# Darboux Covariance: A Hidden Symmetry of Perturbed Schwarzschild Black Holes

Michele Lenzi  Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer
de Can Magrans s/n, 08193 Cerdanyola del Vallès, Spain  Institut d’Estudis
Espacials de Catalunya (IEEC), Edifici Nexus, Carrer del Gran Capità 2-4,
despatx 201, 08034 Barcelona, Spain  Dipartimento di Fisica e Astronomia,
Università di Bologna, via Irnerio 46, 40126 Bologna, Italy  Carlos F.
Sopuerta  Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de
Can Magrans s/n, 08193 Cerdanyola del Vallès, Spain  Institut d’Estudis
Espacials de Catalunya (IEEC), Edifici Nexus, Carrer del Gran Capità 2-4,
despatx 201, 08034 Barcelona, Spain

###### Abstract

Starting from the infinite set of possible master equations for the
perturbations of Schwarzschild black holes, with master functions linear in
the metric perturbations and their first-order derivatives, we show that of
all them are connected via Darboux transformations. These transformations
preserve physical quantities like the quasinormal mode frequencies and the
infinite hierarchy of Korteweg-de Vries conserved quantities, revealing a new
hidden symmetry in the description of the perturbations of Schwarzschild black
holes: Darboux covariance.

Introduction. General relativistic perturbation theory of spherically-
symmetric spacetimes is of paramount importance since it applies to a wide
variety of physical phenomena: From structure formation in the homogeneous and
isotropic standard cosmological models [1, 2, 3, 4] to the dynamics of
perturbed Schwarzschild black holes (BHs) and spherical relativistic stars [5,
6, 7, 8] (see [9] for the impact on gravitational wave physics). In the case
of BHs [10, 11, 12, 13], perturbation theory describes scattering processes
[14, 15, 16] and quasinormal modes [5, 6, 17] (QNMs), which are crucial for
the last stage of the emission of gravitational radiation from BH binary
coalescence.

Master Functions and Equations. Spherically-symmetric spacetimes have a warped
geometry and as such the metric has the form: $g^{(4)}=h\times_{r}\Omega$,
where $h_{ab}$ [$x^{a}=(t,r)$] is a Lorentzian metric, $r$ is the area radial
coordinate, and $\Omega_{AB}$ [$x^{A}=(\theta,\varphi)$] is the metric of the
unit 2-sphere. For a Schwarzschild BH in Schwarzschild coordinates:
$h_{ab}={\rm diag}(-f,1/f)$ with $f=1-2M/r$ and $M$ is the BH mass. The warped
geometry allows us to decompose the metric perturbations in spherical
harmonics in such a way that modes with different harmonic numbers $(\ell,m)$
and different parity (odd/even parity) decouple from each other. We can find
linear combinations of the metric perturbations and their first-order
derivatives, the master functions $\Psi^{\rm even/odd}_{\ell m}$, so that the
perturbative Einstein equations become wave equations of the form:
$(\square_{2}-V_{\ell}/f)\Psi^{\rm even/odd}_{\ell m}=0$, where
$\square_{2}\Psi=h^{ab}\Psi_{:ab}$ and $V_{\ell}(r)\equiv V(r)$ is the
$\ell$-dependent potential. Considering the BH exterior, where there is a
timelike Killing vector, $t^{a}=\partial/\partial t$, and using the radial
tortoise coordinate, $dx/dr=1/f(r)$, the master equations become

$\left(-\partial^{2}_{t}+L_{V}\right)\Psi=\left(-\partial^{2}_{t}+\partial^{2}_{x}-V\right)\Psi=0\,,$
(1)

where $L_{V}=\partial^{2}_{x}-V$ is the well-known Schrödinger time-
independent operator. Physical quantities like the gravitational-wave fluxes
of energy and momenta can be estimated exclusively from the master functions,
which are gauge-invariant. Other non-gauge invariant quantities like the self-
force [18, 19] require the reconstruction of all metric perturbations, which
depends on the choice of perturbative gauge.

We recently found [20] all possible master equations assuming the master
functions are linear combinations (with coefficients depending only on $r$) of
the metric perturbations and their first-order derivatives. There are two
branches of possible pairs of potentials/master functions, $\\{(V,\Psi)\\}$:
(i) The standard branch. We call it the standard branch because it contains a
single potential for each parity: The Regge-Wheeler potential $V_{\rm RW}$ for
odd-parity perturbations [10] and the Zerilli potential $V_{\rm Z}$ for even-
parity perturbations [11]. The most general master function is a linear
combination (with constant coefficients) of two master functions:

${}_{S}\Psi^{\rm even}_{\rm odd}=\mathcal{C}_{1}\Psi^{\rm ZM}_{\rm
CPM}+\mathcal{C}_{2}\Psi^{\rm NE}_{\rm RW}\,,$ (2)

where $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are two arbitrary constants. In
the odd-parity case the two master functions turn out to be the well-known
Regge-Wheeler [10] and Cunningham-Price-Moncrief master functions [21, 22,
23]: $(\Psi_{\rm RW},\Psi_{\rm CPM})$. In the even-parity case, the two master
functions are the Zerilli-Moncrief master function $\Psi_{\rm ZM}$ [12, 11,
24] and another function that to our knowledge was unknown, which we called
$\Psi_{\rm NE}$ (new even-parity master function). All of these master
functions are gauge-invariant. The two master functions in each parity are
related by a time derivative: $t^{a}\Psi^{\rm ZM}_{{\rm CPM}\,,a}=2\,\Psi^{\rm
NE}_{\rm RW}\,$. (ii) The Darboux branch. In this branch, for each parity,
there is an infinite set of possible potentials and master functions,
$\\{(V,\Psi)\\}$. The set of possible potentials are determined by a non-
linear second-order ordinary differential equation. Then, for each potential
the master function can be written as [20]

${}_{D}\Psi^{\rm even}_{\rm odd}=\mathcal{C}_{1}\Psi^{\rm ZM}_{\rm
CPM}+\mathcal{C}_{2}\left(\Sigma^{\rm even}_{\rm odd}\,\Psi^{\rm ZM}_{\rm
CPM}+\Phi^{\rm even}_{\rm odd}\right)\,,$ (3)

where $\Sigma^{\rm even}_{\rm odd}=\Sigma^{\rm even}_{\rm odd}(x)$ is a
function that contains the integral of the potential; $\Phi^{\rm even}_{\rm
odd}=\Phi^{\rm even}_{\rm odd}(t,r)$ are linear combinations of the metric
perturbations and their first-order derivatives, but only the combination with
$\Psi^{\rm ZM}_{\rm CPM}$ in Eq. (3) is a true master function.

Darboux Covariance. To understand this landscape of master equations, or pairs
$\\{(V,\Psi)\\}$, let us first consider the standard branch, where we have
only the Regge-Wheeler and Zerilli potentials. It was first noted by
Chandrasekhar [25, 26] that these potentials, for both Schwarzschild and
Reissner-Nordström BHs, lead to the same transmission and reflection
coefficients (see also [27, 28]) as well as the same spectra of QNMs
frequencies. However, it has only been recently realized [29] (see also [30])
that this is a consequence of the master equations being related by a Darboux
transformation (DT) [31, 32] (see also [33, 34]).

Two any master equations, characterized by pairs $(v,\varphi)$ and $(V,\Psi)$,
are related by a DT if the two pairs are related by a transformation of the
form

$\displaystyle\Psi=\varphi_{,x}+g\,\varphi\,,\quad V=v+2\,g_{,x}\ $ (4)

where the DT generating function, $g(x)$, must satisfy the following Riccati
equation

$\displaystyle g_{,x}-g^{2}+v={\cal C}\,,$ (5)

where ${\cal C}$ is an arbitrary constant. We can write the DT generating
function as $g=(V+v)_{,x}/(2(V-v))$. Then, the consistency between the
expressions for $g(x)$ and $g_{,x}(x)$ is a second-order non-linear equation

$\left(\frac{\delta V_{,x}}{\delta V}\right)_{,x}+2\left(\frac{v_{,x}}{\delta
V}\right)_{,x}-\delta V=0\,,$ (6)

where $\delta V=V-v\,$. This is precisely the equation that any potential in
the Darboux branch should satisfy [20], where $v=V^{\rm Z}_{\rm RW}$. Hence,
all master equations in the Darboux branch are connected via a DT to the
standard branch, with DT generating functions given by:

$\displaystyle g^{\rm even}_{\rm odd}=\frac{1}{2}\int dx\,\left(V-V^{\rm
Z}_{\rm RW}\right)\,,$ (7)

while the two parities in the standard branch are connected by a DT with
generating function:

$g^{\rm odd\rightarrow even}=\frac{1}{2}\int dx\,\left(V_{\rm Z}-V_{\rm
RW}\right)=-g^{\rm even\rightarrow odd}\,.$ (8)

In conclusion, we have an infinite set of master equations linked by DTs,
showing the existence of a hidden symmetry in the perturbations of
spherically-symmetric BHs: Darboux covariance [33].

In this work we have adopted a view of the DT that is more general than the
original one, as introduced by Crum [35], based on Sturm-Liouville problems
and where the generating function of the DT is constructed from an
eigenfunction. Instead, we apply the DT to wave-type equations (1) and
consider generating functions that only have to satisfy Eq. (5). We can make
contact with the Crum approach by working in the frequency domain and study
single-frequency solutions: $\Psi(t,r)=e^{i\omega t}\psi(x;\omega)$, which
obey a time-independent Schrödinger equation

$L_{V}\psi(x;\omega)=-\omega^{2}\psi(x;\omega)\,.$ (9)

Given a solution $\psi_{o}(x;\omega_{o})$ with eigenvalue $-\omega^{2}_{o}$,
the function $g(x)=-(\ln\psi_{o})_{,x}$ generates a DT that transform Eq. (9)
into another equation of the same form with the same eigenvalue $-\omega^{2}$,
therefore showing the isospectral character of the DT. The Riccati equation
(5) is automatically satisfied with ${\cal C}=-\omega^{2}_{o}$, and so is Eq.
(6). The new master function from (4), say $\phi$, can be written as
$\phi=W[\psi,\psi_{o}]/\psi_{o}$ where $W[\psi,\psi_{o}]$ denotes the
Wronskian of $\psi$ and $\psi_{o}$. It turns out that the DT generating
function between the Regge-Wheeler and Zerilli-Moncrief master equations, Eq.
(8), can be constructed from one of the algebraically-special solutions of the
Regge-Wheeler equation [36, 37] (see [29]), namely

$\psi_{o}=\frac{\lambda(r)}{2}\mbox{e}^{-i\omega_{\ast}x}\,,\quad\omega_{\ast}=-i\frac{n_{\ell}(n_{\ell}+1)}{3M}\,,$
(10)

where $n_{\ell}=(\ell+2)(\ell-1)/2$ and $\lambda(r)=2n_{\ell}+6M/r$. The
generating function itself is:

$G(x)=\frac{6M\,f(r)}{\lambda(r)r^{2}}\,.$ (11)

Following [27, 28], the Regge-Wheeler and Zerilli potentials can be written in
terms of $G(x)$ as

$V^{\rm Z}_{\rm RW}=\pm G_{,x}+\alpha
G+G^{2}\,,\quad\alpha=\frac{1}{6M}\frac{(\ell+2)!}{(\ell-2)!}\,,$ (12)

which can be seen as a Riccati equation for $G(x)$. This form of the
potentials is reminiscent of Supersymmetric Quantum Mechanics (SUSY QM) [38,
39, 40] where the quantum description of systems with double degeneracy of
energy levels is realized. This is related to the fact that the Schrödinger
equation (9), for two DT-related potentials [Eq. (4)], can be written in the
form

$(\partial^{2}_{x}-V_{\pm})\psi=-\hat{\omega}^{2}\psi\,,\quad V_{\pm}=\pm
g_{,x}+g^{2}\,,$ (13)

where $g(x)$ plays the role of the SUSY QM superpotential, $V_{\pm}$ are
partner potentials, $\hat{\omega}^{2}=\omega^{2}-{\cal C}$ is the energy
eigenvalue, and we can introduce ladder operators ${A}=\partial_{x}-g\,$ and
${A}^{\dagger}=-\partial_{x}-g\,$ that factorize the Hamiltonians
$H_{-}=A\cdot{A}^{\dagger}$ and $H_{+}={A}^{\dagger}\cdot{A}\,$
($H_{\pm}=-\partial^{2}_{x}+V_{\pm}$). In the standard branch: $g=G+\alpha/2$
and ${\cal C}=-\alpha^{2}/4\,$. This factorization is the key ingredient of
the intertwining operator method used in [41] (see also [42, 43]) to study
properties of the infinite set of possible potentials that includes the Regge-
Wheeler and Zerilli ones.

DTs and the Korteweg-de Vries Equation. In the frequency-domain we can
establish the connection with inverse scattering theory following the work by
Gardner, Green, Miura and Kruskal [44] (see also [45, 46]), where they
discovered a way to solve the initial-value problem for the Korteweg-de Vries
(KdV) equation [47]

$V_{,\tau}=6VV_{,x}-V_{,xxx}\,.$ (14)

by identifying $V$ with the potential of the time-independent Schrödinger
equation (9). We now show how this connection reveals interesting properties
of our Darboux-covariant master equations in the frequency domain. The
spectrum of $L_{V}$ is twofold [48, 49]: It has a continuous part, the
scattering states, and a discrete part made out of a finite number of discrete
negative eigenvalues. In the case of the Schwarzschild BH, the potentials
$V^{\rm ZM}_{\rm RW}(x)$ are positive everywhere and decay to zero at both
ends ($x\rightarrow\pm\infty$). Therefore, there are no discrete normalizable
states. Let us now deform the Schrödinger equation by introducing the KdV time
$\tau$ in the following way: $V(x)\rightarrow V(\tau,x)$,
$\psi(x)\rightarrow\psi(\tau,x)$, and $\omega\rightarrow\omega(\tau)\,$. If
$V(\tau,x)$ follows the KdV flow we can show that

$\displaystyle[\partial^{2}_{x}-(V-\omega^{2})]\Xi=-(\omega^{2})_{,\tau}\psi\,,$
(15)

where $\Xi(\tau,x)=\psi_{,\tau}+V_{,x}\psi-2(V+2\omega^{2})\psi_{,x}$. In the
hypothetical case of bound states (not our case), and assuming that $\psi$ and
$V$ decay sufficiently fast at $x\rightarrow\pm\infty$, one can show, by
multiplying by $\psi$ and integrating over $x\in(-\infty,\infty)$, that
$(\omega^{2})_{,\tau}=0\,$. For non-normalizable states we can adopt an
approach due to Lax [50] consisting in the introduction of a pair of
operators, $P_{V}$ and $L_{V}$ (Lax pair), defined by

$\displaystyle\psi_{,\tau}=P_{V}\psi=-4\psi_{,xxx}+6V\psi_{,x}+3V_{,x}\psi\,,$
(16)

and Eq. (9) respectively. A remarkable fact about this Lax pair is that the
relation between differential operators, $dL_{V}/d\tau=[P_{V},L_{V}]\,$,
yields the KdV equation [Eq. (14)]. Following [33], one can show that the pair
of equations $(L_{V}+\omega^{2},-\partial_{\tau}+P_{V})\psi=0$ is invariant
under a DT provided the DT generating function satisfies Eq. (5) and is KdV-
deformed according to

$\displaystyle g_{,\tau}=-g_{,xxx}+6(V+g_{,x})g_{,x}\,.$ (17)

On the other hand, using the KdV-deformation of $\psi$, Eq. (16), we rewrite
Eq. (15) in the form

$(V_{,\tau}-6VV_{,x}+V_{,xxx}-(\omega^{2})_{,\tau})\psi=0\,.$ (18)

Therefore, if $(V,\psi)$ are KdV-deformed according to Eqs. (14) and (16), we
must have $\psi\,(\omega^{2})_{,\tau}=0$, which means that $\omega$ is
preserved by the KdV flow. This argument can be applied to the discrete and
continuous spectra as well as to the QNM frequencies.

DTs and the KdV Hierarchy. It was shown by Lax [50] that equations that are
equivalent to a relation between a Lax pair of operators, like the KdV
equation, have an infinite set of first integrals. Gardner showed [51] that
these first integrals are associated with symmetries of the KdV equation that
yield higher-order KdV equations, and all of them can be formulated as a
Hamiltonian system with infinite degrees of freedom. Zakharov and Fadeev [52]
showed that the hierarchy of KdV equations leads to a completely integrable
Hamiltonian system that admits canonical action-angle variables constructed
from the scattering data of the Schrödinger equation. Here, we use this point
of view to study the KdV hierarchy of first integrals for the infinite set of
master equations for BH perturbations.

The scattering states of the continuum spectrum coming either from
$x\rightarrow-\infty$ or from $x\rightarrow+\infty$ towards the potential
barrier described by $V$ are part reflected and part transmitted. For plane
waves coming from $x\rightarrow\infty$, the solution of the Schrödinger
equation has the Jost asymptotic behavior [53]:

$\displaystyle\psi~{}\rightarrow~{}\left\\{\begin{array}[]{lc}e^{i\omega
x}&\mbox{for~{}}x\rightarrow-\infty\,,\\\\[5.69054pt] a(\tau,\omega)e^{i\omega
x}+b(\tau,\omega)e^{-i\omega
x}&\mbox{for~{}}x\rightarrow+\infty\,,\end{array}\right.$ (21)

where the complex coefficients $a(\tau,\omega)$ and $b(\tau,\omega)$, which
fully determine the S-matrix, satisfy: $|a|^{2}-|b|^{2}=1\,$. The reflection
coefficient is $R(\tau,\omega)=b(\tau,\omega)/a(\tau,\omega)$, and in our
case, it completely characterizes the scattering data so that the mapping
$V(\tau,x)\rightarrow s(\tau)$ is uniquely invertible [54]. The transmission
coefficient is $T(\tau,\omega)=1/a(\tau,\omega)$ so that
$|R|^{2}+|T|^{2}=1\,$. Under the KdV flow they evolve as [44]: $T_{,\tau}=0$
and $R_{,\tau}=8i\omega^{3}R\,$, which implies:

$a_{,\tau}=0\,,\quad\mbox{and}\quad b_{,\tau}=8i\omega^{3}b\,.$ (22)

In the inverse scattering method, given the initial value of the potential
$V(\tau=0,x)$ we construct the associated scattering data, $s(0)$, evolve it
according the KdV flow, thus obtaining $s(\tau)$, and we recover $V(\tau,x)$
from $s(\tau)$ using the Gelfand-Levitan-Marchenko method [55, 56, 57].

Let us look at the consequences of the conservation law for $a(\tau,\omega)$
[Eq. (22)] under the KdV flow. Following [52], let us write

$\psi(\tau,x,\omega)=\exp\left\\{i\omega
x+\int^{x}_{-\infty}\\!\\!dx^{\prime}\Phi(\tau,x^{\prime},\omega)\right\\}\,,$
(23)

so that $a(\tau,\omega)$ becomes [see Eq. (21)]

$\displaystyle a(\tau,\omega)=\lim_{x\rightarrow\infty}e^{-i\omega
x}\psi=\exp\left\\{\int_{-\infty}^{+\infty}\\!\\!\\!\\!\\!dx^{\prime}\Phi(\tau,x^{\prime},\omega)\right\\}\,.$
(24)

It turns out [52] that $\ln a(\tau,\omega)$ admits an expansion in inverse
powers of $\omega$ for $|\omega|\rightarrow\infty$. We can then write

$\Phi(\tau,x,\omega)=\sum_{n=1}^{\infty}\frac{f_{n}(\tau,x)}{(2i\omega)^{n}}\,.$
(25)

Therefore, $a_{,\tau}=0$ implies that each coefficient $f_{n}(\tau,x)$ yields
a conserved quantity, the KdV integrals [46]:
$I_{n}(\tau)=\int^{+\infty}_{-\infty}dxf_{n}(\tau,x)$ with $dI_{n}/d\tau=0\,$.
After inserting Eq. (23) into the Schrödinger equation we get

$\Phi_{,x}+2i\omega\Phi+\Phi^{2}=V\,.$ (26)

This is a complex Riccati equation. Introducing the expansion for $\Phi$ here
we find that $f_{1}(\tau,x)=V(\tau,x)$ and a recursion for the rest of
coefficients $f_{n}(\tau,x)$ ($n>1$), which turn out to be differential
polynomials in $V(\tau,x)$:

$\frac{df_{n}}{dx}+f_{n+1}+\sum_{m=1}^{n-1}f_{m}f_{n-m}=0\,.$ (27)

It is convenient to split $\Phi$ into its real and imaginary parts

$\displaystyle\Phi=\sum_{N=1}^{\infty}\frac{f_{2N}}{(2i\omega)^{2N}}+\sum_{M=0}^{\infty}\frac{f_{2M+1}}{(2i\omega)^{2M+1}}=\chi_{\rm
R}+i\chi_{\rm I}\,.$ (28)

From Eq. (26), $\chi_{\rm R}(\tau,x)$ and $\chi_{\rm I}(\tau,x)$ satisfy

$\displaystyle\chi_{{\rm R},x}-2\omega\chi_{\rm I}+\chi^{2}_{\rm
R}-\chi^{2}_{\rm I}=V\,,$ (29) $\displaystyle\chi_{{\rm I},x}+2\omega\chi_{\rm
R}+2\chi_{\rm R}\chi_{\rm I}=0\,,$ (30)

and from here we get an expression for $\chi_{\rm R}$

$\chi_{\rm R}=-\frac{1}{2}\frac{d}{dx}\ln(\chi_{\rm I}+\omega)\,,$ (31)

that is, $\chi_{\rm R}$ is a gradient involving only $\chi_{\rm I}$. This,
together with the decaying behaviour of the potential $V$, which follows from
the decaying properties of $V_{\rm RW}$ and $V_{\rm Z}$ and the Riccati
equation (5), implies the known result [52, 54] that all the even KdV
integrals, $I_{2N}$, vanish. To study the odd KdV integrals, let us integrate
Eq. (26) over the real line $x\in(-\infty,+\infty)$ and use the decaying
properties of our potentials and derivatives to obtain:

$2i\omega\int^{+\infty}_{-\infty}dx\Phi+\int^{+\infty}_{-\infty}dx\Phi^{2}=\int^{+\infty}_{-\infty}dx\,V\,.$
(32)

For the standard branch, the potential $V=V^{\rm Z}_{\rm RW}$ admits the form
in Eq. (12). Therefore, using the behaviour of $G(x)$ at
$x\rightarrow\pm\infty$, the right-hand side of Eq. (32) becomes

$\int^{+\infty}_{-\infty}dx\,V=\int^{+\infty}_{-\infty}dx\,(\alpha
G+G^{2})\,,$ (33)

and hence Eq. (32) is the same for the whole standard branch. Any potential of
the Darboux branch can be written as $V=V^{\rm even}_{\rm odd}=V^{\rm Z}_{\rm
RW}+2g_{,x}$. Then, using again the decaying properties of the DT generating
functions, we deduce that any potential of the Darboux branch also satisfies
Eq. (33), thus Eq. (32) is universal. We can write it in terms of $(\chi_{\rm
R},\chi_{\rm I})$ and then use Eq. (31) and

$\displaystyle\chi_{\rm R}\chi_{\rm I}=-\frac{1}{2}\partial_{x}\left(\chi_{\rm
I}-\omega\ln(\chi_{\rm I}+\omega)\right)\,,$ (34)

which is a total derivative. Then, we arrive at

$-2\omega\int^{+\infty}_{-\infty}\\!\\!\\!\\!\\!\\!\\!\\!dx\,\chi_{\rm
I}+\int^{+\infty}_{-\infty}\\!\\!\\!\\!\\!\\!\\!dx(\chi^{2}_{\rm
R}-\chi^{2}_{\rm I})=\int^{+\infty}_{-\infty}\\!\\!\\!\\!\\!\\!\\!dx(\alpha
G+G^{2})\,.$ (35)

When we introduce the expansions for $\chi_{\rm R}$ and $\chi_{\rm I}$ [Eq.
(28)] this becomes a universal recurrent relation for the odd KdV integrals.
Then, we conclude that all the KdV integrals associated with the potentials of
the infinite set of master equations are the same. The first indication of
this result was provided by Chandrasekhar’s work [28, 14], where he showed
evidence that the KdV integrals should be the same for a pair of potentials of
the form in Eq. (12) (but not necessary related to BH perturbations), although
a full proof was not given.

It is interesting to mention that this infinite set of KdV integrals, which
makes the KdV equation completely integrable [52], has been connected to a
recurrence between the infinite KdV hierarchy of equations, initially
suggested by Lenard [58], and is rooted in the fact that the KdV equation
admits a bi-Hamiltonian structure [51, 59]. On the other hand, Gelfand and
Dickii [60] showed that these conserved quantities are connected with trace
formulae for half-integer powers of the operator $L_{V}$.

DTs and QNMs. We have seen that all the possible potentials associated with
the infinite set of master equations found in [20] have the same set of KdV
integrals when studying the continuous spectrum associated with the Sturm-
Liouville problem that emerges when one considers scattering problems. QNMs
are not associated with a Sturm-Liouville problem, they rather appear as
scattering resonances [61], poles in the meromorphic continuation of the
resolvent/Green function (related to $L_{V}$ in our case). They can also be
seen as the poles of the S-matrix and the associated residues [62]. We can use
the argument given by Chandrashekar [14] to show that the frequencies of QNMs
are the same for all possible potentials, provided they have similar decaying
behaviour at $x\rightarrow\pm\infty$. This is the case for our set of
potentials by virtue the Riccati equation (5) for the DT generating function.
Finally, thanks to Eq. (18) we can state that QNM frequencies and damping
times are preserved by the KdV flow provided the potential is KdV-deformed and
the radial master function $\psi$ is KdV-deformed according to Eq. (16). Apart
from these results, it would be interesting to explore the structure of the
resolvent associated with our time-independent master equations and their
behaviour and properties under the KdV flow.

Conclusions. The general structure of master functions and equations has
revealed a hidden symmetry in the theory of perturbations of (spherically
symmetric) BHs, Darboux covariance. The implications are diverse and here we
have shown that, given the decaying properties of the potentials at both
infinities (horizon and spatial infinity), DTs preserve the spectrum of
scattering processes and QNMs. A large part of the developments shown in this
work can be extended to other spherically-symmetric backgrounds and even to
other theories of gravity. The main changes may come from different boundary
conditions and their implications for the asymptotic behaviour of the
potentials.

Acknowledgments. CFS is supported by contracts ESP2017-90084-P and
PID2019-106515GB-I00/AEI/10.13039/501100011033 (Spanish Ministry of Science
and Innovation) and 2017-SGR-1469 (AGAUR, Generalitat de Catalunya). We thank
the COST Action CA16104 Gravitational waves, black holes and fundamental
physics (GWverse) for a Short Term Scientific Mission award to ML. ML also
acknowledges support from an “Angelo della Riccia” fellowship.

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# On the difference of initial logarithmic coefficients for the class of
univalent functions

Milutin Obradović  Department of Mathematics, Faculty of Civil Engineering,
University of Belgrade, Bulevar Kralja Aleksandra 73, 11000, Belgrade, Serbia
<EMAIL_ADDRESS>and  Nikola Tuneski  Department of Mathematics and
Informatics, Faculty of Mechanical Engineering, Ss. Cyril and Methodius
University in Skopje, Karpoš II b.b., 1000 Skopje, Republic of North
Macedonia<EMAIL_ADDRESS>
###### Abstract.

In this paper we give estimates of the differences $|\gamma_{3}|-|\gamma_{2}|$
and $|\gamma_{4}|-|\gamma_{3}|$ for the class of functions $f$ univalent in
the unit disc and normalized by $f(0)=f^{\prime}(0)-1=0$. Here, $\gamma_{2}$,
$\gamma_{3}$ and $\gamma_{4}$ are the initial logarithmic coefficients of the
function $f$.

###### Key words and phrases:

univalent, Grunsky coefficients, logarithmic coefficients, difference

###### 1991 Mathematics Subject Classification:

30C45, 30C50, 30C55

##  1\. Introduction and preliminaries

As usual, let $\mathcal{A}$ be the class of functions $f$ that are analytic in
the open unit disc ${\mathbb{D}}=\\{z:|z|<1\\}$ of the form

(1) $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots,$

and let $\mathcal{S}$ be the subclass of $\mathcal{A}$ consisting of functions
that are univalent in ${\mathbb{D}}$.

One of the problems in the theory of univalent functions is finding sharp
estimates of logarithmic coefficient, $\gamma_{n}$, of a univalent function
$f\in\mathcal{S}$, defined by

(2) $\log\frac{f(z)}{z}=2\sum_{n=1}^{\infty}\gamma_{n}z^{n}.$

From the relations (1) and (2), after equating the coefficients we receive the
next initial logarithmic coefficients:

(3)
$\gamma_{1}=\frac{a_{2}}{2},\quad\gamma_{2}=\frac{1}{2}\left(a_{3}-\frac{1}{2}a_{2}^{2}\right),\quad\text{and}\quad\gamma_{3}=\frac{1}{2}\left(a_{4}-a_{2}a_{3}+\frac{1}{3}a_{2}^{3}\right).$

Relatively little exact information is known about the logarithmic
coefficients. The natural conjecture $|\gamma_{n}|\leq 1/n$, inspired by the
Koebe function (whose logarithmic coefficients are $1/n$) is false even in
order of magnitude (see Duren [1, Section 8.1]), and true only for the class
of starlike functions ([7]). For the class $\mathcal{S}$ the sharp estimates
of single logarithmic coefficients are known only for $\gamma_{1}$ and
$\gamma_{2}$, namely,

$|\gamma_{1}|\leq
1\quad\mbox{and}\quad|\gamma_{2}|\leq\frac{1}{2}+\frac{1}{e}=0.635\ldots.$

In their papers [4] and [5], the authors gave estimates $|\gamma_{3}|\leq
0.5566178\ldots$ and $|\gamma_{4}|\leq 0.51059\ldots$ for the class
$\mathcal{S}.$

In this paper we will give the estimates for the differences
$|\gamma_{3}|-|\gamma_{2}|$ and $|\gamma_{4}|-|\gamma_{3}|$ in the class
$\mathcal{S}$. The sharp estimates of $|\gamma_{2}|-|\gamma_{1}|$ were
established in [3], with a simple proof given in [6] using different
technique.

In this paper our main tool will be a method based on Grunsky’s coefficients,
used previously in [4] and [5]. This method is different than the method used
in[3], or in [6].

We will use mainly the notations and results given in the book of N.A. Lebedev
([2]).

Let $f\in\mathcal{S}$ and let

$\log\frac{f(t)-f(z)}{t-z}=\sum_{p,q=0}^{\infty}\omega_{p,q}t^{p}z^{q},$

where $\omega_{p,q}$ are so-called Grunsky’s coefficients with property
$\omega_{p,q}=\omega_{q,p}$. For those coefficients we have the next Grunsky’s
inequality ([1, 2]):

(4)
$\sum_{q=1}^{\infty}q\left|\sum_{p=1}^{\infty}\omega_{p,q}x_{p}\right|^{2}\leq\sum_{p=1}^{\infty}\frac{|x_{p}|^{2}}{p},$

where $x_{p}$ are arbitrary complex numbers such that last series converges.

Further, it is well-known that if function

(5) $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$

belongs to $\mathcal{S}$, then also

$f_{2}(z)=\sqrt{f(z^{2})}=z+c_{3}z^{3}+c_{5}z^{5}+\cdots$

belongs to the class $\mathcal{S}$. For the function $f_{2}$ we have the
appropriate Grunsky’s coefficients of the form $\omega_{2p-1,2q-1}$, and the
inequality (4) has the form

(6)
$\sum_{q=1}^{\infty}(2q-1)\left|\sum_{p=1}^{\infty}\omega_{2p-1,2q-1}x_{2p-1}\right|^{2}\leq\sum_{p=1}^{\infty}\frac{|x_{2p-1}|^{2}}{2p-1}.$

As it has been shown in [2, p.57], if $f$ is given by (1) then the
coefficients $a_{2}$, $a_{3}$, $a_{4}$ and $a_{5}$ are expressed by Grunsky’s
coefficients $\omega_{2p-1,2q-1}$ of the function $f_{2}$ given by (5) in the
following way:

(7) $\begin{split}a_{2}&=2\omega_{11},\\\
a_{3}&=2\omega_{13}+3\omega_{11}^{2},\\\
a_{4}&=2\omega_{33}+8\omega_{11}\omega_{13}+\frac{10}{3}\omega_{11}^{3},\\\
a_{5}&=2\omega_{35}+8\omega_{11}\omega_{33}+5\omega_{13}^{2}+18\omega_{11}^{2}\omega_{13}+\frac{7}{3}\omega_{11}^{4},\\\
0&=3\omega_{15}-3\omega_{11}\omega_{13}+\omega_{11}^{3}-3\omega_{33},\\\
0&=\omega_{17}-\omega_{35}-\omega_{11}\omega_{33}-\omega_{13}^{2}+\frac{1}{3}\omega_{11}^{4}.\end{split}$

We note that in the book of Lebedev [2] there exists a typing mistake for the
coefficient $a_{5}$. Namely, instead of the term $5\omega_{13}^{2}$, there is
$5\omega_{15}^{2}$.

##  2\. Maine results

######  Theorem 1.

Let $\gamma_{2}$, $\gamma_{3}$ and $\gamma_{4}$ be the logarithmic
coefficients of function $f\in\mathcal{S}$. Then

$|\gamma_{3}|-|\gamma_{2}|\leq\frac{1}{\sqrt{5}}\quad\text{and}\quad|\gamma_{4}|-|\gamma_{3}|\leq\frac{1}{\sqrt{7}}.$

###### Proof.

For our consideration we need the following facts. From (6), choosing
$x_{2p-1}=0$ when $p=3,4,\ldots$, we have

$\begin{split}&|\omega_{11}x_{1}+\omega_{31}x_{3}|^{2}+3|\omega_{13}x_{1}+\omega_{33}x_{3}|^{2}+5|\omega_{15}x_{1}+\omega_{35}x_{3}|^{2}\\\
+&7|\omega_{17}x_{1}+\omega_{37}x_{3}|^{2}\leq|x_{1}|^{2}+\frac{|x_{3}|^{2}}{3}.\end{split}$

If additionally, $x_{1}=1$ and $x_{3}=0$, then

$|\omega_{11}|^{2}+3|\omega_{13}|^{2}+5|\omega_{15}|^{2}+7|\omega_{17}|^{2}\leq
1,$

and from here also

$|\omega_{11}|^{2}+3|\omega_{13}|^{2}+5|\omega_{15}|^{2}\leq 1,$
$|\omega_{11}|^{2}+3|\omega_{13}|^{2}\leq 1,$

and

$|\omega_{11}|^{2}\leq 1.$

From the previous inequalities we have

(8) $\begin{split}|\omega_{11}|&\leq 1,\\\
|\omega_{13}|&\leq\frac{1}{\sqrt{3}}\sqrt{1-|\omega_{11}|^{2}},\\\
|\omega_{15}|&\leq\frac{1}{\sqrt{5}}\sqrt{1-|\omega_{11}|^{2}-3|\omega_{13}|^{2}},\\\
|\omega_{17}|&\leq\frac{1}{\sqrt{7}}\sqrt{1-|\omega_{11}|^{2}-3|\omega_{13}|^{2}-5|\omega_{15}|^{2}}.\end{split}$

Also, from the fifth relation in (7) we obtain

(9)
$\omega_{33}=\omega_{15}-\omega_{11}\omega_{13}+\frac{1}{3}\omega_{11}^{3}.$

Now, for the first estimate of the theorem, using (3) and (7), we have

(10)
$\begin{split}|\gamma_{3}|-|\gamma_{2}|&=\frac{1}{2}\left|a_{4}-a_{2}a_{3}+\frac{1}{3}a_{2}^{3}\right|-\frac{1}{2}\left|a_{3}-\frac{1}{2}a_{2}^{2}\right|\\\
&=|\omega_{33}+2\omega_{11}\omega_{13}|-\left|\omega_{13}+\frac{1}{2}\omega_{11}^{2}\right|,\end{split}$

and after applying $|\omega_{11}|\leq 1$, brings

$\begin{split}|\gamma_{3}|-|\gamma_{2}|&\leq|\omega_{33}+2\omega_{11}\omega_{13}|-|\omega_{11}|\cdot\left|\omega_{13}+\frac{1}{2}\omega_{11}^{2}\right|\\\
&\leq\left|(\omega_{33}+2\omega_{11}\omega_{13})-(\omega_{11})\left(\omega_{13}+\frac{1}{2}\omega_{11}^{2}\right)\right|\\\
&=\left|\omega_{33}+\omega_{11}\omega_{13}-\frac{1}{2}\omega_{11}^{3}\right|.\end{split}$

From here, having in mind the relation (9) and inequalities (8), we receive

(11)
$\begin{split}|\gamma_{3}|-|\gamma_{2}|&\leq\left|\omega_{15}-\frac{1}{6}\omega_{11}^{3}\right|\leq|\omega_{15}|+\frac{1}{6}|\omega_{11}|^{3}\\\
&\leq\frac{1}{\sqrt{5}}\sqrt{1-|\omega_{11}|^{2}-3|\omega_{13}|^{2}}+\frac{1}{6}|\omega_{11}|^{3}\\\
&=:\Phi(|\omega_{11}|,|\omega_{13}|),\end{split}$

where

$\Phi(u,v)=\frac{1}{\sqrt{5}}\sqrt{1-u^{2}-3v^{2}}+\frac{1}{6}u^{3}$

with domain

$\Omega=\left\\{(u,v):0\leq u\leq 1,\,0\leq
v\leq\frac{1}{\sqrt{3}}\sqrt{1-u^{2}}\right\\}.$

It remains to find the maximal value of $\Phi$ on the domain $\Omega$.

Is easy to verify that the function $\Phi(u,v)$ has no interior singular
points in the domain $\Omega$ ($\Phi^{\prime}_{v}(u,v)=0$, if, and only of,
$v=0$). On the edges:

$\begin{split}\Phi(u,0)&=\frac{1}{\sqrt{5}}\sqrt{1-u^{2}}+\frac{1}{6}u^{3}\leq\Phi(0,0)=\frac{1}{\sqrt{5}}=0.44721\ldots,\\\
\Phi(0,v)&=\frac{1}{\sqrt{5}}\sqrt{1-3v^{2}}\leq\frac{1}{\sqrt{5}}=0.44721\ldots\\\
\Phi\left(u,\frac{1}{\sqrt{3}}\sqrt{1-u^{2}}\right)&=\frac{1}{6}u^{3}\leq\frac{1}{6}=0.1(6).\end{split}$

Using all previous facts and (11), we finally conclude that

$|\gamma_{3}|-|\gamma_{2}|\leq\frac{1}{\sqrt{5}}.$

As for second estimate of this theorem, from (2) we receive

$\gamma_{4}=\frac{1}{2}\left(a_{5}-a_{2}a_{4}-\frac{1}{2}a_{3}^{2}+a_{2}^{2}a_{3}-\frac{1}{4}a_{2}^{4}\right),$

and by using the relations (7),

(12)
$\gamma_{4}=\frac{1}{2}\left(2\omega_{35}+3\omega_{13}^{2}+4\omega_{11}\omega_{33}+4\omega_{11}^{2}\omega_{13}-\frac{5}{6}\omega_{11}^{4}\right).$

Next, from the last two relations in (7) we can express $\omega_{33}$ and
$\omega_{35}$ and apply them in (12). After some calculations, we receive

$\gamma_{4}=\omega_{17}+\omega_{11}\omega_{15}+\omega_{11}^{2}\omega_{13}+\frac{1}{2}\omega_{13}^{2}+\frac{1}{4}\omega_{11}^{4}.$

Now, using the above expression for $\gamma_{4}$, the expression for
$\gamma_{3}$ obtained within (10),
$\gamma_{3}=\omega_{33}+2\omega_{11}\omega_{13}$, and the expressions for
$\omega_{33}$ and $\omega_{35}$ from the last two lines from (7), having in
mind that $|\omega_{11}|\leq 1$, we receive:

(13)
$\begin{split}|\gamma_{4}|-|\gamma_{3}|&\leq|\gamma_{4}|-|\omega_{11}||\gamma_{3}|\\\
&\leq|\gamma_{4}-\omega_{11}\gamma_{3}|\\\
&=\left|\omega_{17}+\frac{1}{2}\omega_{13}^{2}-\frac{1}{12}\omega_{11}^{4}\right|\\\
&\leq|\omega_{17}|+\frac{1}{2}|\omega_{13}|^{2}+\frac{1}{12}|\omega_{11}|^{4}\\\
&\leq\frac{1}{\sqrt{7}}\sqrt{1-|\omega_{11}|^{2}-3|\omega_{13}|^{2}-5|\omega_{15}|^{2}}+\frac{1}{2}|\omega_{13}|^{2}+\frac{1}{12}|\omega_{11}|^{4}\\\
&\leq\frac{1}{\sqrt{7}}\sqrt{1-|\omega_{11}|^{2}-3|\omega_{13}|^{2}}+\frac{1}{2}|\omega_{13}|^{2}+\frac{1}{12}|\omega_{11}|^{4}\\\
&=:\Psi(|\omega_{11}|^{2},|\omega_{13}|^{2}),\end{split}$

where

$\Psi(s,t)=\frac{1}{\sqrt{7}}\sqrt{1-s-3t}+\frac{1}{2}t+\frac{1}{12}s^{2},$

$0\leq s\leq 1$, $0\leq t\leq\frac{1}{3}(1-s)$. It is easily verified that the
function $\Psi(s,t)$ attains its maximum $\frac{1}{\sqrt{7}}$ for $s=t=0$.
Finally, from (13),

$|\gamma_{4}|-|\gamma_{3}|\leq\frac{1}{\sqrt{7}}.$

∎

Previous theorem leads to the conjecture for the difference of the moduli of
two consecutive logarithmic coefficients of univalent functions.

######  Conjecture 1.

If $\gamma_{n}$ is logarithmic coefficient of function $f\in\mathcal{S}$, then

$|\gamma_{n}|-|\gamma_{n-1}|\leq\frac{1}{\sqrt{2n-1}},\quad n=3,4,\ldots.$

## References

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