# Double-slit experiment at fermi scale: coherent photoproduction in heavy-ion
collisions

W. Zha<EMAIL_ADDRESS>University of Science and Technology of
China, Hefei, China  L. Ruan  Brookhaven National Laboratory, New York, USA
Z. Tang<EMAIL_ADDRESS>University of Science and Technology of China,
Hefei, China  Z. Xu  Brookhaven National Laboratory, New York, USA  Shandong
University, Jinan, China  S. Yang  Brookhaven National Laboratory, New York,
USA

###### Abstract

The double-slit experiment has become a classic thought experiment, for its
clarity in expressing the central puzzle of quantum mechanics – wave-particle
complementarity. Such wave-particle duality continues to be challenged and
investigated in a broad range of entities with electrons, neutrons, helium
atoms, C60 fullerenes, Bose-Einstein condensates and biological molecules. All
existing experiments are performed with the scale larger than angstrom. In
this article, we present a double-slit scenario at fermi scale with new
entities – coherent photon products in heavy-ion collisions. Virtual photons
from the electromagnetic fields of relativistic heavy ions can fluctuate to
quark-antiquark pairs, scatter off a target nucleus and emerge as vector
mesons. The two colliding nuclei can take turns to act as targets, forming a
double-slit interference pattern. Furthermore, the “which-way” information can
be partially solved by the violent strong interactions in the proposed
scenario, which demonstrates a key concept of quantum mechanics –
complementary principle.

Relativistic heavy ions carry giant electromagnetic field which can be
equivalent into a field of quasi-real virtual photons Krauss, Greiner, and
Soff (1997). When two ions collide, the photon from the field of one nucleus
may fluctuate into a virtual quark-antiquark pair which scatters elastically
from the other nucleus, emerging as a real vector meson. The elastic
scattering occurs via short-range strong force, which imposes a restriction on
the production site of vector meson within one of the two ions. This provides
the production process with a choice: either nucleus 1 emits a virtual photon
while nucleus 2 acts as target, or vice versa. If the information is missing
about which nucleus emits a photon or acts as target, the vector meson,
behaving as a wave, originates simultaneously from both colliding nuclei. The
wave-like behaviour and corresponding interference presents itself as a
perfect double-slit experiment of individual particle. The double-slit
experiment has become a classic thought experiment, for its clarity in
expressing the central puzzle of quantum mechanics – wave-particle
complementarity. Such wave-particle duality DE BROGLIE (1923) continues to be
challenged Scully, Englert, and Walther (1991); Wiseman and Harrison (1995);
Lindner  _et al._ (2005); Kiffner, Evers, and Keitel (2006); King, Di Piazza,
and Keitel (2010) and investigated in a broad range of entities with electrons
Jönsson (1961), neutrons Zeilinger  _et al._ (1988), helium atoms Carnal and
Mlynek (1991), C60 fullerenes Arndt  _et al._ (1999), Bose-Einstein
condensates Andrews  _et al._ (1997) and biological molecules Hackermüller
_et al._ (2003). Currently, all existing experiments are performed with the
scale larger than angstrom. In this article, we demonstrate that the coherent
photoproduction in heavy-ion collisions could extend the double-slit scenario
to fermi scale with new entities – vector mesons Bertulani, Klein, and
Nystrand (2005). Furthermore, the violent strong interactions accompanied with
heavy-ion collisions could partially solve the “which-slits” problem, leading
to the appearance of decoherence, which demonstrates a key concept of quantum
mechanics – complementary principle.

In contrast to the light vector mesons, photoproduction of heavy vector mesons
offers an opportunity to directly determine the gluon distributions in
nucleons and nuclei Guzey and Zhalov (2013), which are not directly accessible
in deep inelastic scattering. Due to their clarity in physical picture and
experimental feasibility, we choose J$/\psi$, the most abundant heavy vector
meson, as the entity to illustrate the double-slit scenario in heavy-ion
collisions. The J$/\psi$ photoproduction from the two nuclei is related by a
parity transformation, and the parity of J$/\psi$ is negative, which assigns
the amplitudes from the two directions with opposite signs. This gives the
presented double-slit scenario a unique feature – a phase shift of $\pi$
between the two interference sources. Furthermore the probability of multiple
J$/\psi$ production in one collision is negligible in comparison with that of
single production, which indicates that the interference pattern is build up
one by one in the proposed scenario. This is an important version of double-
slit experiments to clearly demonstrate the wave-particle duality, which
states that all matter exhibits both wave and particle properties.

Figure 1: Amplitude and momentum distribution patterns of coherent J/$\psi$
photoproduction with different scenarios for b = 15 fm at mid-rapidity (y=0).
Panel a: point source scenario, panel b: Woods-Saxion distribution, panel c:
realistic scenario described in the text. Panel d - f show the corresponding
momentum distributions according to the amplitude scenarios in Panel a - c,
respectively.

Figure 2: Amplitude and momentum distribution patterns of coherent J/$\psi$
photoproduction in Au+Au collisions at $\sqrt{s_{\rm{NN}}}=$ 200 GeV for mid-
rapidity (y=0) with different impact parameters. Panel a and d: b = 13.0 fm,
panel b and e: b = 10.0 fm, panel c and f: b = 5.7 fm.

Figure 3: Amplitude and momentum distribution patterns of coherent J/$\psi$
photoproduction in Au+Au collisions at $\sqrt{s_{\rm{NN}}}=$ 200 GeV with b =
10.0 fm at different rapidities. Panel a and d: y = 0.0, panel b and e: y =
0.6, panel c and f: y = 1.4.

Figure 4: Amplitude and momentum distribution patterns of coherent J/$\psi$
photoproduction at mid-rapidity (y=0) in Au+Au collisions at
$\sqrt{s_{\rm{NN}}}=$ 200 GeV with disruption effect from overlap region for
different impact parameters. Panel a and d: b = 13.0 fm, panel b and e: b =
10.0 fm, panel c and f: b = 5.7 fm.  Figure 5: Integrated yields of coherent
J/$\psi$ production with and without “observation” effect as a function of
$N_{\text{part}}$ in Au+Au collisions at $\sqrt{s_{\text{NN}}}$ = 200 GeV (a)
and Pb+Pb collisions at $\sqrt{s_{\text{NN}}}$ = 2.76 TeV (b). Data from the
ALICE experiment Adam  _et al._ (2016) are shown for comparison.

In heavy-ion collisions, the momentum of J$/\psi$ can be reconstructed by the
detectors, which allows us to observe the interference fringes in momentum
space. The probability distribution of J$/\psi$ in momentum space can be
calculated by performing a Fourier transformation to the amplitude in
coordinate space, defined as:

$\frac{d^{2}P}{dp_{x}dp_{y}}=\frac{1}{2\pi}|\int
d^{2}r(A_{1}(y,\textbf{r},-\frac{b}{2})+A_{2}(y,\textbf{r},\frac{b}{2}))e^{i\textbf{p}\cdot\textbf{r}}|^{2},$
(1)

where $A_{1}(y,\textbf{r},-\frac{b}{2})$ and $A_{2}(y,\textbf{r},\frac{b}{2})$
are the amplitudes with rapidity $y$ for J$/\psi$ production at distant r from
the two colliding nuclei, respectively; b is the slit separation, which is the
so-called impact parameter in heavy-ion collisions. We take $\hbar=c=1$ here.
The J$/\psi$ production is coherent over the entire nucleus, for a start, we
approximate the production regions as two point sources, one at the center of
each nucleus. At mid-rapidity ($y=0$), the magnitudes of amplitudes from the
two directions are the same. With the point source assumption, the amplitudes
from nucleus 1 and 2 at mid-rapidity can be written as:

$A_{1}(0,\textbf{r},-\frac{b}{2})=\delta(r-\frac{b}{2}),A_{2}(0,r,\frac{b}{2})=-\delta(r+\frac{b}{2}).$
(2)

Here, $\delta(x)$ is the Dirac delta function. The amplitude distribution in
coordinate space and the corresponding interference pattern in momentum space
with slit separation $b=$ 15 fm are shown in Fig. 1 panel a and d,
respectively. The momentum distribution reveals typical Young’s double-slit
interference pattern with a series of alternating light and dark fringes. The
width of the fringes reflects the distant between the two slits:
$d=\frac{2\pi}{b}$. The straight fringes exhibit the central minimum due to
the opposite sign amplitudes from the two sources. To more realistic case, the
two nuclei (slits) possess certain size and profile at fermi scale. The
density profile of nucleus can be parameterized by the Woods-Saxon
distribution:

$\rho_{A}(r)=\frac{\rho^{0}}{1+\exp[(r-R_{\rm{WS}})/d]}$ (3)

where the radius $R_{\rm{WS}}$ and skin depth $d$ are based on fits to
electron scattering data Barrett and Jackson (1977), and $\rho^{0}$ is the
normalization factor. With
$A(\textbf{r})=T(\textbf{r})=\int^{+\infty}_{-\infty}\rho(\sqrt{\textbf{r}^{2}+z^{2}})dz$,
the amplitude distribution in coordinate space and the corresponding momentum
distribution for Au+Au collisions are shown in panel b and e, respectively.
Due to the profile of slits, typical diffraction rings emerge on top of the
interference fringes in momentum distribution. The amplitudes in coordinate
space are further modified by the spatial photon flux, nuclear shadowing and
coherent length effects. The spatial photon flux induced by ions can be given
by the equivalent photon approximation Krauss, Greiner, and Soff (1997):

$\displaystyle\frac{d^{3}N_{\gamma}(\omega_{\gamma},\vec{x}_{\bot})}{d\omega_{\gamma}d\vec{x}_{\bot}}=\frac{4Z^{2}\alpha}{\omega_{\gamma}}\bigg{|}\int\frac{d^{2}\vec{k}_{\gamma\bot}}{(2\pi)^{2}}\vec{k}_{\gamma\bot}\frac{F_{\gamma}(\vec{k}_{\gamma})}{|\vec{k}_{\gamma}|^{2}}e^{i\vec{x}_{\bot}\cdot\vec{k}_{\gamma\bot}}\bigg{|}^{2}$
(4)
$\displaystyle\vec{k}_{\gamma}=(\vec{k}_{\gamma\bot},\frac{\omega_{\gamma}}{\gamma_{c}})\
\ \ \ \ \omega_{\gamma}=\frac{1}{2}M_{\text{J}/\psi}e^{\pm y}$

where $\vec{x}_{\bot}$ and $\vec{k}_{\gamma\bot}$ are 2-dimensional photon
position and momentum vectors perpendicular to the beam direction, $Z$ the
nuclear charge, $\alpha$ the electromagnetic coupling constant, $\gamma_{c}$
the Lorentz factor of the photon-emitting nucleus, $M_{\text{J}/\psi}$ and $y$
the mass and rapidity of J/$\psi$, and $F_{\gamma}(\vec{k}_{\gamma})$ the
nuclear electromagnetic form factor. $F_{\gamma}(\vec{k}_{\gamma})$ is
obtained via the Fourier transformation of the charge density in the nucleus.
The calculation of scattering amplitude $\Gamma_{\gamma
A\rightarrow\rm{J}/\psi A}(\textbf{r})$ with shadowing effect can be performed
with quantum Glauber Miller  _et al._ (2007) \+ vector meson dominance (VMD)
approach Bauer  _et al._ (1978) coupled with the parameterized forward
scattering amplitude $f_{\gamma p\rightarrow\rm{J}/\psi N}(0)$ Klein  _et al._
(2017) as input:

$\Gamma_{\gamma A\rightarrow\rm{J}/\psi A}(\textbf{r})=\frac{f_{\gamma
p\rightarrow\rm{J}/\psi N}(0)}{\sigma_{\rm{J}/\psi N}}\times
2\times[1-\rm{exp}(-\frac{\sigma_{\rm{J}/\psi N}}{2}\times
T^{\prime}(\textbf{r})].$ (5)

Using the optical theorem and VMD relation, the total cross section for
J$/\psi N$ scattering is given by:

$\sigma_{\rm{J}/\psi N}=\frac{f_{\rm{J}/\psi}}{4\sqrt{\alpha}C}f_{\gamma
p\rightarrow\rm{J}/\psi N}(0),$ (6)

where $f_{\rm{J}/\psi}$ is J$/\psi$-photon coupling and C is a correction
factor for the non-diagonal coupling through higher mass vector mesons Hufner
and Kopeliovich (1998). To account for the coherent length effect,
$T^{\prime}(\textbf{r})$ is defined as:

$T^{\prime}(\textbf{r})=\int^{+\infty}_{-\infty}dz\rho(\sqrt{\textbf{r}^{2}+z^{2}})e^{iq_{L}z},q_{L}=\frac{M_{\rm{J}/\psi}e^{y}}{2\gamma}$
(7)

where $q_{L}$ is the longitudinal momentum transfer required to produce a real
J$/\psi$. With these effects, the amplitude from one direction can be written
as:

$A_{1}(y,\textbf{r},-\frac{b}{2})=\Gamma_{\gamma A\rightarrow\rm{J}/\psi
A}(\textbf{r}-\frac{\textbf{b}}{2})\times\sqrt{\frac{d^{2}N_{\gamma}(\textbf{r}-\frac{\textbf{b}}{2})}{d^{2}r}},$
(8)

which makes the slits become asymmetric. The amplitude distributions of the
asymmetric slits in Au+Au collisions at $\sqrt{s_{\rm{NN}}}=$ 200 GeV for mid-
rapidity (y=0) are shown in Fig. 1 panel c and the resulting interference
pattern is depicted in panel f. The corresponding diffraction pattern does not
show typical symmteric diffraction rings due to the asymmetric profile of
slits, and the interference fringes become curving.

The coherent photoproduction has been studied in detail in ultra-peripheral
collisions (UPCs) Bertulani, Klein, and Nystrand (2005), in which the impact
parameter is larger than twice the nuclear radius. The destructive
interference of $\rho^{0}$ in transverse momentum for UPCs has been proposed
by Klein and Nystrand Klein and Nystrand (2000), and verified by the STAR
collaboration Abelev  _et al._ (2009). However, no special azimuthal direction
can be determined in UPCs, which prevents us to observed the two-dimensional
interference fringes. Fortunately, in hadronic heavy-ion collisions (HHICs),
when the two nuclei overlap, the reaction plane, spanned by the impact
parameter and the beam axis, can be estimated from the azimuthal anisotropy of
produced particles due to the asymmetric geometry in overlap region. Herein,
the disruption to interference action from the violent strong interactions in
overlap region is ignored, which would be detailedly discussed in the later.
Furthermore, in HHICs, the impact parameter (centralities) could be precisely
determined, which allows us to vary the slit separation in the presented
scenario. The profile of amplitudes and the corresponding momentum
distributions in Au+Au collisions at $\sqrt{s_{\rm{NN}}}=$ 200 GeV with
different slit separations at mid-rapidity (y=0) are shown in Fig. 2. As
demonstrated in the figure, the band width of interference fringes becomes
wider when the slit separation gets smaller. And the diffraction pattern also
changes due to the different profile of slits with different slit separations.
In the proposed scenario, the relative brightness between the two slits could
also be adjusted by detecting the coherently produced J$/\psi$’s at different
rapidities. Figure 3 shows the amplitude distributions and the corresponding
momentum distributions in Au+Au collisions at $\sqrt{s_{\rm{NN}}}=$ 200 GeV
with the slit separation $b=10$ fm for J$/\psi$ production at different
rapidities. When going to forward rapidities, due to the dominant production
from one direction, the interference fringes merge into each other, which
makes the diffraction pattern dominant over the momentum distributions.

A well-known double-slit thought experiment predicts that if particle
detectors are positioned at the slits to solve the “which way” problem, the
interference pattern will disappear, which illustrates the complementary
principle that matter can behave as either particle or wave, but can not
possess both at the same time. Conventional wisdom says that, in HHICs, the
strong interactions and possible quark-gluon-plasma (QGP) Braun-Munzinger and
Stachel (2007) in overlap region detect the coherent produced J$/\psi$’s and
determine the “which way” information, leading to decoherence, which prohibits
the coherent photoproduction. However, ALICE Adam  _et al._ (2016) and STAR
Zha (2017) collaborations observed significant anomaly excess of J$/\psi$
production in HHICs at very low transverse momentum ($p_{T}<$ 0.3 GeV/c),
which could be qualitatively described by coherent photoproduction mechanism
Zha  _et al._ (2018). The strong evidence of existence of coherent
photoproduction in HHICs focus us to consider the disruption from overlap
region more carefully. The strong interactions are short-range, so they can
not observe or detect the coherent photoproduction out of the hot medium
region. The “which way” information is only obtained within the extent of
strong interactions, which means that the “which slit” problem is only
partially solved or the two slits are only sectionally blocked by the
interactions but not all. Herein, we adopt a simple assumption that the slits
are only blocked in the initial overlap region. The amplitude distributions of
the sectional blocked slits and the corresponding momentum distribution
patterns in Au+Au collisions at $\sqrt{s_{\rm{NN}}}=$ 200 GeV with different
impact parameters at mid-rapidity are depicted in Fig. 4. In comparison with
the interference and diffraction patterns shown in Fig. 2, the diffraction
rings and interference fringes change significantly due to the partially
blocking of the slits. The integrated yields of coherent J/$\psi$ production
as a function of $N_{\rm{part}}$ with the whole slits and sectional blocked
slits (“observation” effect) are shown in Fig. 5 for Au+Au collisions at
$\sqrt{s_{\text{NN}}}$ = 200 GeV (a) and Pb+Pb collisions at
$\sqrt{s_{\text{NN}}}$ = 2.76 TeV (b). The integrated production rate is
significantly reduced by the “observation” effect in semi-central and central
collisions. And in comparison with the experimental measurements from ALICE
Adam  _et al._ (2016) and STAR Zha (2017), the data seem to favor the
calculation with “observation” effect , however could not distinguish the two
scenarios due to limited statistics.

By making use of the coherent photoproduction in HHICs, one can take Young’s
famous experiment one step further at fermi scale and create a truly one-by-
one double-slit set-up comprised of new entities – vector meson. The slit
separations and relative brightness between slits can be adjusted in presented
scenario. In addition, the strong interactions in the overlap region detect
the coherent products and partially solve the “which-way” information, which
serves as a good double-slit scenario to test the “observation” effect on
wave-particle duality and further demonstrates the complementary principle.

We thank Prof. Spencer Klein and Pengfei Zhuang for useful discussions. This
work was funded by the National Natural Science Foundation of China under
Grant Nos. 11775213, 11505180, and 11375172, the U.S. DOE Office of Science
under contract No. DE-SC0012704, and MOST under Grant No. 2014CB845400 and
2016YFE0104800.

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# Symmetry and uniqueness of minimizers of Hartree type equations with
external Coulomb potential

Vladimir Georgiev  Dipartimento di Matematica, Università di Pisa
Largo Bruno Pintecorvo 5, 56127 Pisa, Italy
e-mail<EMAIL_ADDRESS>George Venkov  Faculty of Applied Mathematics
and Informatics, Technical University of Sofia
Kliment Ohridski 8, 1756 Sofia, Bulgaria
e-mail<EMAIL_ADDRESS>
###### Abstract

In the present article we study the radial symmetry of minimizers of the
energy functional, corresponding to the repulsive Hartree equation in external
Coulomb potential. To overcome the difficulties, resulting from the ”bad” sign
of the nonlocal term, we modify the reflection method and then, by using
Pohozaev integral identities we get the symmetry result.

###### keywords:

Hartree equations , minimizers , symmetry , variational methods , nonlinear
solitary waves

###### MSC:

35J50 , 35J60 , 35Q55

††journal: Journal of Differential Equationslabel2label2footnotetext: The
first author was supported by the Italian National Council of Scientific
Research (project PRIN No. 2008BLM8BB ) entitled: ”Analisi nello spazio delle
fasi per E.D.P.”

##  1 Introduction

Solitary waves associated with the Hartree type equation in external Coulomb
potential are solutions of type

$\chi(x)e^{-i\omega t},\quad x\in\mathbb{R}^{3},t\in\mathbb{R},$

where $\omega>0$ and $\chi$ satisfies the nonlinear elliptic equation

$\displaystyle\ \ \ \ \
-\Delta\chi(x)+\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}dy}{|x-y|}\
\chi(x)-\frac{\chi(x)}{|x|}+\omega\chi(x)=0.$ (1)

The natural energy functional associated with this problem is (see [4])

$\displaystyle\mathcal{E}(\chi)=\frac{1}{2}\|\nabla\chi\|^{2}_{L^{2}}+\frac{1}{4}A(|\chi|^{2})-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx,$ (2)

where we shall denote

$A(f)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{f(x)f(y)}{|x-y|}dydx.$
(3)

The corresponding minimization problem is associated with the quantity

$I_{N}=\min\\{\mathcal{E}(\chi);\chi\in H^{1},\|\chi\|^{2}_{L^{2}}=N\\}.$ (4)

The existence of positive minimizers $\chi_{0}(x)$, such that

$\mathcal{E}(\chi_{0})=I_{N},\ \ \|\chi_{0}\|^{2}_{L^{2}}=N,$

is established by Cazenave and Lions in [4] by the aid of the concentration
compactness method.

For a given $\omega>0$, the constrained minimization problem (4) can be
compared with the unconstrained minimization problem

$S^{min}_{\omega}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\},$

where $S_{\omega}(\chi)$ is the corresponding action functional, defined by

$S_{\omega}(\chi)=\mathcal{E}(\chi)+\frac{\omega}{2}\|\chi\|^{2}_{L^{2}}.$ (5)

There are different results on the symmetry (and uniqueness) of the
minimizers. The basic result due to Gidas, Ni and Nirenberg [9] implies the
radial symmetry of the minimizers associated with the semilinear elliptic
equation

$\Delta u+f(u)=0,$

provided suitable assumptions on the function $f(u)$ are satisfied and the
scalar function $u$ is positive. As in the previous result due to Serrin [19],
the proof is based on the maximum principle and the Hopf’s lemma.

Therefore, the first natural question is to ask if the linear operator

$P_{\omega}=-\Delta-\frac{1}{|x|}+\omega$

in (1), satisfies the weak maximum principle in the sense that

$u\in H^{2},\ P_{\omega}(u)=g\geq 0,\Longrightarrow u\geq 0.$ (6)

The above maximum principle is incomplete, since additional behavior of $u$
and $g$ at infinity has to be imposed, namely, we shall suppose that

$(1+|x|)^{-M}e^{\sqrt{\omega}|x|}u\in
H^{2},\quad(1+|x|)^{-M}e^{\sqrt{\omega}|x|}g\in H^{2},$ (7)

for some real number $M>0.$

Note, that the energy levels of the hydrogen atom are described by the
eigenvalues $\omega_{k}>0$ of the eigenvalue problem

$\Delta e_{k}(x)+\frac{e_{k}(x)}{|x|}=\omega_{k}e_{k}(x),\ \ e_{k}(x)\in
H^{2}.$

One has

$\omega_{k}=\frac{1}{4(k+1)^{2}},\quad k=0,1,...$

and $e_{0}(x)=ce^{-|x|/2},c>0.$ The first observation is that all
eigenfunctions $e_{k}(x)$, $k\geq 1$, are expressed in terms of Laguerre
polynomials of $|x|$, having exactly $k$ roots. This fact guarantees that the
maximum principle is not valid for $\omega=\omega_{k}.$ More precisely, we can
show the following.

######  Lemma 1.

The weak maximum principle (6) is valid if an only if

$\omega\geq\frac{1}{4}.$

This result can be compared with the existence of action minimizers for the
corresponding functional $S_{\omega}$, obtained by Lions for $0<\omega<1/4$
(see for details [14]).

######  Theorem 2.

We have the properties:

a) for any $\omega>0,$ the inequality

$\min_{\chi\in H^{1}}S_{\omega}(\chi)=S^{min}_{\omega}>-\infty$

holds;

b) if $0<\omega<1/4$, then $S^{min}_{\omega}<0;$

c) if $0<\omega<1/4$, then there exists a positive function $\chi(x)\in
H^{1},$ such that

$S_{\omega}(\chi)=S^{min}_{\omega}.$

Our main goal of this paper is to clarify if the positive minimizers of
$S_{\omega}$ are radially symmetric and unique. The above results show that we
have to consider the domain $0<\omega<1/4,$ where the key tool of Gidas, Ni
and Nirenberg (i.e. the maximum principle for the corresponding linear
operator) is not applicable.

The symmetry of the energy functional (even with constraint conditions) can
not imply, in general, the radial symmetry of the minimizers. This phenomena
was discovered and studied in the works [6], [7] and [8] in the scalar case.

Some sufficient conditions that guarantee the symmetry of minimizers have been
studied by Lopes in [15], by means of the reflection method that (for the case
of plane $x_{1}=0$) uses the functions

$u_{1}(x)=\left\\{\begin{array}[]{ll}u(\hat{x}),\
\hat{x}=(-x_{1},x_{2},\cdots,x_{n}),&\hbox{if $x_{1}>0$;}\\\ u(x),&\hbox{if
$x_{1}<0$}\end{array}\right.$

and

$u_{2}(x)=\left\\{\begin{array}[]{ll}u(\hat{x}),\
\hat{x}=(-x_{1},x_{2},\cdots,x_{n}),&\hbox{if $x_{1}<0$;}\\\ u(x),&\hbox{if
$x_{1}>0$.}\end{array}\right.$

If the functional to be minimized has the form

$E(u)=\frac{1}{2}\|\nabla u\|^{2}_{L^{2}}+\int_{\mathbb{R}^{n}}F(u(x))\ dx,$

then we have the relation

$E(u_{1})+E(u_{2})=2E(u)$

and this enables one to obtain the symmetry of minimizer, when $F(u)$ is a
combination of functions of type $|u|^{p},p\geq 2$.

The reflection method works effectively when $u(x)$ is a vector-valued
function and constraint conditions (as in the problem (4)) are involved too.

Recently, the reflection method was generalized in [16] and [17] for very
general situations and one example of possible application is the functional
of type

$E(u)=\frac{1}{2}\|\nabla u\|^{2}_{L^{2}}+\int_{\mathbb{R}^{n}}F(u(x))\
dx-A(|u|^{2}),$

involving nonlocal term as in (2). This Choquard type functional has the
specific property

$E(u_{1})+E(u_{2})\leq 2E(u),$

exploiting the negative sign of the nonlocal term $A(|u|^{2})$.

An analogous result for the scalar case can be obtained by means of the
Schwarz symmetrization (or spherical decreasing rearrangement [12])
$u^{*}(|x|)$ of the non-negative $u\in H^{1}.$ Indeed, we have the equality

$\int_{\mathbb{R}^{n}}F(u(x))\ dx=\int_{\mathbb{R}^{n}}F(u^{*}(x))\ dx,$

as well as the inequalities

$\|\nabla u\|^{2}_{L^{2}}\geq\|\nabla u^{*}\|^{2}_{L^{2}},\ \ \ A(|u|^{2})\leq
A(|u^{*}|^{2}),$

so, we get

$E(u^{*})\leq E(u)$

and one can use the property that $u$ is minimizer.

The functional in (2) is a typical example, when reflection method and Schwarz
symmetrization meet essential difficulty to be applied directly.

The main goal of this work is to find an approach to establish the symmetry of
the minimizer for functionals of Hartree type (2), involving nonlocal terms
with ”bad” sign.

To state this main result, we shall try first to connect the minimizers of the
constraint problem (4) (associated with the energy functional
$\mathcal{E}(\chi)$) with the minimization of the action functional
$S_{\omega}(\chi)$. Similar relation for local type interactions is discussed
in chapter IX of [3]. Then, we shall establish that the minimizer of Theorem 2
is a radially symmetric function.

######  Theorem 3.

The solution $\chi(x)$ from Theorem 2 is a radially symmetric function for

$\frac{1}{16}<\omega<\frac{1}{4}.$

######  Remark 1.

The result of Theorem III.1 in [4] treats more general case of potentials of
type

$V(x)=-\sum_{j=1}^{K}\frac{Z}{|x-x_{j}|},$

while in our case we have

$V(x)=-\frac{Z}{|x|}.$

Therefore, the energy functional $\mathcal{E}(\chi)$ is rotationally invariant
in our case. From Theorem 2 and Theorem 3 one can see that the solution
$\chi_{0}(x)$ of (4) is radially symmetric and unique (up to a multiplication
with complex number $z$, with $|z|=1$).

As it was mentioned above the energy (and therefore the action) is a
functional involving the nonlocal term with ”bad” sign. To explain the main
idea to treat this case, we recall the rotational symmetry of the energy (and
action) functional. Therefore, if $\chi$ is the action minimizer from Theorem
2, it is sufficient to show that the solution is symmetric with respect to
$x_{1}$-plane, for any choice of the $x_{1}$-direction. In other words, we
consider $\hat{\chi}(x)=\chi(\hat{x})$, with $\hat{x}=(-x_{1},x_{2},x_{3})$
and we aim to prove that $\chi=\hat{\chi}.$

To show this, we shall consider the two terms

$\chi_{\pm}=\frac{\chi\pm\hat{\chi}}{2}.$

So, our goal is to verify the inequality

$S_{\omega}(\chi_{+})+S_{\omega}(\chi_{-})\leq S_{\omega}(\chi)$ (8)

and see that the condition $\chi\neq\hat{\chi}$ implies
$S_{\omega}(\chi_{-})>0.$

The form of the functional $S_{\omega}$ suggests one, in order to verify (8),
to use an appropriate version of the Clarkson inequality for the quadratic
form $A(f)$. Namely, we can prove that the following inequality

$A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)\leq\frac{A(f^{2})+A(g^{2})}{2}$

holds true. Unfortunately, the usual Clarkson inequality in the form given
above, is too rough to serve as a tool for proving (8). Therefore, we shall
use a refined version of Clarkson inequality (see Lemma 5 below) in the form

$\displaystyle
A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)\leq\frac{A(f^{2})+A(g^{2})}{8}$
$\displaystyle+\frac{3\sqrt{A(f^{2})A(g^{2})}}{4}.$

The final step is to treat the uniqueness of positive minimizers. of the
problem

$S_{\omega}^{min}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\}.$ (9)

Our proof can not follow the Lieb’s uniqueness proof for the ground state
solution of the Choquard equation [11]. In general, the Lieb’s proof strongly
depends on the specific features of the nonlocal nonlinear equation (1) and
differs from the corresponding results for semilinear elliptic equation given
by Kwong in [10]. Indeed, once the radial symmetry is established, one can use
Pohozaev identities and reduce the nonlocal nonlinear elliptic problem (1) to
an ordinary differential equation of the type

$u^{\prime\prime}(r)+W(r)u(r)+4\pi\int^{r}_{0}\left(\frac{1}{s}-\frac{1}{r}\right)u^{2}(s)dsu(r)=\omega
u(r),$

where

$W_{\chi}(r)=\frac{1}{r}-4\pi\int^{\infty}_{0}\chi^{2}(s)sds.$

The positive sign in front of the nonlinear term is the main obstacle to apply
Sturm type argument and derive the uniqueness of positive solutions to this
ordinary differential equation. However, for $\frac{1}{16}<\omega<\frac{1}{4}$
we can apply the approach based on the refined Clarkson inequality and using
the orthogonal projection on the eigenspace of the first eigenvalue of the
operator $\Delta+1/|x|$, we can establish the following result.

######  Theorem 4.

Let $\frac{1}{16}<\omega<\frac{1}{4}$. Then, the solution $\chi$ of
minimization problems (9) is unique.

Let’s mention that the results in Theorems 3 and 4 can be compared with the
results in [1], where the uniqueness of minimizers for the constrained
variational problem (4) is studied. To show the relations between action
minimization and (4) one has to apply the uniqueness of action minimizers or
alternatively the uniqueness of minimizers of constrained variational problem.

The plan of the work is the following. In Section 2 we consider the maximum
principle for the linear Schrödinger equation with Coulomb potential and prove
Lemma 1. The proof of Theorem 3, stating that the minimizers are radially
symmetric is presented in Section 3 by the aid of a refined version of
Clarkson inequality. In Section 4 we establish the Pohozaev integral
relations, corresponding to equation (1), and in Section 5 we prove uniqueness
Theorem 4. Finally, in A we prove for completeness the existence of positive
action minimizers, stated in Theorem 2, while in B the connection between
energy and action minimizers is discussed.

The authors are grateful to Louis Jeanjean for important discussions and
remarks on symmetry of minimizers as well as to the referee for pointing out a
gap in the proof of the Theorem 3.

##  2 Maximum principle for Schrödinger equation with Coulomb potential

The maximum principle, stated in (6) will be verified by the aid of the
substitution

$u=\varphi w,\varphi(x)=\varphi(|x|),$

where $\varphi$ is a radial function, satisfying the property

$-\Delta\varphi-\frac{\varphi}{|x|}+\omega\varphi=h(|x|)\geq 0.$ (10)

Our goal is to construct $\varphi$, so that $\varphi(|x|)>0.$ We have several
possibilities, depending on $\omega.$ If $\omega>1/4$, we shall show that such
a function exists and it is of type

$\varphi(r)=e^{-\beta r}Q(r),\ \ \beta=\sqrt{\omega},\ Q(r)=Ar^{2}+Br+C.$ (11)

If $\omega=1/4$, then we can take simply $\varphi(r)=e^{-r/2}.$ If
$0<\omega<1/4$, we shall see that a function $\varphi$ of type (11) exists,
but $\varphi(r)$ changes the sign for $r>0.$ Hence, this function gives a
counterexample, showing that the weak maximum principle (6) is not fulfilled
in this case.

Therefore, to complete the proof of Lemma 1, we have to explain how the
existence of positive $\varphi(r)$, satisfying (10) will imply the weak
maximum principle and then to construct in different cases the function $Q(r)$
in (11), so that (10) is satisfied.

######  Proof of Lemma 1 1.

After the substitution $u=\varphi w$, we have

$P_{\omega}(u)=-\varphi\Delta w-2\nabla\varphi\nabla
w+P_{\omega}(\varphi)w=\varphi\Delta w+2\nabla\varphi\nabla w+hw.$

If $\varphi(|x|)>0$, then we can write

$-\Delta w-\frac{2}{\varphi}\nabla\varphi\nabla
w+\frac{h}{\varphi}w=\frac{g}{\varphi}.$

Choosing $M=1,$ we see that

$\frac{g}{\varphi}\in H^{2},$

so we can apply the classical maximum principle (since $h\geq 0$) and obtain
$w\geq 0.$ This argument shows that the maximum principle is fulfilled if the
function $\varphi(r)$ satisfies inequality (10) and its polynomial term
$Q(r)>0$ for $r\geq 0$.

To construct $Q$, we substitute $\varphi(r)=e^{-\beta r}Q(r)$ into (10) and
find that

$e^{\beta r}rh(r)=-(2B+C(-2\beta+1))-(6A+B(-4\beta+1))r+(6\beta-1)Ar^{2}.$

We take for simplicity $A=1$ and

$B=C(\beta-1/2),\,C=\frac{12}{(2\beta-1)(4\beta-1)}.$

Then the condition $\beta>1/2$ implies that

$B=\frac{6}{(4\beta-1)}>0,\,C=\frac{12}{(2\beta-1)(4\beta-1)}>0$

so $Q(r)>0$ and

$e^{\beta r}rh(r)=(6\beta-1)r^{2}\geq 0.$

This argument completes the proof of the weak maximum principle for
$\omega>1/4.$

If $1/16<\omega<1/4$, then we can take the same $A,B,C$ and see that

$e^{\beta r}rh(r)=(6\beta-1)r^{2}\geq 0.$

Since $A=1$ and $C<0$ in this case, the function $Q(r)$ changes the sign.

Finally, if $0<\omega<1/16$, then we choose

$A=0,B=-1,C=\frac{1}{1/2-\beta}$

and then

$Q(r)=\frac{2}{1-2\beta}-r,\ \ e^{\beta r}rh(r)=(1-4\beta)r\geq 0.$

Again, it is clear that $Q(r)$ changes the sign, and the proof of the Lemma is
completed.

##  3 Radial symmetry of action minimizers

Even in the non-local case, the problem that action and energy minimizers are
nonnegative functions, is easy to be proved. Indeed, if $\chi(x)\in H^{1}$ is
a real-valued minimizer of the functional

$S_{\omega}(\chi)=\frac{1}{2}\|\nabla\chi\|^{2}_{L^{2}}+\frac{1}{4}A(\chi^{2})-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{\chi(x)^{2}}{|x|}\
dx+\frac{\omega}{2}\|\chi\|^{2}_{L^{2}},$ (12)

then $|\chi(x)|$ satisfies the inequality

$\|\nabla|\chi|\|^{2}_{L^{2}}\leq\|\nabla\chi\|^{2}_{L^{2}},$

as well as the identities

$A(|\chi|^{2})=A(\chi^{2}),\ \ \int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx=\int_{\mathbb{R}^{3}}\frac{\chi(x)^{2}}{|x|}\ dx,$

so $|\chi(x)|\geq 0$ is also a minimizer of $S_{\omega}.$

Let us define the bilinear form

$L_{\omega}(\chi,\psi)=\langle(-\Delta-\frac{1}{|x|}+\omega)\chi,\psi\rangle_{L^{2}},\quad\omega>0$
(13)

and the corresponding quadratic form

$L_{\omega}(\chi)=\langle(-\Delta-\frac{1}{|x|}+\omega)\chi,\chi\rangle_{L^{2}}.$
(14)

The quadratic form $A(\chi)$ defined in (3) generates the corresponding
bilinear form

$A(\chi,\psi)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{\chi(x)\psi(y)}{|x-y|}dydx.$
(15)

Then, the action functional $S_{\omega}$ can be written as

$S_{\omega}(\chi)=\frac{1}{2}L_{\omega}(\chi)+\frac{1}{4}A(\chi^{2}).$ (16)

Also, for any function $\chi$ we shall denote $\hat{\chi}(x)=\chi(\hat{x})$,
where $\hat{x}=(-x_{1},x_{2},x_{3})$ for any choice of our $x_{1}$-axis. It is
easy to check that

$S_{\omega}(\chi)=S_{\omega}(\hat{\chi}),\quad
L_{\omega}(\chi)=L_{\omega}(\hat{\chi}).$ (17)

With our next result, we shall establish Clarkson type inequalities for the
forms $A$ and $L_{\omega}$. In fact, we shall prove the Lemma.

######  Lemma 5.

The following inequalities hold

$L_{\omega}\left(\frac{f+g}{2}\right)+L_{\omega}\left(\frac{f-g}{2}\right)=\frac{L_{\omega}(f)+L_{\omega}(g)}{2},$
(18) $\displaystyle
A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)\leq\frac{A(f^{2})+A(g^{2})}{8}$
$\displaystyle+\frac{3\sqrt{A(f^{2})A(g^{2})}}{4}.$ (19)

######  Proof 1.

It is easy to verify the relation

$\displaystyle
A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)$
$\displaystyle=\frac{1}{16}A(f^{2}+g^{2}+2fg)+\frac{1}{16}A(f^{2}+g^{2}-2fg).$
(20)

Note that from

$A(a+b)+A(a-b)=2A(a)+2A(b),$

equality (1) becomes

$\displaystyle
A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)=\frac{1}{8}\left[A(f^{2}+g^{2})+4A((fg)^{2})\right]$
$\displaystyle=\frac{1}{8}\left[A(f^{2})+A(g^{2})+2A(f^{2},g^{2})+4A((fg)^{2})\right]$
$\displaystyle\leq\frac{A(f^{2})+A(g^{2})}{8}+\frac{3\sqrt{A(f^{2})A(g^{2})}}{4},$
(21)

which proves (5). The first relation (18) in the Lemma, follows directly.

The next result will play the crucial role in the present study. We shall
prove the following Lemma.

######  Lemma 6.

If $L_{\omega}(f)=L_{\omega}(g)$ and $\mu,\nu\geq 0$ satisfy
$2(\mu^{2}+\nu^{2})=1,$ then

$L_{\omega}\left(\mu f+\nu g\right)+L_{\omega}\left(\mu f-\nu
g\right)=L_{\omega}(f).$ (22)

If $A(f^{2})=A(g^{2})$ and $\mu,\nu\geq 0$ satisfy $2(\mu^{2}+\nu^{2})=1,$
then we have

$A\left(\left(\mu f+\nu g\right)^{2}\right)+A\left(\left(\mu f-\nu
g\right)^{2}\right)\leq A(f^{2}).$ (23)

######  Proof 2.

Setting $\mu_{1}=2\mu,$ $\nu_{1}=2\nu$, we apply (18) with $f,g$ replaced by
$\mu_{1}f$ and $\nu_{1}g$ respectively. Thus, we get

$L_{\omega}\left(\frac{\mu_{1}f+\nu_{1}g}{2}\right)+L_{\omega}\left(\frac{\mu_{1}f-\nu_{1}g}{2}\right)=\frac{\mu_{1}^{2}L_{\omega}(f)+\nu_{1}^{2}L_{\omega}(g)}{2}.$
(24)

From $L_{\omega}(f)=L_{\omega}(g)$ and $\mu_{1}^{2}+\nu_{1}^{2}=2$, we
complete the proof of (22).

Similarly, applying (5) and the assumption $A(f^{2})=A(g^{2})$, we find

$\displaystyle
A\left(\left(\frac{\mu_{1}f+\nu_{1}g}{2}\right)^{2}\right)+A\left(\left(\frac{\mu_{1}f-\nu_{1}g}{2}\right)^{2}\right)\leq\frac{\mu_{1}^{4}+\nu_{1}^{4}+6\mu_{1}^{2}\nu_{1}^{2}}{8}A(f^{2})$

or, equivalently

$\displaystyle A\left(\left(\frac{\mu f+\nu
g}{2}\right)^{2}\right)+A\left(\left(\frac{\mu f-\nu
g}{2}\right)^{2}\right)\leq 2(\mu^{4}+\nu^{4}+6\mu^{2}\nu^{2})A(f^{2})$ (25)

Consider now the homogeneous quartic polynomial

$2(\mu^{4}+\nu^{4}+6\mu^{2}\nu^{2})$ (26)

on the circle $\mu^{2}+\nu^{2}=\frac{1}{2}$. Substituting
$\nu^{2}=\frac{1}{2}-\mu^{2}$, we obtain the following estimate

$\displaystyle
2(\mu^{4}+\nu^{4}+6\mu^{2}\nu^{2})=2((\mu^{2}+\nu^{2})^{2}+4\mu^{2}\nu^{2})$
$\displaystyle=\frac{1}{2}+4\mu^{2}-8\mu^{4}=1-\frac{(1-4\mu^{2})^{2}}{2}\leq
1.$ (27)

Then, from (25) and (2) follows the proof of the Lemma.

Turning back to the minimization problem of the action functional
$S_{\omega}$, we observe the following fact. If $\chi(x)$ is a minimizer of
the problem

$\min_{\chi\in H^{1}}S_{\omega}(\chi),$ (28)

then $\hat{\chi}(x)$ and $-\hat{\chi}(x)$ are also minimizers of
$S_{\omega}(\chi)$. Moreover, we have the property.

######  Lemma 7.

Assume that $\chi(x)$ is a minimizer of the problem (28) and one of the
following alternatives:

  1. 1.

$L_{\omega}(\chi-\hat{\chi})\geq 0$;

  2. 2.

$L_{\omega}(\chi+\hat{\chi})\geq 0$

holds. Then $\chi=\hat{\chi}$.

######  Proof 3.

For simplicity, we shall consider the first case only. Suppose
$\chi\neq\hat{\chi}$, then from (18) we have

$L_{\omega}\left(\frac{\chi+\hat{\chi}}{2}\right)+L_{\omega}\left(\frac{\chi-\hat{\chi}}{2}\right)=L_{\omega}(\chi),$
(29)

implying

$L_{\omega}\left(\frac{\chi+\hat{\chi}}{2}\right)\leq L_{\omega}(\chi).$ (30)

On the other hand, it is easy to check that the following Cauchy inequalities

$A(f^{2},g^{2})\leq\sqrt{A(f^{2})A(g^{2})},\quad
A(fg)\leq\sqrt{A(f^{2})A(g^{2})}$ (31)

hold true. Applying now (5), we obtain

$\displaystyle
A\left(\left(\frac{\chi+\hat{\chi}}{2}\right)^{2}\right)+A\left(\left(\frac{\chi-\hat{\chi}}{2}\right)^{2}\right)\leq\frac{A(\chi^{2})+A(\hat{\chi}^{2})}{8}$
$\displaystyle+\frac{3\sqrt{A(\chi^{2})A(\hat{\chi}^{2})}}{4}\leq\frac{A(\chi^{2})+A(\hat{\chi}^{2})}{2}=A(\chi^{2}),$
(32)

which, together with the assumption $\chi\neq\hat{\chi}$ gives that

$A\left(\left(\frac{\chi+\hat{\chi}}{2}\right)^{2}\right)<A(\chi^{2}).$ (33)

Thus, from (30), (33) and the definition (16) it follows

$S_{\omega}\left(\frac{\chi+\hat{\chi}}{2}\right)\leq S_{\omega}(\chi),$ (34)

which contradicts to the assumption that $\chi$ is a minimizer. This proves
the Lemma.

Now, we are ready to prove the radial symmetry of the action minimizer, stated
in Theorem 3.

######  Proof of Theorem 3 1.

Taking into account Lemma 7, we shall take a minimizer $\chi(x)\geq 0$ of
$S_{\omega}$ and shall show that the condition

$\frac{1}{16}<\omega<\frac{1}{4},$

implies that $\chi=\hat{\chi}$ or

$L_{\omega}\left(\chi-\hat{\chi}\right)>0.$ (35)

Let

$\chi(x)=e_{0}(x)+f(x),$

where $e_{0}(x)=ce^{-|x|/2},c>0$ is the eigenvector corresponding to the first
eigenvalue of the operator $\Delta+1/|x|,$ while $\langle
f,e_{0}\rangle_{L^{2}}=0.$ Since $e_{0}$ is a radial function, we have
$\hat{e_{0}}=e_{0},$ so

$\chi-\hat{\chi}=f-\hat{f}=g,\ \ \langle g,e_{0}\rangle_{L^{2}}=0,\ g\neq 0.$

######  Lemma 8.

Let us assume that $g\perp e_{0}$ in $L^{2}$. Then

$L_{\omega}\left(g\right)\geq\left(\omega-\frac{1}{16}\right)\|g\|^{2}_{L^{2}}.$

######  Proof 4.

Note that $g\perp e_{0}$ in $L^{2}$ implies

$g=\sum_{k\geq 1}c_{k}e_{k}+h,$

where $h$ is in the absolutely continuous space of the self-adjoint operator
$\Delta+\frac{1}{|x|}$ in $L^{2}$, while $e_{k}$ are eigenvectors of the same
operator in $\\{g\in L^{2};g\perp e_{0}\\}$ with eigenvalues $\omega_{k}\leq
1/16.$ On the absolutely continuous space the operator has spectrum on
$(-\infty,0)$ and it is non positive, so

$\left\langle\left(\Delta+\frac{1}{|x|}\right)h,h\right\rangle\leq 0.$

Hence, we have

$\left\langle\left(\Delta+\frac{1}{|x|}\right)g,g\right\rangle\leq\sum|c_{k}|^{2}\omega_{k}\leq\frac{1}{16}\left(\sum|c_{k}|^{2}\right)=\frac{1}{16}\|g\|^{2}_{L^{2}}$

and

$L_{\omega}\left(g\right)=-\left\langle\left(\Delta+\frac{1}{|x|}\right)g,g\right\rangle+\omega\|g\|^{2}_{L^{2}}\geq\left(\omega-\frac{1}{16}\right)\|g\|^{2}_{L^{2}}.$

This completes the proof of the Lemma.

Applying the above Lemma, we find

$L_{\omega}\left(\chi-\hat{\chi}\right)=L_{\omega}\left(g\right)\geq\left(\omega-\frac{1}{16}\right)\|g\|^{2}_{L^{2}}>0,$

since $\omega>1/16$ and $g\neq 0.$ Hence, (35) is fulfilled and the proof of
the Theorem is complete.

##  4 Pohozaev identities

In this part we shall establish the so-called Pohozaev identities for (1).
More precisely, we shall prove the following

######  Lemma 9.

If $\chi\in H^{1}(\mathbb{R}^{3})$ and satisfies (1) in
$H^{-1}(\mathbb{R}^{3})$, then the following identities hold

$\|\nabla\chi\|^{2}_{L^{2}}+\omega\|\chi\|^{2}_{L^{2}}=\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx-A(|\chi|^{2}),$ (36)

$\|\nabla\chi\|^{2}_{L^{2}}+3\omega\|\chi\|^{2}_{L^{2}}=2\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx-\frac{5}{2}A(|\chi|^{2}).$ (37)

######  Proof 5.

To prove (36) we multiply equation (1) by $\bar{\chi}$, take the real part and
integrate over $\mathbb{R}^{3}$. To prove (37) we shall use the following
relations

$\nabla\cdot(x|\chi|^{2})=3|\chi|^{2}+2\;\mathrm{Re}\;\chi(x\cdot\nabla\bar{\chi}),$
(38)
$\nabla\cdot\left(x|\nabla\chi|^{2}-2\;\mathrm{Re}\;\nabla\chi(x\cdot\nabla\bar{\chi})\right)=|\nabla\chi|^{2}-2\;\mathrm{Re}\;\Delta\chi(x\cdot\nabla\bar{\chi}),$
(39)
$\nabla\cdot(x\frac{|\chi|^{2}}{|x|})=2\frac{|\chi|^{2}}{|x|}+2\;\mathrm{Re}\;\frac{\chi(x\cdot\nabla\bar{\chi})}{|x|},$
(40)

and

$\displaystyle\nabla\cdot\left(x\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}dy}{|x-y|}|\chi|^{2}\right)=3\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}dy}{|x-y|}|\chi|^{2}$
$\displaystyle\qquad-\int_{\mathbb{R}^{3}}\frac{x(x-y)|\chi(y)|^{2}dy}{|x-y|^{3}}|\chi|^{2}+2\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}dy}{|x-y|}\;\mathrm{Re}\;\chi(x\cdot\nabla\bar{\chi}).$
(41)

Integrating (38)–(5) over $\mathbb{R}^{3}$ implies the equalities

$\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\chi(x\cdot\nabla\bar{\chi})\
dx=-\frac{3}{2}\|\chi\|^{2}_{L^{2}},$ (42)
$\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\Delta\chi(x\cdot\nabla\bar{\chi})\
dx=\frac{1}{2}\|\nabla\chi\|^{2}_{L^{2}},$ (43)
$\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\frac{1}{|x|}\chi(x\cdot\nabla\bar{\chi})\
dx=-\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\ dx,$ (44)
$\displaystyle\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}\chi(x)(x\cdot\nabla\bar{\chi}(x))}{|x-y|}dydx=-\frac{3}{2}A(|\chi|^{2})$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{x(x-y)|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}dydx.$
(45)

On the other hand, observing the symmetry

$\displaystyle\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{x(x-y)|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}\
dydx$
$\displaystyle=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{y(y-x)|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}\
dydx,$ (46)

we calculate

$\displaystyle\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{x(x-y)|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}\
dydx$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{(x-y)^{2}|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}\
dydx=\frac{1}{2}A(|\chi|^{2}).$ (47)

Substituting (5) into (5) we get

$\displaystyle\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}\chi(x)(x\cdot\nabla\bar{\chi}(x))}{|x-y|}dydx=-\frac{5}{4}A(|\chi|^{2}).$
(48)

Finally, multiplying equation (1) by $x\cdot\nabla\bar{\chi}$, taking the real
part, integrating over $\mathbb{R}^{3}$ and using (42), (43), (44) and (48) we
complete the proof of the Lemma.

The Pohozaev identities are useful to treat the uniqueness of the minimizers
(modulo multiplication by complex constant $z$ with $|z|=1$). Indeed, let
$\chi_{1}$ and $\chi_{2}$ are minimizers of the problem

$S_{\omega}^{min}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\}.$ (49)

Since

$S_{\omega}(\chi)=\frac{1}{2}\|\nabla\chi\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi\|^{2}_{L^{2}}-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx+\frac{1}{4}A(|\chi|^{2}),$

we can apply the Pohozaev identities of Lemma 9. In this way we find

$S_{\omega}(\chi)=-\frac{1}{4}A(|\chi|^{2})$ (50)

and

$A(|\chi_{1}|^{2})=A(|\chi_{2}|^{2}),\
L_{\omega}(\chi_{1})=L_{\omega}(\chi_{2}),$ (51)

where $L_{\omega}(\chi)$ is defined according to (14).

##  5 Uniqueness of minimizers

In this section we shall prove the uniqueness result of Theorem 4. The
classical approach for proving the uniqueness of minimizers is to reduce the
initial nonlinear equation to an ordinary differential equation, using the
radial symmetry. Uniqueness of positive ground state solutions for nonlinear
Schrödinger equation on ${\mathbb{R}}^{n}$ with local nonlinearities of the
form $|u|^{p}u$ for $0<p<\frac{4}{n-2}$, is a well-known fact, due to Kwong
[10]. The proof in this case relies on Sturm comparison theorems, but it
cannot be applied directly to nonlocal equations, such as (1). For the
attractive Choquard equation, Lieb in [11] prove uniqueness of energy
minimizer by using Newton’s theorem for radial function $f(x)=f(|x|)$, that is

$\displaystyle\int\frac{f(|y|)}{|x-y|^{n-2}}dy=\int\frac{f(|y|)}{\max\\{|x|,|y|\\}}dy.$
(52)

.

The repulsive sign of the Hartree term in (1) is again the main obstacle for
applying directly the standard technique.

######  Proof of Theorem 4 1.

Let $\chi_{1}$ and $\chi_{2}$ are non negative minimizers of the problem

$S_{\omega}^{min}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\}.$

Since they are radial functions, one can rewrite the elliptic equation (1),
using Newton’s theorem (52), as an ordinary differential equation of the form

$-\chi^{\prime\prime}(r)-\frac{2}{r}\chi^{\prime}(r)-\frac{\chi(r)}{r}+4\pi\int^{\infty}_{0}\frac{\chi^{2}(s)s^{2}ds}{\max\\{r,s\\}}\chi(r)+\omega\chi(r)=0.$
(53)

The above equation can be rewritten in the form

$-\chi^{\prime\prime}(r)-\frac{2}{r}\chi^{\prime}(r)-W(r)\chi(r)+4\pi\int^{r}_{0}\chi^{2}(s)\left(\frac{1}{r}-\frac{1}{s}\right)s^{2}ds\chi(r)+\omega\chi(r)=0,$

where

$W(r)=\frac{1}{r}-4\pi\int^{\infty}_{0}\chi^{2}(s)sds.$

If we set $u(r)=r\chi(r)$, then from the identity

$\chi^{\prime\prime}(r)+\frac{2}{r}\chi^{\prime}(r)=\frac{u^{\prime\prime}(r)}{r},$

the last equation becomes

$u^{\prime\prime}(r)+W(r)u(r)-4\pi\int^{r}_{0}\left(\frac{1}{r}-\frac{1}{s}\right)u^{2}(s)dsu(r)=\omega
u(r).$ (54)

This observation shows that the assumption $\chi(x)$ is a non negative
minimizer implies $u(r)>0$ for $r>0.$ Hence $\chi_{1}(x)$ and $\chi_{2}(x)$
are positive functions.

Our goal is to use the projection of $\chi_{1}$ and $\chi_{2}$ on the one
dimensional eigenspace

$E_{0}=\\{\alpha e^{-|x|/2},\alpha\in(-\infty,\infty)\\}$

is the eigenvector corresponding to the first eigenvalue $\omega_{0}=1/4$ of
the operator $\Delta+1/|x|.$ First, we have to observe that $\chi_{1}$ is not
orthogonal to $E_{0}.$ Indeed, if $\chi_{1}\perp E_{0}$, then Lemma 8 implies

$L_{\omega}(\chi_{1})\geq\left(\omega-\frac{1}{16}\right)\|\chi_{1}\|^{2}_{L^{2}}>0.$

The relation (16) guarantees now $S_{\omega}(\chi_{1})>0$ and this contradicts
the relation (50). The contradiction shows that $\chi_{1}$ (and also
$\chi_{2}$) is not orthogonal to $E_{0}.$

Let

$\chi_{1}=\mu_{1}\alpha e^{-|x|/2}+f_{1},\ \ \chi_{2}=\mu_{2}\alpha
e^{-|x|/2}+f_{2},$

where $\alpha e^{-|x|/2}\in E_{0},$ with $\alpha>0$ and $f_{1},f_{2}\perp
E_{0}.$ Note that $\mu_{1},\mu_{2}>0$, since $\chi_{1},\chi_{2}$ and $e_{0}$
are positive functions. We can choose $\alpha>0$, such that

$2(\mu_{1}^{2}+\mu_{2}^{2})=1,$ (55)

used as assumption in Lemma 6. The other assumption

$A(|\chi_{1}|^{2})=A(|\chi_{2}|^{2}),\
L_{\omega}(\chi_{1})=L_{\omega}(\chi_{2}),$

is already established in (51).

Applying Lemma 6, we find the identity

$L_{\omega}\left(\mu_{2}\chi_{1}+\mu_{1}\chi_{2}\right)+L_{\omega}\left(\mu_{2}\chi_{1}-\mu_{1}\chi_{2}\right)=L_{\omega}(\chi_{1}),$

as well as the inequality

$A\left(\left(\mu_{2}\chi_{1}+\mu_{1}\chi_{2}\right)^{2}\right)+A\left(\left(\mu_{2}\chi_{1}-\mu_{1}\chi_{2}\right)^{2}\right)\leq
A(\chi_{1}^{2}).$

Then, we have the relation

$\mu_{2}\chi_{1}-\mu_{1}\chi_{2}=\mu_{2}f_{1}-\mu_{1}f_{2}=g\perp E_{0}.$

If $g=0$, then $\chi_{1}=\mu_{1}\chi_{2}/\mu_{2}$ and one can use the ODE (54)
and the corresponding integral identities (36) and (37), to show that
$\chi_{1}=\chi_{2}.$ If $g\neq 0,$ then one can apply Lemma 8 and find

$L_{\omega}\left(\mu_{2}\chi_{1}-\mu_{1}\chi_{2}\right)\geq\left(\omega-\frac{1}{16}\right)\|g\|^{2}_{L^{2}}>0.$

Hence,

$S(\mu_{2}\chi_{1}+\mu_{1}\chi_{2})=\frac{1}{2}L_{\omega}(\mu_{2}\chi_{1}+\mu_{1}\chi_{2})+\frac{1}{4}A((\mu_{2}\chi_{1}+\mu_{1}\chi_{2})^{2})<S_{\omega}(\chi_{1})$

and this is a contradiction. The contradiction shows that $\chi_{1}=\chi_{2}$
and this completes the proof of Theorem 4.

##  Appendix A Existence of action minimizers

The existence of action minimizers for Hartree type equation is already
established in [14]. For completeness, we shall sketch the proof.

To show the boundedness from below of $S_{\omega}$, we shall prove the
following inequalities involving homogeneous Sobolev norms

$\|f\|_{\dot{H}^{s}(\mathbb{R}^{3})}=\|(-\Delta)^{s/2}f\|_{L^{2}(\mathbb{R}^{3})},\
\ s>-3/2.$

######  Lemma 10.

For any $p_{1}\in[3,6]$ and $p_{2}\in[2,3]$ we have the estimates

$\left(\int_{|x|\leq 1}|\chi(x)|^{p_{1}}dx\right)^{1/p_{1}}\leq
C\|\chi\|^{\theta_{1}}_{\dot{H}^{1}}\|\chi^{2}\|^{(1-\theta_{1})/2}_{\dot{H}^{-1}}$
(56) $\left(\int_{|x|\geq 1}|\chi(x)|^{p_{2}}dx\right)^{1/p_{2}}\leq
C\|\chi\|^{\theta_{2}}_{L^{2}}\|\chi\|^{\theta_{3}}_{\dot{H}^{1}}\|\chi^{2}\|^{(1-\theta_{2}-\theta_{3})/2}_{\dot{H}^{-1}},$
(57)

where

$\theta_{1}=\frac{5}{3}-\frac{4}{p_{1}},\ \
\theta_{2}=\frac{4(3-p_{2})}{p_{2}},\ \ \theta_{3}=\frac{p_{2}-2}{p_{2}}.$

######  Remark 2.

The assumptions $p_{1}\in[3,6]$ and $p_{2}\in[2,3]$ guarantee that all
parameters $\theta_{1},\theta_{2},\theta_{3},\theta_{2}+\theta_{3}$ are in the
interval $[0,1].$

######  Remark 3.

The relation

$\|f\|^{2}_{\dot{H}^{-1}}=\langle(-\Delta)^{-1}f,f\rangle_{L^{2}}=\frac{1}{4\pi}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{f(x)f(y)}{|x-y|}dydx$

implies

$\|\chi^{2}\|^{2}_{\dot{H}^{-1}}=\frac{1}{4\pi}A(\chi^{2}).$

######  Proof 6.

For $p_{1}=6$ the inequality (56) becomes

$\left(\int_{|x|\leq 1}|\chi(x)|^{6}dx\right)^{1/6}\leq
C\|\chi\|_{\dot{H}^{1}}$

and this is the standard Sobolev embedding. For $p_{1}=3$ we have to verify
the following estimate

$\left(\int_{\mathbb{R}^{3}}|\chi(x)|^{3}dx\right)^{1/3}\leq
C\|\chi\|^{1/3}_{\dot{H}^{1}}\|\chi^{2}\|^{1/3}_{\dot{H}^{-1}}.$ (58)

This inequality follows from

$\left|\int f(x)g(x)dx\right|\leq\|f\|_{\dot{H}^{1}}\|g\|_{\dot{H}^{-1}}$

with $f(x)=|\chi(x)|,$ $g(x)=|\chi(x)|^{2}=\chi^{2}(x)$ and the observation
that

$\||\chi|\|_{\dot{H}^{1}}=\|\chi\|_{\dot{H}^{1}}.$

Interpolation between $p_{1}=6$ and $p_{1}=3$ proves (56).

The inequality (57) for $p_{2}=3$ follows from (58).

For $p_{2}=2$ (57) reduces to the simple inequality

$\left(\int_{|x|\geq 1}|\chi(x)|^{2}dx\right)^{1/2}\leq C\|\chi\|_{L^{2}}.$

An interpolation argument implies (57) and completes the proof of the Lemma.

After this Lemma we can show that the action functional is bounded from below.

######  Lemma 11.

For any $\omega>0$ the inequality

$\min_{\chi\in H^{1}}S_{\omega}(\chi)=S^{min}_{\omega}>-\infty$

holds. For $0<\omega<1/4$ we have $S^{min}_{\omega}<0.$

######  Proof 7.

The only negative term in $S_{\omega}$ is

$-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\ dx.$

Decomposing the integration domain into $|x|\leq 1$ and $|x|>1$ we apply
Hölder inequality and obtain

$\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\ dx\leq C\left(\int_{|x|\leq
1}|\chi(x)|^{p_{1}}dx\right)^{2/p_{1}}+C\left(\int_{|x|>1}|\chi(x)|^{p_{2}}dx\right)^{2/p_{2}},$

where $p_{1}>3>p_{2}$. Applying Lemma 10 as well as the Young inequality

$X^{\theta_{1}}Y^{\theta_{2}}Z^{\theta_{3}}\leq\varepsilon X+\varepsilon
Y+C_{\varepsilon}Z,$

with

$\theta_{j}\in(0,1),\theta_{1}+\theta_{2}+\theta_{3}=1,$

we get

$\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx\leq\varepsilon\|\chi\|^{2}_{L^{2}}+\varepsilon\|\nabla\chi\|^{2}_{L^{2}}+C_{\varepsilon}\sqrt{A(\chi^{2})}.$

This estimate implies

$S_{\omega}(\chi)\geq\frac{1-\varepsilon}{2}\|\nabla\chi\|^{2}_{L^{2}}+\frac{\omega-\varepsilon}{2}\|\chi\|^{2}_{L^{2}}+\frac{1}{4}A(\chi^{2})-C_{\varepsilon}\sqrt{A(\chi^{2})}.$

Choosing $\varepsilon>0$ so small that $\varepsilon<\min(1,\omega),$ we find

$S_{\omega}(\chi)\geq\frac{1}{4}A(\chi^{2})-C_{\varepsilon}\sqrt{A(\chi^{2})}\geq-2C_{\varepsilon}^{2}.$

To finish the proof we take $\chi_{\delta}(x)=\delta e^{-|x|/2},$ such that

$\left(\Delta+\frac{1}{|x|}\right)\chi_{\delta}=\frac{1}{4}\chi_{\delta}.$

Then

$2S_{\omega}(\chi_{\delta})=(\omega-1/4)\|\chi_{\delta}\|^{2}_{L^{2}}+A(\chi_{\delta}^{2})/2.$

Since

$\|\chi_{\delta}\|^{2}_{L^{2}}=C_{0}\delta^{2},\ \
A(\chi_{\delta}^{2})/2=O(\delta^{4}),$

the condition $\omega\in(0,1/4)$ implies $2S_{\omega}(\chi_{\delta})<0$ and
this completes the proof.

######  Proof of Theorem 2 1.

Take a minimizing sequence $\chi_{k}\in H^{1},$ so that

$\lim_{k\rightarrow\infty}S_{\omega}(\chi_{k})=S_{\omega}^{min}<0.$ (59)

The argument of the proof of Lemma 11 guarantees that there exists a constant
$C>0,$ so that

$\|\chi_{k}\|_{H^{1}}\leq C.$ (60)

One can find $\chi_{*}(x)\in H^{1}$ so that (after taking a subsequence)
$\chi_{k}$ tends weakly in $H^{1}$ to $\chi_{*}.$ Using the inequality

$\int_{|x|>R}\frac{|\chi(x)|^{2}}{|x|}\ dx\leq\frac{C}{R},$

and the compactness of the embedding $L^{p}(|x|<R)\hookrightarrow
H^{1}(|x|<R),$ when $2\leq p<6,$ we see that (choosing a suitable subsequence)

$\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{3}}\frac{|\chi_{k}(x)|^{2}}{|x|}\
dx=\int_{\mathbb{R}^{3}}\frac{|\chi_{*}(x)|^{2}}{|x|}\ dx.$ (61)

Then we introduce $\varphi_{k},\ \varphi_{*}$ so that

$\Delta\varphi_{k}=-4\pi\chi_{k}^{2}(x),\ \
\Delta\varphi_{*}=-4\pi\chi_{*}^{2}(x).$

One can show that $\varphi_{k}$ tends weakly to $\varphi_{*}$ in
$\dot{H}^{1}.$ We have also the identities

$A(\chi_{k}^{2})=\int\varphi_{k}(x)\chi^{2}_{k}(x)dx=\frac{1}{4\pi}\|\nabla\varphi_{k}\|^{2}_{L^{2}}$

and

$A(\chi_{*}^{2})=\int\varphi_{*}(x)\chi^{2}_{*}(x)dx=\frac{1}{4\pi}\|\nabla\varphi_{*}\|^{2}_{L^{2}}$

so we obtain

$S_{\omega}(\chi_{k})=\frac{1}{2}\|\nabla\chi_{k}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{k}\|^{2}_{L^{2}}+\frac{1}{4}A(\chi_{k}^{2})-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{k}(x)|^{2}}{|x|}\
dx$
$=\frac{1}{2}\|\nabla\chi_{k}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{k}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{k}\|^{2}_{L^{2}}-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{k}(x)|^{2}}{|x|}\
dx.$

Using (59) and (61), we get

$\lim_{k\rightarrow\infty}S_{\omega}(\chi_{k})+\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{k}(x)|^{2}}{|x|}\
dx$
$=\lim_{k\rightarrow\infty}\frac{1}{2}\|\nabla\chi_{k}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{k}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{k}\|^{2}_{L^{2}}=S_{\omega}^{min}+\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{*}(x)|^{2}}{|x|}\
dx.$

It is well - known that for any sequence $f_{k}$ in a Hilbert space $H$
tending weakly (in $H$) to $f_{*}\in H$, one has

$\liminf_{k\rightarrow\infty}\|f_{k}\|_{H}\geq\|f_{*}\|_{H}$ (62)

and

$\lim_{k\rightarrow\infty}\|f_{k}-f_{*}\|_{H}=0\ \ \Longleftrightarrow\ \
\lim_{k\rightarrow\infty}\|f_{k}\|_{H}=\|f_{*}\|_{H}.$ (63)

From (62) we have

$S_{\omega}^{min}+\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{*}(x)|^{2}}{|x|}\
dx\geq\|\nabla\chi_{*}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{*}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{*}\|^{2}_{L^{2}}$

and a strict inequality is impossible since this will contradicts the
definition of $S_{\omega}^{min}.$ Hence

$\lim_{k\rightarrow\infty}\frac{1}{2}\|\nabla\chi_{k}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{k}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{k}\|^{2}_{L^{2}}=$
$=\frac{1}{2}\|\nabla\chi_{*}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{*}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{*}\|^{2}_{L^{2}}$

and applying (63) we conclude that

$\lim_{k\rightarrow\infty}\|\chi_{k}-\chi_{*}\|_{H^{1}}=0.$

This completes the proof of the Theorem.

##  Appendix B Connection between the action and energy minimization problems

Consider the minimization problem

$S_{\omega}^{min}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\},$ (64)

associated with the action functional $S_{\omega}(\chi)$ and the
Lions–Cazenave minimization problem

$I_{N}=\min\\{\mathcal{E}(\chi);\chi\in H^{1},\|\chi\|^{2}_{L^{2}}=N\\}.$ (65)

As we have seen before, for every $\omega\in(1/16,1/4)$, there exists (at most
one) solution $\chi_{\omega}\in H^{1}(\mathbb{R}^{3})$, which is positive and
radially symmetric, and such that

$S_{\omega}(\chi_{\omega})=S_{\omega}^{min}.$ (66)

Let us denote

$N(\omega)=\|\chi_{\omega}\|^{2}_{L^{2}}.$ (67)

The above definition of the function $N(\omega)$ poses the question if

$S_{\omega}^{min}=I_{N(\omega)}+\frac{\omega}{2}N(\omega).$

For completeness, in this section we shall prove the following Lemma.

######  Lemma 12.

If $\chi_{1}$ is a solution of (65) with $N=N(\omega)$, then $\chi_{1}$
satisfies the equation

$\displaystyle-\Delta\chi_{1}(x)+\int_{\mathbb{R}^{3}}\frac{\chi_{1}^{2}(y)dy}{|x-y|}\
\chi_{1}(x)-\frac{\chi_{1}(x)}{|x|}+\omega\chi_{1}(x)=0.$ (68)

and

$S_{\omega}(\chi_{1})=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\}.$

######  Proof 8.

To prove the Lemma we shall follow the idea of the proof of Corollary 8.3.8 in
[3]. It is obvious, that the relation

$S_{\omega}(\chi_{1})=\mathcal{E}(\chi_{1})+\frac{\omega}{2}N(\omega),$

guarantees that $\chi_{1}$ is a minimizer of the problem

$\min_{\|\chi\|^{2}_{L^{2}}=N(\omega)}S_{\omega}(\chi).$

Since,

$S_{\omega}(\chi_{1})=\min_{\|\chi\|^{2}_{L^{2}}=N(\omega)}S_{\omega}(\chi)\geq\min
S_{\omega}(\chi)=S_{\omega}(\chi_{\omega}),$

we can use (67) and see that this inequality becomes equality, so

$S_{\omega}(\chi_{1})=\min_{\|\chi\|^{2}_{L^{2}}=N(\omega)}S_{\omega}(\chi)=\min
S_{\omega}(\chi)=S_{\omega}(\chi_{\omega}).$

Now, the uniqueness result of Theorem 4 implies $\chi_{1}=\chi_{\omega}$ and
completes the proof.

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# Shared Sequencing and Latency Competition as a Noisy Contest

Akaki Mamageishvili and Jan Christoph Schlegel

###### Abstract

We study shared sequencing for different chains from an economic angle. We
introduce a minimal non-trivial model that captures cross-domain arbitrageurs’
behavior and compare the performance of shared sequencing to that of separate
sequencing. While shared sequencing dominates separate sequencing trivially in
the sense that it makes it more likely that cross-chain arbitrage
opportunities are realized, the investment and revenue comparison is more
subtle: In the simple latency competition induced by First Come First Serve
ordering, shared sequencing creates more wasteful latency competition compared
to separate sequencing. For bidding-based sequencing, the most surprising
insight is that the revenue of shared sequencing is not always higher than
that of separate sequencing and depends on the transaction ordering rule
applied and the arbitrage value potentially realized.

##  1 Introduction

Cryptocurrency trading is one of the biggest use cases of blockchains that
support smart contracts. Decentralized exchanges (DEXes) that run on different
chains handle daily exchange volumes of billions of dollars equivalent. A lot
of (potential) trading volume in DEXes is generated through arbitrage trading
between different exchanges on the same chain, as well as between exchanges on
different chains or between DEXes and centralized exchanges. While arbitrage
opportunities on the same chain can be exploited through the atomic execution
of bundles of transactions, cross-domain arbitrage is generally riskier
(harder) to capture. We propose a game theoretic model that captures the
different nature of same-domain versus cross-domain arbitrage on blockchains.
One of our main motivation for the model is recent proposals around shared
sequencing for rollups.

Rollup chains are layer-two chains built on top of Ethereum for the purpose of
scaling the Ethereum main chain. They are offering lower fees and faster
execution of transactions. Because of these properties, DEXes built on them
already attract significant trading volume. Rollups have a designated
operator, called a sequencer, which receives transactions from users and
schedules these transactions for execution. Shared sequencing schemes propose
to jointly process transactions for several rollups, with the ultimate aim of
improving user experience. For example, with a shared sequencer the users will
be able to schedule their transaction bundles atomically even if they include
transactions on different rollup chains: either all transactions will be
scheduled and executed or none. A secondary aim of shared sequencing is
improved economics through lower operating and maintenance costs and through
collecting more of the increased arbitrage value captured by traders. More
specifically, users send bundles of transactions to the shared sequencer. The
sequencer distributes them to corresponding chains for execution. This gives
flexibility to cross-domain arbitrageurs to get their transactions scheduled
on different chains simultaneously. A proposed platform for shared sequencing
is for example Espresso Systems, [7]. One fundamental economic question that
we address in this paper is whether shared-sequencing services really lead to
improved value capture as advertised by their proponents.

As, in any trading activity, it is important how sequencers order
transactions. Similarly to traditional finance exchanges, rollup sequencers
usually use a first come first serve (FCFS) policy, see, e.g., [2]. This often
generates latency competition, where parties invest into latency reduction to
be competitive in arbitrage trading.111While the latency focus of competition
is special to rollups, the phenomenon of transaction-ordering induced
competition in blockchains is much broader and often described under the
umbrella term of MEV [1]. FCFS competition may even affect the sequencer’s
operation. For example, Arbitrum’s sequencer distributes its feed to other
nodes in a fair (random) order, that is exploited by parties creating many
nodes. Recently, there have been proposals to extract some fraction of MEV for
rollups, see [3]. In this approach, transaction senders bid per resource unit
(in the case of chains using Ethereum’s EVM, per-gas) that the transaction
consumes. Our game-theoretic model allows us to study FCFS-based transaction
ordering, as well as bidding-based ordering. We are interested, in particular,
in the latency investment resp. bidding expenditure by traders in the shared
sequencers versus the multiple sequencer case and derive these quantities in
equilibrium.

We consider two versions of our model: the baseline version assumes that
traders’ expenditure on executing a transaction is irreversible and
independent of whether they succeed in capturing the arbitrage. This is for
example the case if traders compete in latency investment with an FCFS policy.
In other cases, it might be reasonable to assume instead that traders can
partially recoup their cost if their arbitrage trade fails. More precisely, if
bidding for inclusion is done per resource unit of the chain (per gas), the
user may save a significant fraction of their cost by adding conditional
statements to the transaction that check whether the price moved away from the
target and not executing the transaction if it does. This approach may save a
significant portion of gas usage in case of losing the race. However, it does
(slightly) increase the overall cost by adding one conditional statement,
which is added independently of the outcome. If the race is lost sufficiently
often, which is the case in our model with few symmetric players, such a
strategy is dominant and will be applied. Therefore, we also consider a
version of our model where the loser of the race for the arbitrage trade only
pays a fraction of the bid.

###  1.1 Related Literature

Cross-domain extraction in the context of blockchains was first discussed in
[5]. The authors argue that to extract value, arbitrageurs should execute
trades on different chains sequentially and bridge assets between them. While
a useful abstraction for the formalism, in a competitive setting trades should
be executed in parallel, and assets may be bridged between the chains after
all trades are executed. [6] conduct an empirical analysis on cross-domain
MEV. We base our modeling on this assumption of competing for the fastest
execution of trades on all involved chains. A more recent survey article in
the field of cross-chain MEV is [4], which describes different approaches and
implementation details.

##  2 Model and Results

There are two traders, indexed by $i\in\\{1,2\\}$, who compete to execute an
arbitrage trade.222We comment on the case of more than two traders and of more
than two sequencers in the discussion section. The trade, if successful, has a
value of $v>0$ to both of the traders. There are two chains. Successful
execution of the trade requires to have two transactions included in the
sequencer(s) earlier than the other trader. We consider two situations:

  * •

In the first, both transactions are processed by the same sequencer, modeling
the scenario where the same sequencer serves both chains and DEXes on them. In
this case, the trader can send the transaction in a single message that
arrives at the sequencer with the same timestamp.

  * •

In the second, the two transactions need to be sent to two independent
sequencers, serving the two different chains and therefore, DEXes on them.

The competition is modeled as follows: each trader $i$ can invest into a
signal $s_{i}$ at a cost of $C(s_{i})$, where $C$ is a strictly increasing and
strictly convex function, and receives a random term $\epsilon_{i}$. Trader
$i$ gets their transaction included earlier than $j$ if and only if

$s_{i}+\epsilon_{i}>s_{j}+\epsilon_{j}.$

The interpretation of $s_{i}$ and $\epsilon_{i}$ can be different, depending
on the choice of the cost function $C(s_{i})$. We have in particular the
following scenarios in mind:

  1. 1.

The sequencer applies an FCFS transaction ordering policy. The signal $s_{i}$
represents an investment into latency reduction that leads to a timestamp
$t(s_{i}+\epsilon_{i})$, where $t$ is a strictly decreasing function. Here
$\epsilon_{i}$ represents random fluctuations in the timestamps around the
average.

  2. 2.

The sequencer uses a combination of bidding and timestamp-based ordering such
as the TimeBoost proposal in [3], where $s_{i}$ is the monetary bid,
$\epsilon_{i}$ is the timestamp (assuming that the latency technology is
fixed).

  3. 3.

The sequencer runs a competitive block-building market similar to the one on
Ethereum. The term $s_{i}$ represents expenditure on fees, tips, and direct or
indirect payments to intermediaries (searchers, builders, etc.) and the random
term $\epsilon_{i}$ represents frictions and other uncertain factors in the
market.

Since our model is additive in the random terms, it suffices to specify a
distribution for the difference in signals. Define the random variable
$\Delta_{ij}=\epsilon_{i}-\epsilon_{j}$. Then, the probability of winning the
single sequencer competition for $i$ is

$F_{\Delta}(s_{i}-s_{j}),$

where $F_{\Delta}$ is the distribution of $\Delta_{ij}$. We assume that
$F_{\Delta}$ is symmetric around $0$ and uni-modal with $F_{\Delta}(0)=1/2$.
That is, players are symmetric conditional on making equal investments in the
signal. Therefore, the probability of winning the single sequencer competition
for player $j$ is:

$F_{\Delta}(s_{j}-s_{i}).$

For the two sequencer case, if the delta in the first sequencer
$\Delta_{ij}^{1}$ and the delta in the second sequencer $\Delta_{ij}^{2}$ are
independent and identically distributed, and each bidder produces the same
signal $s_{i}$ for both sequencers, we have accordingly that the probability
of winning the two sequencer competition for $i$ is:

$F_{\Delta^{2}}(s_{i}-s_{j})=F^{2}_{\Delta}(s_{i}-s_{j}),$

where $F_{\Delta^{2}}$ is the distribution of the maximum of $\Delta^{1}_{ij}$
and $\Delta_{ij}^{2}$.

### Symmetric Pure Strategy Equilibria

Since our model is symmetric for the two traders ($F_{\Delta}$ is symmetric,
and the value $v$ is the same for both), we are interested in symmetric pure
strategy Nash equilibria, which means that the two traders choose equal
average signals $s_{i}=s_{j}$ which are mutual best responses to the other
player’s strategy. For the one sequencer case, a trader’s best response solves

$\max_{s_{i}\geq 0}F_{\Delta}(s_{i}-s_{j})v-C(s_{i}).$

Assuming that the optimal signal is characterized by the first order condition
and that the equilibrium is symmetric, we have candidates for equilibrium
signals $s^{*}$ characterized by:

$C^{\prime}(s^{*})=f_{\Delta}(0)v.$

The equation is obtained by deriving the expected payoff function with respect
to the signal $s_{i}$, setting $s_{i}=s_{j}$, and equating it to zero. Then,
the equilibrium signal is

$s^{*}=s^{*}_{i}=s^{*}_{j}=(C^{\prime})^{-1}(f_{\Delta}(0)v).$ (1)

Similarly for the two sequencer case, we have the optimization problem

$\max_{s_{i}\geq 0}F_{\Delta}^{2}(s_{i}-s_{j})v-2C(s_{i}).$ (2)

Assuming that the optimal signal is characterized by the first order condition
and that the equilibrium is symmetric, we have candidates for equilibrium
signals $s^{*}$ characterized by:

$C^{\prime}(s^{*})=f_{\Delta}(0)F_{\Delta}(0)v=f_{\Delta}(0)v/2.$ (3)

Then, the equilibrium signal is

$s^{*}=(C^{\prime})^{-1}({f(0)v}/{2}).$ (4)

In the following subsections, we consider different specifications of $C$
based on different sequencer designs.

###  2.1 Latency Competition

First, we consider a scenario where the platform uses an FCFS policy to order
transactions. This is a standard choice in traditional financial exchanges as
well as in most rollup sequencers. FCFS policies often lead to latency
competition. That is, players invest in latency improvement. The
interpretation of our model in this case is as follows: the signal can be
interpreted as an average latency that can be improved by investment. The
random noise gives fluctuation around the average.

A straightforward parametric specification of the cost function is to assume
constant cost elasticity (which can be estimated from data), $C(s)=s^{\beta}$,
where $\beta>1$ is the cost elasticity. The timestamp $t_{i}$ is a decreasing
function (lower timestamps win the competition) of investment and the random
term, $t_{i}(s_{i}+\epsilon_{i})$. For example we can choose
$t(s_{i}+\epsilon_{i})=\frac{1}{s_{i}+\epsilon_{i}}$ so that $i$ wins if
$t_{i}<t_{j}\Leftrightarrow s_{i}+\epsilon_{i}>s_{j}+\epsilon_{j}$.

######  Proposition 1.

For the single sequencer case with a high enough value of the trade, $v\geq
2\left(\frac{\beta}{2f(0)}\right)^{\beta}$, there exists a unique pure
strategy Nash equilibrium where both players invest

$\left(\frac{f(0)v}{\beta}\right)^{\beta/(\beta-1)}$

into latency reduction. For low value of the trade,
$v<2\left(\frac{\beta}{2f(0)}\right)^{\beta},$ both players in equilibrium do
not invest in latency reduction.

###### Proof.

Using the previously derived Equation 1 for the optimal signaling strategy, we
obtain:

$s=\left(\frac{f(0)v}{\beta}\right)^{1/(\beta-1)}.$

This induces an investment cost of:

$C(s)=\left(\frac{f(0)v}{\beta}\right)^{\beta/(\beta-1)},$

which is smaller than the expected value from the arb $v/2$ if and only if

$\frac{v}{2}\geq\left(\frac{f(0)v}{\beta}\right)^{\beta/(\beta-1)}\Leftrightarrow
v\geq 2\left(\frac{\beta}{2f(0)}\right)^{\beta}.$

∎

By the same kind of proof for the case of two sequencers, we obtain:

######  Proposition 2.

For the two sequencer case with a high enough value of the trade, $v\geq
8\left(\frac{\beta}{4f(0)}\right)^{\beta}$, there exists a unique pure
strategy Nash equilibrium where both players invest

$2\left(\frac{f(0)v}{2\beta}\right)^{\beta/(\beta-1)}$

into latency reduction. For low value of the trade,
$v<8\left(\frac{\beta}{4f(0)}\right)^{\beta},$ both players in equilibrium do
not invest in latency reduction.

From the protocol perspective, investment in latency can be seen as a waste,
as it is not captured by the protocol e.g. to fund protocol development and/or
other users’ subsidies.

A straightforward calculation shows that this waste with a shared sequencer is
always larger than with two separate sequencers as long as the value of trade
is large enough so that investing into latency happens in the two sequencer
case.

######  Proposition 3.

For a high enough value of trade $v$ (so that traders invest in latency at
all), the waste with a shared sequencer is always higher than the waste with
two separate sequencers.

###### Proof.

From the previous two propositions, the cost is higher under the shared
sequencer if and only if $1\geq 2^{-\frac{1}{\beta-1}}$, which holds for any
$\beta>1$. ∎

One particularly natural case is that $\Delta$ is normally distributed with
mean $0$ and standard deviation $\sigma$, in which case
$f(0)=\tfrac{1}{\sqrt{2\pi}\sigma}.$

######  Example 1.

For the normally distributed case, the investment into latency in the one
sequencer case is

$\left(\frac{v}{\sqrt{2\pi}\beta\sigma}\right)^{\beta/(\beta-1)},$

and for the two sequencer case is

$2\left(\frac{v}{2\sqrt{2\pi}\beta\sigma}\right)^{\beta/(\beta-1)}.$

Thus, investment is increasing in the value $v$, decreasing in the standard
deviation of noise $\sigma$, and decreasing in cost elasticity $\beta$.

####  2.1.1 Constant Cost Elasticity for Bidding

The previously analyzed cost function can be given a different interpretation
as the cost of bidding in a bidding-based transaction ordering scheme. This
could for example be a block-based system where bidders bid for positions in
the block, but random factors in the block building market perturb the
ranking. Alternatively, it could be a hybrid bidding scheme, similar to
TimeBoost (which we analyze in the next section) in which time advantage is
bought. In the latter case, we would need to impose an upper bound on bidding
needed for the liveness of the protocol. In the case of bidding instead of
latency investment, expenditure by traders is captured by the protocol so that
the comparison flips (revenue instead of waste) and a shared sequencer is
preferable.

######  Proposition 4.

For constant cost elasticity and high enough value of trade $v$ (so that
traders invest in latency at all), the bidding revenue with a shared sequencer
is always at least as high as the bidding revenue with two separate
sequencers.

The previous result assumed that there is no upper bound on the signal that
bidders can produce. A natural variant is a cap $\bar{s}$ on $s$. For low
values of $U$ we have an opposite result, in particular, the following holds:

######  Proposition 5.

For $\bar{s}<\left(\frac{vf(0)}{2\beta}\right)^{1/(1-\beta)}$, the revenue
with a shared sequencer is lower than the total revenue with separate
sequencers.

###### Proof.

In the case of a low enough value of $\bar{s}$, the only pure Nash equilibrium
of the single sequencer game is for both players to bid $\bar{s}$ in which
case the revenue per bidder is $\bar{s}^{\beta}$. In the case of shared
sequencers, the revenue per bidder is the minimum of
$2(\frac{f(0)v}{2\beta})^{\beta/(\beta-1)}$ and $2\bar{s}^{\beta}$ which is
larger than $\bar{s}^{\beta}$ if $\bar{s}$ is small enough. ∎

###  2.2 TimeBoost

In this section, we consider the TimeBoost proposal, [3]. In the proposal,
transactions are ordered by a combination of time stamps and bids. More
precisely, transactions are ordered by a score:

$s-t,$

where $t$ is the timestamp of the arrival of a bid at the sequencer and $0\leq
s<g$ is the ”time boost” which is a function of a bid submitted together with
the transaction. Here the parameter $g>0$ bounds the maximal time boost to
guarantee the finality of the ordering policy. Following the formula of the
original TimeBoost proposal [3], we get the following fee for producing a
boost of $s$:

$C(s_{i})=\frac{cs_{i}}{g-s_{i}},$

where $c>0$ is a parameter representing an approximate normalized marginal
cost. Given the cost function for the TimeBoost proposal, the marginal cost is

$C^{\prime}(s_{i})=\frac{cg}{(g-s_{i})^{2}},$

so that our equilibrium condition becomes:

$\frac{cg}{(g-s^{*})^{2}}=f(0)v\Rightarrow
s^{*}=g-\frac{\sqrt{cg}}{\sqrt{f(0)v}},$

which for the normal distribution becomes

$s^{*}=g-\frac{\sqrt{\sqrt{2\pi}cg\sigma}}{\sqrt{v}}.$

To guarantee the existence of a pure strategy symmetric Nash equilibrium, we
need to make sure that the candidate for the optimal signal through the FOC is
positive, and gives positive profit (otherwise the best response to $s^{*}$
would be $0$ instead of $s^{*}$). This gives the following conditions:

######  Proposition 6.

For the single sequencer case with parameters $f(0),c,g,v$ such that

$1+\tfrac{c}{v}+\tfrac{v}{4c}\geq gf(0)$

there exists a pure strategy Nash equilibrium where both players produce a
signal of

$s^{*}=\begin{cases}g-\sqrt{\frac{cg}{vf(0)}},\quad&\text{ if
}v>\frac{c}{gf(0)}\\\ 0,&\text{ if }v\leq\frac{c}{gf(0)}\end{cases}$

with cost

$\max\\{\sqrt{cgf(0)v}-c,0\\}.$

###### Proof.

From Equation 1 with $C^{\prime}(s)=\frac{cg}{(g-s)^{2}}$ we obtain the
expression for the optimal signal $s^{*}$. Substituting the optimal signal in
the cost function $C(s)=\frac{cs}{g-s}$ gives the expression for the cost. The
parameter restriction comes from the requirement that in equilibrium profit
$v/2-C(s^{*})$ needs to be non-negative. ∎

By the same logic for the case two sequencers

######  Proposition 7.

For the two sequencer case with parameters $f(0),c,g,v$ such that

$1+\tfrac{c}{v}+\tfrac{v}{4c}\geq\tfrac{gf(0)}{2}$

there exists a pure strategy Nash equilibrium where both players produce a
signal

$s^{*}=\begin{cases}g-\sqrt{\frac{2cg}{vf(0)}},\quad&\text{ if
}v>\frac{2c}{gf(0)}\\\ 0,&\text{ if }v\leq\frac{2c}{gf(0)}\end{cases}$

in each of the two sequencers with cost

$\max\\{\sqrt{2cgf(0)v}-2c,0\\}.$

######  Example 2.

For the normally distributed case, the revenue in the one sequencer case is

$\sqrt{\frac{cgv}{\sqrt{2\pi}\sigma}}-c,$

and for the two sequencer case is

$\sqrt{\frac{2cgv}{\sqrt{2\pi}\sigma}}-2c.$

Thus, investment is increasing in the value $v$, increasing in the standard
deviation of noise $\sigma$, increasing in the time boost parameter. The
effect of an increase in the parameter $c>0$ is increasing for small $c$ and
decreasing for large $c$.

It is interesting to compare the expenditure on signaling in both cases. We
can interpret the fraction $\frac{c}{g}$ as an approximation to the marginal
cost of bidding. The following proposition then states that the expenditure is
larger in the two sequencer case, if the marginal cost of bidding is low, the
variance in noise is small or the valuation is large. In these cases, the two
sequencer case produces more revenue. Otherwise, the shared sequencer produces
more revenue.

######  Proposition 8.

For the time boost policy with normally distributed $\Delta$, the bidding
revenue for two separate sequencers is larger than the revenue under a shared
sequencer if and only

$0.068447\frac{v}{\sigma}\geq\frac{c}{g},$

i.e. if the marginal cost of bidding is small, the value of the trade is high
or the variance in timestamps is low.

###### Proof.

We have

$\sqrt{2\frac{cgv}{\sqrt{2\pi}\sigma}}-2c\geq\sqrt{\frac{cgv}{\sqrt{2\pi}\sigma}}-c\Leftrightarrow
0.414214\sqrt{\frac{cgv}{\sqrt{2\pi}\sigma}}\geq c\Leftrightarrow
0.068447\frac{v}{\sigma}\geq\frac{c}{g}.$

∎

The previous analysis took the parameter $c$ as given. Next, we consider the
revenue when choosing the parameter in an optimal way. The assumption is that
the sequencer sets these parameters ex-ante assuming $v$ is distributed
according to a CDF $G$ on the non-negative reals.333The same analysis can be
conducted for the optimal choice of $g$. However, choosing $g$ differently has
other implications beyond revenue as it influences the time until the finality
of a transaction.

######  Proposition 9.

Under time boost, assuming $4\geq gf(0)$, the ex-ante revenue for a shared
sequencer is the same as the revenue from two separate sequencers if the
parameter $c>0$ is optimally chosen.

###### Proof.

The parameter restriction $4\geq gf(0)$ guarantees that a pure strategy Nash
equilibrium exists for any $c>0$ in both cases. Let $G$ be the distribution of
values $v$. For the shared sequencer case the revenue optimisation problem is
the following:

$\max_{c>0}\int_{\tfrac{c}{gf(0)}}^{\infty}(\sqrt{cgf(0)v}-c)dG(v).$

For the two sequencer case, the optimisation problem is:

$\max_{c>0}\int_{\tfrac{2c}{gf(0)}}^{\infty}(\sqrt{2cgf(0)v}-2c)dG(v).$

Now note that with the substitution $\tilde{c}:=2c$ the second problem becomes
equivalent to the first problem and therefore has the same objective value

∎

##  3 Extensions

In this section, we consider extensions to the base model presented in the
paper so far.

###  3.1 More than 2 Players and Chains

The results can be straightforwardly extended to the case of multiple chains,
as long as the distributions of deltas are independent between chains. In that
case the probability of winning for player $i$ is given by
$F_{\Delta}^{n}(s_{i}-s_{j})$ and an analogous argument as previously the
first order condition

$C^{\prime}(s^{*})=f_{\Delta}(0)F^{n-1}_{\Delta}(0)v=f_{\Delta}(0)v/2^{n-1}.$

which leads to the equilibrium signal

$s^{*}=(C^{\prime})^{-1}({f(0)v}/{2^{n-1}}).$ (5)

Note that the signal is exponentially decreasing in $n$.

The generalization to more than two players is more complicated, as now the
probability of winning does not only depend on the difference of two signaling
strategies, but rather on all marginal distributions of signals. However,
qualitatively we expect similar results as in the two player case.

###  3.2 Smaller cost in case of losing

We have assumed that in case the trader does not win, he still incurs the full
cost of bidding. It is natural to consider the extension where in case of
losing, the player pays only an $\alpha$ fraction of the total bid. Then, the
optimization problem with one sequencer becomes:

$\max_{s_{i}\geq
0}F_{\Delta}(s_{i}-s_{j})(v-C(s_{i}))+(1-F_{\Delta}(s_{i}-s_{j}))(-\alpha
C(s_{i})).$ (6)

The first summand corresponds to the expected gains of winning the race, and
the second summand corresponds to the expected losses of losing the race. The
first order condition on (6) with respect to $s$ gives:

$f_{\Delta}(s_{i}-s_{j})v-(1-\alpha)(f_{\Delta}(s_{i}-s_{j})C(s_{i})+F_{\Delta}(s_{i}-s_{j})C^{\prime}(s_{i}))-\alpha
C^{\prime}(s_{i})=0.$

Plugging in $s_{i}=s_{j}$ and $F_{\Delta}(0)=\frac{1}{2}$ gives an equation
from which we can solve for $s_{i}$:

$f_{\Delta}(0)v-(1-\alpha)(f_{\Delta}(0)C(s_{i})+1/2C^{\prime}(s_{i}))-\alpha
C^{\prime}(s_{i})=0.$ (7)

Note that the case $\alpha=1$ corresponds to the equilibrium condition (1).
For the constant cost elasticity previously analyzed in Section 2.1,
$C(s)=s^{\beta}$, we can show that the equilibrium solution is decreasing in
$\alpha$ provided that the cost elasticity is large enough. In particular, the
signal with positive $\alpha$ is always higher than the solution from (1):

######  Proposition 10.

The equilibrium signal $s$ is decreasing in $\alpha$ for $\beta\geq 2$.

###### Proof.

Without loss of generality, assume that $f_{\Delta}(0)=1$ and $v=1$, a
normalization. After plugging in the values for $f_{\Delta}(0)$, $v$ and
simplification, the equation (7) becomes

$T(\alpha,s_{i}):=1-(1-\alpha)(s_{i}^{\beta}+1/2\beta
s_{i}^{\beta-1})-\alpha\beta s_{i}^{\beta-1}=0.$ (8)

By the implicit function theorem, we get:

$\frac{\partial T/\partial s_{i}}{\partial
T/\partial\alpha}=\frac{-(1-\alpha)\beta
s_{i}^{\beta-1}-1/2(1-\alpha)\beta(\beta-1)s_{i}^{\beta-2}-\alpha\beta(\beta-1)s_{i}^{\beta-2}}{s_{i}^{\beta}+1/2\beta
s_{i}^{\beta-1}-\beta s_{i}^{\beta-1}}.$

The nominator of the right-hand side of the above expression is always
negative. The denominator: $s_{i}^{\beta}-\frac{1}{2}\beta s_{i}^{\beta-1}$ is
negative for $\beta$ large enough, since $s_{i}\leq\beta/2$ holds for any
$\beta\geq 2$. The reason is that if the value is $v=1$, the investment in the
equilibrium can not be more than $1$. That is, $s_{i}$ that solves (8)
equation is decreasing in $\alpha$ parameter. ∎

Similarly to the shared sequencer case, we can show that the equilibrium
solution for the two separate sequencers is decreasing in $\alpha$ if the cost
elasticity is large enough. The optimization problem in case of 2 sequencers
is:

$\displaystyle\max_{s_{i}\geq 0}$ $\displaystyle
F_{\Delta}^{2}(s_{i}-s_{j})(v-2C(s_{i}))-2F_{\Delta}(s_{i}-s_{j})(1-F_{\Delta}(s_{i}-s_{j}))(1+\alpha)C(s_{i})-$
$\displaystyle(1-F_{\Delta}(s_{i}-s_{j}))^{2}2\alpha C(s_{i}).$

The first term corresponds to the expected gains of winning both races and
therefore, obtaining an arbitrage value. The second term corresponds to the
expected losses of winning one race and losing the other. In this case, full
cost in the first race and fractional cost in the second race are incurred.
The third term corresponds to the expected losses of losing both races. In
this case, fractional costs in both races are incurred. The first order
condition is:

$\displaystyle
2F_{\Delta}(s_{i}-s_{J})f_{\Delta}(s_{i}-s_{J})(v-2C(s_{i}))+F^{2}_{\Delta}(s_{i}-s_{j})(-2C^{\prime}(s_{i}))-$
$\displaystyle(2f_{\Delta}(s_{i}-s_{j})-4F_{\Delta}(s_{i}-s_{j})f_{\Delta}(s_{i}-s_{j}))(1+\alpha)C(s_{i})-$
$\displaystyle
2(F_{\Delta}(s_{i}-s_{j})-F^{2}_{\Delta}(s_{i}-s_{j}))(1+\alpha)C^{\prime}(s_{i})-$
$\displaystyle 2(1-F_{\Delta}(s_{i}-s_{j}))(-f_{\Delta}(s_{i}-s_{j}))2\alpha
C(s_{i})-(1-F_{\Delta}(s_{i}-s_{j}))^{2}2\alpha C^{\prime}(s_{i})=0,$

which after plugging in $s_{i}=s_{j}$ and $F(0)=\frac{1}{2}$ becomes

$\displaystyle f_{\Delta}(0)(v-2C(s_{i}))-(1+\alpha)C^{\prime}(s_{i})+2\alpha
f_{\Delta}(0)C(s_{i})=0.$ (9)

Note that the case $\alpha=1$ corresponds to the equilibrium condition (3). A
similar calculation as before establishes the monotonicity of the equilibrium
signal. Comparing the equilibrium cost between shared and separate sequencing
in this extended model is less tractable than in the baseline model. However,
we can obtain numerical comparison results by solving equations (7) and (9),
for given $C$, $F$ and $\alpha$.

### Other Cost Functions, Noise Models and Empirical Work

We currently explore other extensions of the model as well. A common design
proposal for transaction ordering is frequent ordering auction design where
transactions are processed in batches according to bids attached to the
transactions. Randomness comes into play because whether or not a transaction
makes it into the batch is determined by time stamps. Towards the end of a
batch, this introduces randomness about which transactions make it into the
current or the next batch. An ordering auction formulation would require a
different cost function specification, as well as a non-additive noise model.
Another extension of our model that we currently explore is to calibrate the
parameters of our model according to real-world data. This gives a precise
sense of the magnitude of the effects studied.

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