalues. If however V(x, P )describes a potential

barrier, the suitable boundary conditions depend on the

application in mind.

Quasi-normal modes in the context of black holes are

usually deﬁned as purely outgoing solutions of the time

dependent wave equation, which in the time independent

Eq. (1) correspond to diverging solutions for Ψ(x)for

x→ ±∞. For an extended introduction to quasi-normal

modes and other techniques to compute them we refer

the interested reader to Refs. [10–14] and continue with

reviewing a few basics in the following.

The two potentials that describe gravitational pertur-

bations around the Schwarzschild black hole in general

relativity are known as the Regge-Wheeler potential [1]

and the Zerilli potential [3]. It can be shown that both

potentials are isospectral to each other, although their

analytic structure is diﬀerent [43,44]. In the following

we continue with the RW potential, which is given by

VRW(r(x), M, l) = 1−2M

rl(l+ 1)

r2−6M

r3.(2)

Here Mis the mass of the black hole and l(l+ 1) is a

separation constant with l≥2. The familiar form of the

Schrödinger equation with potential term only appears

in the so-called tortoise coordinate x, deﬁned via

x=r+ 2Mln r

2M−1,(3)

which makes a full analytic treatment more involved.

Note that exact analytic solutions have been found in

terms of conﬂuent Heun functions [45], but the quasi-

normal mode spectrum cannot be written in terms of

simple functions.

B. Review of the Analytic Method

In Refs. [28,30–32] the following structure of the

Schrödinger Eq. (1) was noticed and utilized. If one con-

siders the transformation x→ −ix and is able to trans-

form the original parameters Pof the potential to a new

set of parameters P0=π(P)such that

V(x, P ) = V(−ix, P 0),(4)

one can use the spectrum of bound states Ω2(P)≡

−En(P)of the potential well to compute the quasi-

normal modes of the potential barrier via

ωn(P)≡Ωn(π−1(P)).(5)

The success of the method depends on whether the ana-

lytic form of the bound states En(P)is known. Because

an application to complicated potentials usually does not

allow for an analytic computation of the spectrum, but

the analytic form is needed to apply Eq. (5), the method

cannot be used. In the following we show how this major

shortcoming can be circumvented and how the method

can be used for potentials where the bound states can be

computed numerically.

C. Numerical Bound State Method

The recipe of the numerical bound state method is as

follows. It is assumed that one has a precise numerical

method available to compute the bound states En(P)as

function of a given set of parameters P. One straightfor-

ward approach is the shooting method, see e.g. Ref. [46]

for an overview. The method is based on integrating

Ψ(x)as initial value problem from two distant points

for a given choice of Eand computing the Wronskian

of both solutions at an intermediate point. This process

can be formulated as root ﬁnding problem, because the

Wronskian vanishes when the initial guess for Eis an

eigenvalue En. Because the method is widely used we

refer to Appendix Afor more details. In Appendix Bwe

review the direct shooting method for the quasi-normal

mode case and provide supplementary information.

Next it is assumed that En, for each nrespectively1,

can be represented by a Taylor series around a given pa-

rameter set P0

En(P)≈X

Tk.(6)

Up to second order the series can be written in the com-

pact form

En(P)≈T0+T1+T2,(7)

with

T0=E0

n,(8)

T1=grad(En)P=P0P−P0,(9)

T2=1

2P−P0H(En)P=P0P−P0.(10)

1We suppress the ndependency on each Tkfor simplicity.

Here the gradient and Hessian operators are deﬁned as

grad(En)i=dEn

dPi

,(11)

H(En)ij =d2En

dPidPj

.(12)

All derivatives can be obtained numerically, e.g. in terms

of higher order ﬁnite diﬀerences, by using the numerical

method to compute bound states in the vicinity of P0.

Note that it is in principle also possible to extend the

multi-dimensional Taylor series to higher order.

As last step we consider the transformations. If the

inverse transformation π−1(P)is known, one can sim-

ply compute the approximate form of the quasi-normal

modes at any P0from inserting it in the Taylor ansatz

Eq. (7). Note that the transformations typically extend

the parameters to the complex plane and that they do

not have to be close to the expansion point of the Taylor

series. In this case the convergence of the Taylor se-

ries needs to be carefully studied and higher order terms

may become necessary. We emphasize that the validity

of the Taylor series, or any other expression found for

the bound state spectrum, is crucial and non-trivial for

arbitrary potentials.

An important simpliﬁcation of the multi-dimensional

Taylor series can be obtained if the inverse transforma-

tion π−1is just the identity. In that case all terms as-

sociated with (Pi−P0

i)are zero when evaluated at the

expansion point of the Taylor series, while the non-trivial

transformations give non-zero contributions

π−1(Pi)−P0

iPi=P0

in= 0,for π−1(Pi) = Pi,

6= 0,for π−1(Pi)6=Pi.(13)

This has the important and practical simpliﬁcation that

the higher order terms of the multi-dimensional Tay-

lor series that one actually has to consider only depend

on the variables whose transformations are non-trivial.

Those parameters may change the spectrum at the ze-

roth order term, but because the quasi-normal modes

are computed exactly for π−1(P0)all higher order terms

vanish.

III. APPLICATIONS

In this section we apply the numerical bound state

method to diﬀerent potentials. In Sec. III A we ﬁrst dis-

cuss the transformation properties of the RW potential

and how approximate potentials are used. In Sec. III B

we then apply the numerical method to the well known

PT potential, because the results are known analytically

and all steps can therefore be understood carefully. In

Sec. III C and Sec. III D the method is applied to the

Breit-Wigner potential and a piecewise combination.

All shown derivatives in this section are computed us-

ing ﬁnite diﬀerences with 9or 11 point stencils [47] and

diﬀerent step sizes indicated in the caption of each table

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Quasi-NormalModesfromBoundStates:TheNumericalApproachSebastianH.Völkel1,2,31MaxPlanckInstituteforGravitationalPhysics(AlbertEinsteinInstitute),D-14476Potsdam,Germany2SISSA-ScuolaInternazionaleSuperiorediStudiAvanzati,viaBonomea265,34136Trieste,ItalyandINFNSezionediTrieste3IFPU-InstituteforFundamenta...

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