11institutetext:  22institutetext: Lomonosov Moscow State University and
“Vega” Institute, Moscow, Russia
22email<EMAIL_ADDRESS>33institutetext: Lomonosov Moscow State
University and Laboratoire de Mathématiques, Université de Franche-Comté,
Besançon, France
33email<EMAIL_ADDRESS>
# Ruin problems with investments on a finite interval: PIDEs and their
viscosity solutions

Viktor Antipov  Yuri Kabanov

(Received: date / Accepted: date)

###### Abstract

The study deals with the ruin problem when an insurance company invests its
reserve in a risky asset whose the price dynamics is given by a geometric Lévy
process. Considering the ruin probability as a of the capital reserve we
obtain for it a partial integro-differential equation understood in a
viscosity sense and prove a result on the uniqueness of the viscosity solution
for a corresponding boundary value problem.

###### Keywords:

Ruin probabilities Risky investments Actuarial models with investments
Viscosity solutions Integro-differential equations for ruin probabilities

###### MSC:

60G44

JEL Classification G22 $\cdot$ G23

##  1  Introduction

In the classical models of collective risk theory it was assumed that
insurance companies keep their reserves in cash or put it into a bank account,
typically, with zero interest rate. The dynamic of the reserve was described,
in the majority of cases by a compound Poisson process or, in more advanced
frameworks, by a Lévy process. The use of the latter is justified by the fact
that the majority of claims are small with respect to the total volume, the
interarrival times between small claims are small with respect to the planning
horizon and the limit theorems can be applied.

In more realistic recent models it is assumed that insurance companies operate
in a financial environment and their reserves, totally, or partially, are
invested into risky assets having various types of price dynamic used in
quantitative finance. In this paper we consider a rather general model where
the price dynamic and the insurance business activity are described by two
independent Lévy processes. In this case the capital reserve evolution solves
a linear stochastic equation and referred to as the generalized
Ornstein–Uhlenbeck process driven by the mentioned Lévy processes. To the
moment there is quite a lot of works devoted to ruin problems with
investments, mainly on asymptotic behavior of the exit probability of the
capital reserve process from the positive half-axis. Usually papers deal with
the asymptotic behavior of the ultimate ruin probability when the initial
capital tends to infinity, see, e.g., KP2020  and references therein.
”Ultimate” means: on an infinite time interval. For very specific cases it is
proven that the ultimate ruin probability is smooth and satisfies, in the
classical sense, an integro-differential equation, see KP , KPukh . Though
from the practical point of view the ruin probability with investments on a
finite interval is more important, we do not know any works on it.

It the present paper we obtain a partial integro-differential equation (PIDE)
for more general functionals of the capital reserve. This equation is
understood in the viscosity sense developing further the ideas from the paper
Belk . A uniqueness theorem for a boundary value problem is proven. Note that
the viscosity approach allows us to avoid quite a delicate question of
smoothness of the studied functions with respect to the initial capitaL

##  2 The model

In the description of the model we follow KP2020  and use the notations of
stochastic calculus.

We are given a filteblue probability space $(\Omega,{\cal F},{\bf F}=({\cal
F}_{r})_{r\geq 0},{\bf P})$ with two independent Lévy processes $R$ and $P$,
having the Lévy triplets $(a,\sigma^{2},\Pi)$ and
$(a_{P},\sigma_{P}^{2},\Pi_{P})$, and admitting, respectively, the canonical
decompositions

$R_{t}=at+\sigma W_{t}+xI_{\\{|x|\leq
1\\}}*(\mu-\nu)_{t}+xI_{\\{|x|>1\\}}*\mu_{t}$ (1)

and

$P_{t}=a_{P}t+\sigma_{P}W^{P}_{t}+xI_{\\{|x|\leq
1\\}}*(\mu^{P}-\nu^{P})_{t}+xI_{\\{|x|>1\\}}*\mu^{P}_{t},$ (2)

where $\mu$ and $\mu^{P}$ are jump measures, $\nu(dt,dx)=\Pi(dx)dt$ and
$\nu^{P}(dt,dx)=\Pi_{P}(dx)dt$, $W$ and $W^{P}$ are Wiener processes. Let
denote by $R^{d}$ and $P^{d}$ purely discontinuous martingale components of
$R$ and $P$, i.e. the processes

$R^{d}:=xI_{\\{|x|\leq 1\\}}*(\mu-\nu),\qquad P^{d}:=xI_{\\{|x|\leq
1\\}}*(\mu^{P}-\nu^{P}).$

We consider the process $X=X^{t,u}$, defined on the interval $[t,\infty[$ by
the formula

$X^{t,u}_{r}=u+\int_{]t,r]}X^{t,u}_{s-}dR_{s}+P_{r}-P_{t},$ (3)

or, in the traditional differential form,

$dX^{t,u}_{r}=X^{t,u}_{r-}dR_{r}+dP_{r},\quad X^{t,u}_{t}=u.$ (4)

We assume the condition $\Pi(]\infty,-1])=0$ implying that the stochastic
(Doléans) exponential $S_{t}:={\cal E}(R)$ is strictly positive (a.s.).

In the actuarial context $X^{t,u}$ is the process describing the evolution of
the reserve of an insurance company starting its activity at time $t>0$ with
the initial capital $u>0$, having the time horizon $T>t$, and investing the
total of its reserve in a risky asset with the price process $S$ solving the
stochastic equation $dS=S_{-}dR$, $S_{0}=1$. That is, $S$ is a geometric Lévy
process; in the particular case, where $R$ is a Wiener process with drift, $S$
is a geometric Brownian motion, the most popular model for the stock price.

Recall that in the classical models the process $P$, describing the business
activity of the insurance company is, usually, a compound Poisson process with
drift. The latter can be represented in the form

$P_{r}=cr+\sum_{i=1}^{N_{r}}\xi_{i},$

where $N$ is a Poisson process independent on the i.i.d. sequence $\xi_{i}$.
The case where $c>0$ and $\xi_{i}<0$ corresponds to the non-life insurance.
The model $c<0$ and $\xi_{i}>0$ serves to describe the life insurance business
where the company pays the annuity with the intensity $|c|$ and gets benefits
$\xi_{i}$ or the capital of a venture capital paying salary and selling
innovations. Note also that in the modern literature the generalized
Ornstein–Uhlenbeck model suggested by Paulsen in Paul-93  attracted a lot of
attention since a general Lévy process $P$ is viewed as a reasonable
approximation of the claim flow of a large company with many small customers.

Let $\tau^{t,u}:=\inf\\{s>t\colon X^{t,u}_{s}\leq 0\\}$. For a measurable
function $V:{\mathbb{R}}\to[0,1]$ vanishing on $]-\infty,0]$ we can define the
function

$\Psi(t,u)=\Psi(t,u;T):={\bf E}V(X_{\tau^{t,u}}^{t,u})I_{\\{\tau^{t,u}<T\\}}.$
(5)

If $V=I_{]0,\infty[}$, the value $\Psi(t,u)={\bf P}(\tau^{t,u}<T)$ for $u>0$
is the probability of the exit of $X^{t,u}$ from $]0,\infty[$ before $T$. In
particular, in the actuarial terminology, $\Psi(0,u)$ is the ruin probability
on the time interval $[0,T]$ for the company starting its activity at time
zero with the initial capital $u$. In the case, where $T=\infty$, and the
$V=I_{]0,\infty[}$ function $\Psi(0,u;\infty)$, $u>0$, is called ultimate ruin
probability. It is the latter which is studied in the absolute majority of
works on ruins with investments.

##  3 Partial integro-differential equation

Let ${\cal L}:={\cal L}^{d}_{R}+{\cal L}^{i}_{P}+{\cal L}^{d}_{R}+{\cal
L}^{i}_{P}$, where

$\displaystyle{\cal L}^{d}_{R}\Psi(t,u)$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\sigma^{2}u^{2}\Psi_{uu}(t,u)+au\Psi_{u}(t,u),$ (6)
$\displaystyle{\cal L}^{i}_{R}\Psi(t,u)$ $\displaystyle:=$
$\displaystyle\int(\Psi(t,u(1+z))-\Psi(t,u)-\Psi_{u}(t,u)uzI_{\\{|z|\leq
1\\}})\Pi(dz),$ (7) $\displaystyle{\cal L}^{d}_{P}\Psi(t,u)$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\sigma_{P}^{2}\Psi_{uu}(t,u)+a_{P}\Psi_{u}(t,u),$ (8)
$\displaystyle{\cal L}^{i}_{P}\Psi(t,u)$ $\displaystyle:=$
$\displaystyle\int(\Psi(t,u+z)-\Psi(t,u)-\Psi_{u}(t,u)zI_{\\{|z|\leq
1\\}})\Pi_{P}(dz).$ (9)

###### Theorem 3.1

The function $\Psi(t,u)$ on $]0,T[\times]0,\infty[$ is a viscosity solution of
the partial integro-differential equation

$\Psi_{t}(t,u)+{\cal L}\Psi(t,u)=0.$ (10)

The above equation is a symbolical one since we do not know whether the given
function $\Psi$ (defined in (5)) has derivatives. E.g., the ruin probability
it is not differentiable even in the case of the Cramér model ($R=0$, $P$ is a
compound Poisson process with drift, the equation (10) is of the first order)
if the distributions of claims is discrete. The idea of viscosity solutions is
to substitute in the equation the function $\Psi$ by a suitably chosen smooth
test functions. More precisely, one needs to verify at each point of the
domain each element of a suitably defined set of test functions is a
supersolution of (10) and for another set it is a subsolution.

Let denote by $C_{b}^{1,2}(t,u)$ the set of bounded continuous functions
$f:[0,T[\times{\mathbb{R}}\to{\mathbb{R}}$ continuously differentiable in the
time variable and two times continuously differentiable in the phase variable
in some neighborhood of the point $(t,u)\in[0,T[\times]0,\infty[$ and equal to
zero for $u\in]-\infty,0]$. For $f\in C_{b}^{1,2}(t,u)$ the values ${\cal
L}_{R}f(t,u)$ and ${\cal L}_{P}f(t,u)$ are well-defined.

For the operator ${\cal L}$ we introduce the following definitions.

A function $\Psi:]0,T[\times{\mathbb{R}}\to[0,1]$ is called viscosity
supersolution if for every point $(t,u)\in]0,T[\times]0,\infty[$ and every
function $f\in C_{b}^{1,2}(t,u)$ such that $f(t,u)=\Psi(t,u)$ and $f\leq\Psi$
the inequality $f_{t}(t,u)+{\cal L}f(t,u)\leq 0$ holds.

A function $\Psi:[0,T[\times{\mathbb{R}}\to[0,1]$ is called viscosity
subsolution if for every point $(t,u)\in[0,T[\times]0,\infty[$ and every
function $f\in C_{b}^{1,2}(t,u)$ such that $f(t,u)=\Psi(t,u)$ and $f\geq\Psi$
the inequality $f_{t}(t,u)+{\cal L}f(t,u)\geq 0$ holds.

A function $\Psi:[0,T[\times{\mathbb{R}}\to[0,1]$ is viscosity solution if
$\Psi$ is simultaneously a viscosity super- and subsolution.

At last, a function $\Psi\in C_{b}^{1,2}([0,T]\times[0,1])$ is called
classical supersolution if the inequality $(\partial/\partial t+{\cal
L})\Psi\leq 0$ holds on on $[0,T[\times]0,\infty[$. We add the adjective
strict when $(\partial/\partial t+{\cal L})\Psi<0$ on $[0,T]\times]0,\infty[$.

###### Lemma 1

The ruin probability $\Psi$ as a function of $(t,u)$ is a viscosity
supersolution.

Proof. Let $f\in C_{b}^{1,2}(t,u)$ be such that $f(t,u)=\Psi(t,u)$ and
$f\leq\Psi$ on $[0,T]\times{\mathbb{R}}$. Take $\epsilon\in]0,u/2[$ and
$h\in]0,T-t[$ such that that the rectangular
$[t,t+h]\times[u-\epsilon,u+\epsilon]$ is inside a ball in which the function
$f$ is in $C^{1,2}$.

Let $M=(M_{s})_{s\geq t}$ be the local martingale with

$dM_{s}=dX^{c}_{s}+X_{s-}dR^{d}_{s}+dP^{d}_{s},\qquad M_{t}=0,$

where $X^{c}$ is the continuous martingale component of $X$,

$dX^{c}_{s}:=\sigma X_{s-}dW_{s}+\sigma_{P}dW^{P}_{s},\qquad d\langle
X^{c}\rangle_{s}=\sigma^{2}X^{2}_{s}ds+\sigma_{P}^{2}ds.$

Define the stopping time $\tau_{h}=\tau^{\epsilon}_{h}\leq t+h$ by putting

$\tau_{h}:=\inf\big{\\{}s>t\colon
X^{t,u}_{s}\notin[u-\epsilon\,,u+\epsilon]\big{\\}}\wedge\inf\\{s>t\colon|M_{s}|\geq
1\\}\wedge(t+h).$

By the Ito formula in its standard form applied to the semimartingale
$X=X^{t,u}$

$\displaystyle f(\tau_{h},X_{\tau_{h}})$ $\displaystyle=$ $\displaystyle
f(t,u)+\int_{]t,\tau_{h}]}f_{u}(s,X_{s-})dX_{s}+\int_{]t,\tau_{h}]}f_{t}(s,X_{s-})ds$
$\displaystyle+\frac{1}{2}\int_{]t,\tau_{h}]}f_{uu}(s,X_{s-})d\langle
X^{c}_{s}\rangle$
$\displaystyle+\sum_{s\in]t,\tau_{h}]}\big{(}f(s,X_{s-}+\Delta
X_{s})-f(s,X_{s-})-f_{u}(s,X_{s-})\Delta X_{s}\big{)}.$

Representing the integral with respect to $dX$ as

$\int_{]t,\tau_{h}]}f_{u}(s,X_{s-})dM_{s}+\int_{]t,\tau_{h}]}f_{u}(s,X_{s-})(aX_{s-}+a_{P})ds+J^{R}+J^{P},$

where

$\displaystyle J^{R}$ $\displaystyle:=$ $\displaystyle\int_{]t,\tau_{h}]}\int
f_{u}(s,X_{s-})X_{s-}zI_{\\{|z|>1\\}}\mu(ds,dz),$ $\displaystyle J^{P}$
$\displaystyle:=$ $\displaystyle\int_{]t,\tau_{h}]}\int
f_{u}(s,X_{s-})zI_{\\{|z|>1\\}}\mu^{P}(ds,dz).$

and the sum of jumps as

$\displaystyle\int_{]t,\tau_{h}]}\int\big{(}f(s,X_{s-}(1+z))-f(s,X_{s-})-f_{u}(s,X_{s-})X_{s-}zI_{\\{|z|\leq
1\\}}\big{)}\mu(ds,dz)$
$\displaystyle+\int_{]t,\tau_{h}]}\int\big{(}f(s,X_{s-}+z)-f(s,X_{s-})-f_{u}(s,X_{s-})zI_{\\{|z|\leq
1\\}}\big{)}\mu^{P}(ds,dx)-J^{R}-J^{P},$

we rewrite the right-hand side of the Ito formula in the form

$\displaystyle
f(t,u)+\int_{]t,\tau_{h}]}f_{u}(s,X_{s-})(dX^{c}_{s}+X_{s-}dR^{d}+dP^{d})$
$\displaystyle+\int_{]t,\tau_{h}]}\Big{(}f_{t}(s,X_{s-})+f_{u}(s,X_{s-})(aX_{s-}+a_{P})+\frac{1}{2}f_{uu}(s,X_{s-})(\sigma^{2}X^{2}_{s-}+\sigma_{P}^{2})\Big{)}ds$
$\displaystyle+\int_{]t,\tau_{h}]}\int\big{(}f(s,X_{s-}(1+z))-f(s,X_{s-})-f_{u}(s,X_{s-})X_{s-}zI_{\\{|z|\leq
1\\}}\big{)}\mu(ds,dz)$
$\displaystyle+\int_{]t,\tau_{h}]}\int\big{(}f(s,X_{s-}+z)-f(s,X_{s-})-f_{u}(s,X_{s-})zI_{\\{|z|\leq
1\\}}\big{)}\mu^{P}(ds,dz)$

Taking the expectation we get that

${\bf E}f(\tau_{h},X_{\tau_{h}})=f(t,u)+{\bf
E}\int_{t}^{\tau_{h}}\big{(}f_{t}(s,X_{s})+{\cal L}(s,X_{s}))ds.$

Recall that $f(t,u)=\Psi(t,u)$ and $f\leq\Psi$. The crucial observation is
that

$\Psi(t,u)={\bf E}\,\Psi(\tau_{h},X^{t,u}_{\tau_{h}})$

due to the strong Markov property of $X$. Combining these facts, we get that

${\bf E}\int_{t}^{\tau_{h}}\big{(}f_{t}(s,X_{s})+{\cal
L}f(s,X_{s})\big{)}ds\leq 0.$

Almost surely $\tau_{h}(\omega)=t+h$ when $h$ is sufficiently small (the
threshold for which this equality holds depends on $\omega$). Using the
Lebesgue theorem on dominated convergence we get that

${\cal L}f(t,u)=\lim_{h\downarrow 0}\frac{1}{h}{\bf
E}\int_{t}^{\tau_{h}}\big{(}f_{t}(s,X_{s})+{\cal L}f(s,X_{s})\big{)}ds\leq 0$

and the lemma is proven. $\Box$

###### Lemma 2

The function $\Psi$ is a viscosity subsolution.

Proof. One argue as above but with the inequality $f\geq\Psi$. $\Box$

###### Remark 1

If the function $\Psi\in C^{1,2}$, then the equation ${\cal L}\Psi(t,u)=0$
holds in the classical sense.

##  4 Parabolic jets

Let $f$ and $g$ be functions defined in a neighborhood of the origin of
${\mathbb{R}}^{2}$. Denoting $(h_{1},h_{2})$ a generic point of
${\mathbb{R}}^{2}$, we shall write $f(.)\lessapprox g(.)$ if
$f(h_{1},h_{2})\leq g(h_{1},h_{2})+o(|h_{1}|+|h_{2}|^{2})$ as $|h|\to 0$. The
notations $f(.)\gtrapprox g(.)$ and $f(.)\approx g(.)$ have the obvious
meaning.

For $(b,p,A)\in{\mathbb{R}}^{2}\times{\mathbb{R}}_{+}$ we consider the
function

$Q_{b,p,A}(t,x):=bt+px+(1/2)Ax^{2},\quad(t,x)\in{{\mathbb{R}}}^{2},$

linear in $t$ and quadratic in $x$. Define the parabolic super\- and subjets
of a function $v$ at the point $(t,x)\in K:=]0,T[\times]0,\infty[$:

$\displaystyle{\cal P}^{+}v(t,x)$ $\displaystyle:=$
$\displaystyle\\{(b,p,A)\in{\mathbb{R}}^{3}:\ v((t,x)+.)\lessapprox
v(t,x)+Q_{b,p,A}(.)\\},$ $\displaystyle{\cal P}^{-}v(t,x)$ $\displaystyle:=$
$\displaystyle\\{(b,p,A))\in{\mathbb{R}}^{3}:\ v((t,x)+.)\gtrapprox
v(t,x)+Q_{b,p,A}(.)\\}.$

The following lemma asserts that for $v\in C(K)$ and any element
$(b,p,A)\in{\cal P}^{+}v(t,x)$, $(t,x)\in K$, one can associate a test
function $f$ dominating $v$ on $K$ and uniformly close to $v$, having at the
point $(t,x)$ the same value as $v$, belonging to $C^{1,2}$ in a neighborhood
of $(t,x)$ and having at this point the first derivative in $t$ equal to $b$
and the first and second derivatives in $x$, equal, respectively, to $p$ and
$A$. Moreover, $f=v$ outside any arbitrary small neighborhood of $(t,x)$.

###### Lemma 3

Let $v\in C(K)$, $(t,x)\in K$, and $(b,p,A)\in{\cal P}^{+}v(t,x)$. Then for
${\varepsilon},\delta\in]0,1]$ there exist a real $r>0$ depending on
${\varepsilon}$ and a function $f\in C^{1,2}(t,x)$ depending on
${\varepsilon}$ and $\delta$ with the following properties:

$(i)$ $f(t,x)=v(t,x)$, $f_{t}(t,x)=b$, $f_{x}(t,x)=p$, $f_{xx}(t,x)=A$;

$(ii)$ $v\leq f\leq v+{\varepsilon}$ on $K$;

$(iii)$ $f((t,x)+h)=v(t,x)+Q_{b,p,A}(h)+\rho(h)$ for $h\in{\cal O}_{r}(0,0)$,
where $C^{1,2}$-function $\rho:{{\mathbb{R}}}^{2}\to{{\mathbb{R}}}$ with
compact support is such that

$\lim_{|h|\to 0}\frac{\rho(h)}{|h_{1}|+|h_{2}|^{2}}=0,$ (11)

$(iv)$ $f=v$ on the set $K\setminus{\cal O}_{r^{\prime}}(t,x)$ where
$r^{\prime}:=r+\delta$.

Proof. Take $r\in]0,1[$ such that the ball ${\cal O}_{2r}(t,x)\subset K$. By
definition of ${\cal P}^{+}v(t,x)$,

$\varrho(h):=v((t,x)+h)-v(t,x)-Q_{b,p,A}(h)\leq(|h_{1}|+|h_{2}|^{2})\varphi(h),$

where $\varphi(h)\to 0$ as $h\to 0$. We consider on ${\cal
O}_{2r}(0,0)\cap{\mathbb{R}}_{+}^{2}$ the function

$\widehat{\varphi}(u):=\sup_{\\{h:\ |h_{1}|\leq u_{1},|h_{2}|\leq
u_{2}\\}}\frac{\varrho^{+}(h)}{|h_{1}|+|h_{2}|^{2}}\leq\sup_{\\{h:\
|h_{1}|\leq u_{1},|h_{2}|\leq u_{2}\\}}\varphi^{+}(u).$

Obviously, $\widehat{\varphi}$ is continuous, increasing in $u_{1}$, $u_{2}$,
and $\widehat{\varphi}(u)\to 0$ as $u\to 0$. We extend $\widehat{\varphi}$ to
a positive continuous function on ${{\mathbb{R}}}^{2}_{+}$ with
$\widehat{\varphi}(u)=0$ for $|u|\geq 1$ and associate with it the function
$\Upsilon:{\mathbb{R}}^{2}_{+}\to{\mathbb{R}}_{+}$ by putting

$\Upsilon(u)=\Upsilon(u_{1},u_{2}):=\int_{u_{1}}^{2u_{1}}\widehat{\varphi}(\xi,u_{2})d\xi+\frac{2}{3}\int_{u_{2}}^{2u_{2}}\int_{\eta}^{2\eta}\widehat{\varphi}(u_{1},\xi)d\xi
d\eta.$

Then

$(u_{1}+u_{2}^{2})\widehat{\varphi}(u)\leq\Upsilon(u)\leq(u_{1}+u_{2}^{2})\widehat{\varphi}(4u).$

It follows that the function $\rho:h\mapsto\Upsilon(|h_{1}|,|h_{2}|)$ has a
compact support, belongs to $C^{1,2}({\cal O}_{r}(0,0))$, it vanishes at the
origin together with the partial derivatives of the first order in $h_{1}$ and
$h_{2}$ and the second partial derivative in $h_{2}$. For $h\in{\cal
O}_{r}(0,0)$ we have that

$v((t,x)+h)-v(t,x)-Q_{b,p,A}(h)\leq(|h_{1}|+|h_{2}|^{2})\widehat{\varphi}(h)\leq\Upsilon(|h_{1}|,|h_{2}|)=\rho(h).$

Thus, the function

$f^{0}:(s,y)\mapsto v(t,x)+Q_{b,p,A}(s-t,y-x)+\rho(s-t,y-x)$

dominates $v$ on the ball ${\cal O}_{r}(t,x)$. Without loss of generality,
diminishing $r$ if necessary, we may assume that $f^{0}\leq v+{\varepsilon}$
on the ball ${\cal O}_{r^{\prime}}(t,x)$ for some $r^{\prime}>r$.

Let us consider a continuous function
$\xi_{r,r^{\prime}}:{{\mathbb{R}}}^{2}\to[0,1]$ such that
$\xi_{r,r^{\prime}}=1$ on ${\cal O}_{r}(0,0)$ and $\xi_{r,r^{\prime}}=0$
outside ${\cal O}_{r^{\prime}}(0,0)$. The function

$f:(s,y)\mapsto
f^{0}(s,y)\xi_{r,r^{\prime}}(s-t,y-x)+v(s,y)\big{(}1-\xi_{r,r^{\prime}}(s-t,y-x)\big{)}$

has all the needed properties. $\Box$

The corresponding assertion for ${\cal P}^{-}v(x)$ holds with obvious changes
in the formulation.

The following result is a scalar version of the Ishii lemma.

###### Lemma 4

Let ${\cal O}$ be an open subset of ${\mathbb{R}}_{+}$ and $v$ and $\tilde{v}$
be two continuous functions on $]0,T[\times{\cal O}$. Let
$\Delta_{n}(t,x,y):=v(t,x)-\tilde{v}(t,y)-\frac{1}{2}n|x-y|^{2}$, where the
parameter $n>0$. Suppose that the function $\Delta_{n}$ attains a maximum at
$(t_{n},x_{n},y_{n})\in]0,T[\times{\cal O}\times{\cal O}$. Then there are
reals $b_{n}$, $A_{n}$, and $\tilde{A}_{n}$ such that

$(b_{n},n(x_{n}-y_{n}),A_{n})\in\bar{\cal
P}^{+}v(t_{n},x_{n}),\qquad(b_{n},n(x_{n}-y_{n}),\tilde{A}_{n})\in\bar{\cal
P}^{-}\tilde{v}(t_{n},y_{n}),$

and, for all $x,y\in{\cal O}$

$A_{n}x^{2}-\tilde{A}_{n}y^{2}\leq 3n(x-y)^{2}.$ (12)

In particular, $A_{n}\leq\tilde{A}_{n}$.

###### Remark 2

Recall that the symbol $(b_{n},n(x_{n}-y_{n}),A_{n})\in\bar{\cal
P}^{+}v(t_{n},x_{n})$ means that there are sequences $(t_{nk},x_{kn})$ and
$(b_{nk},p_{nk},A_{nk})\in{\cal P}^{+}v(t_{n,k},x_{nk})$, $k\geq 1$, such that

$(t_{nk},x_{kn})\to(t_{n},x_{n}),\quad(b_{nk},p_{nk},A_{nk})\to(b_{n},n(x_{n}-y_{n}),A_{n}),\
\ \ k\to\infty.$

Similar meaning has the symbol
$(b_{n},n(x_{n}-y_{n}),\tilde{A}_{n})\in\bar{\cal
P}^{-}\tilde{v}(t_{n},y_{n})$.

##  5 The maximum principle and uniqueness theorem

###### Theorem 5.1

Let $\psi,\tilde{\psi}$ be continuous functions on $[0,T]\times\bar{R}_{+}$
with values in the interval $[0,1]$ be, respectively, viscosity subsolution
and supersolution of (10) such that

$\psi(t,0)=\tilde{\psi}(t,0),\quad\psi(t,\infty)=\tilde{\psi}(t,\infty),\quad\psi(T,u)=\tilde{\psi}(T,u).$
(13)

Then $\tilde{\psi}\geq\psi$.

Proof. For any $\delta>0$ the function
$\tilde{\psi}^{\delta}(t,u):=\tilde{\psi}(t,u)+\delta/t\geq\delta/t$ is a
strict supersolution of (10). It converges uniformly to infinity as $t\to 0$.
We prove that $\tilde{\psi}^{\delta}\geq\psi$ and get the result by letting
$\delta$ tends to zero.

If the claimed inequality fails, then
$\psi(t_{0},u_{0})=\tilde{\psi}^{\delta}(t_{0},u_{0})+{\varepsilon}_{0}$ for
some $u_{0}\in]0,\infty[$, $t_{0}\in[0,T[$, and ${\varepsilon}_{0}>0$.

1. The first step in a way to get a contradiction consists to use the method of doubling variables and apply the Ishii lemma to each element of the sequence of functions $\Delta_{n}=\Delta^{\delta}_{n}$ attaining their maximum values.

Define a sequence of continuous functions
$\Delta_{n}:]0,T]\times{\mathbb{R}}_{+}^{2}\to{\mathbb{R}}$, $n\geq 0$, by

$\Delta_{n}(t,u,w):=\psi(t,u)-\tilde{\psi}^{\delta}(t,w)-\frac{1}{2}n|u-w|^{2}.$

Note that $\Delta_{n}(t,u,u)=\Delta_{0}(t,u,u)$ for all
$(t,u)\in]0,T]\times{\mathbb{R}}_{+}$ and $\Delta_{0}(t,0,0)\leq 0$.

Let us consider the non-empty closed sets

$\Gamma_{n}:=\\{(t,u,w)\in]0,T]\times{\mathbb{R}}^{2}_{+}\colon\Delta_{n}(t,u,w)\geq{\varepsilon}_{0}\\}.$

Then
$\Gamma_{n+1}\subseteq\Gamma_{n}\subseteq[\delta,T]\times{\mathbb{R}}^{2}_{+}$.
The set $\Gamma_{1}$ is bounded. Indeed, if it is not the case, we can find a
sequence $(s_{m},u_{m},w_{m})\in\Gamma_{1}$ such that $s_{m}\to s$,
$u_{m},w_{m}\to\infty$. Due to assumed continuity of $\psi$ and $\tilde{\psi}$
we get the inequality

$\limsup_{m}\Delta_{1}(s_{m},u_{m},w_{m})\leq-\delta/s<0,$

i.e. a contradiction.

It follows that all functions $\Delta_{n}$ attain their maximal values over
$[0,T]\times{\mathbb{R}}^{2}_{+}$ on the compact subset $\Gamma_{1}$. That is,
there exist $(t_{n},x_{n},y_{n})\in[\delta,T]\times{\mathbb{R}}^{2}_{+}$ such
that

$\displaystyle\Delta_{n}(t_{n},x_{n},y_{n})$ $\displaystyle=$
$\displaystyle\bar{\Delta}_{n}:=\sup_{(t,x,y)\in[0,T]\times{\mathbb{R}}^{2}_{+}}\Delta_{n}(t,x,y)$
$\displaystyle\geq$
$\displaystyle\bar{\Delta}:=\sup_{(t,x)\in[0,T]\times{\mathbb{R}}_{+}}\Delta_{0}(t,x,x)\geq{\varepsilon}_{0}>0.$

This implies that the sequence $n|x_{n}-y_{n}|^{2}$ is bounded. We continue to
argue (without introducing new notations) with a subsequence of indices along
which $(t_{n},x_{n},y_{n})$ converge to some limit
$(\widehat{t},\widehat{x},\widehat{x})$. Necessarily, $n|x_{n}-y_{n}|^{2}\to
0$ (otherwise $\Delta_{0}(\widehat{t},\widehat{x},\widehat{x})>\bar{\Delta}$).
Also, it is easily seen that
$\bar{\Delta}_{n}\downarrow\Delta_{0}(\widehat{t},\widehat{x},\widehat{x})=\bar{\Delta}$.
Thus, $(\widehat{t},\widehat{x})\in K:=]0,T[\times]0,\infty[$.

2. The second step consists in constructing test functions.

By virtue of the Ishii lemma applied to the functions $\psi$ and
$\tilde{\psi}^{\delta}$ at the point $(t_{n},x_{n},y_{n})$ there are numbers
$b_{n}$, $A_{n}$ and $\tilde{A}_{n}$ satisfying (12) such that

$(b_{n},n(x_{n}-y_{n}),A_{n})\in\bar{\cal
P}^{+}\psi(t_{n},x_{n}),\qquad(b_{n},n(x_{n}-y_{n}),\tilde{A}_{n})\in\bar{\cal
P}^{-}\tilde{\psi}^{\delta}(t_{n},y_{n}).$ (14)

According to the remark following the Ishii lemma, there are points
$(t_{nk},x_{nk})\in K$ and $(b_{nk},p_{nk},A_{nk})\in{\cal
P}^{+}\psi(t_{nk},x_{nk})$ such that

$(t_{nk},x_{kn})\to(t_{n},x_{n}),\quad(b_{nk},p_{nk},A_{nk})\to(b_{n},n(x_{n}-y_{n}),A_{n}),$

and $(t^{\prime}_{nk},y_{kn})\in K$,
$(b^{\prime}_{nk},p^{\prime}_{nk},\tilde{A}^{\prime}_{nk})\in{\cal
P}^{-}\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk})$ such that

$(t^{\prime}_{nk},y_{kn})\to(t_{n},y_{n}),\quad(b^{\prime}_{nk},p^{\prime}_{nk},\tilde{A}^{\prime}_{nk})\to(b_{n},n(x_{n}-y_{n}),\tilde{A}_{n}),\quad
k\to\infty.$

From the elements of jets ${\cal P}^{+}\psi(t_{nk},x_{nk})$ and ${\cal
P}^{-}\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk})$ we built functions
$f^{nk}$ and $\tilde{f}^{nk}$ to test, respectively, that $\psi$ is a
viscosity subsolution of the equation (10) at the point $(t_{nk},x_{nk})$ and
$\tilde{\psi}^{\delta}$ is a viscosity supersolution at the point
$(t^{\prime}_{nk},y_{nk})$.

Take arbitrary positive ${\varepsilon}_{n},\delta_{n}\to 0$ as $n\to\infty$
and
${\varepsilon}_{nk}\downarrow{\varepsilon}_{n},\delta_{nk}\downarrow\delta_{n}$
as $k\to\infty$ (the particular choice of parameters will be further). By
virtue of Lemma 3 there are $r_{nk}>0$ and $f^{nk}\in
C_{b}^{1,2}(t_{nk},x_{nk})$ with the properties $\psi\leq
f^{nk}\leq\psi+{\varepsilon}_{nk}$,

$\displaystyle f^{nk}(t_{nk},x_{nk})$ $\displaystyle=$
$\displaystyle\psi(t_{nk},x_{nk}),\quad f^{n}_{t}(t_{nk},x_{nk})=b_{nk},\quad
f^{nk}_{x}(t_{nk},x_{nk})=p_{nk},$ $\displaystyle
f^{nk}_{xx}(t_{nk},x_{nk})=A_{nk},$

$f^{nk}((t_{nk},x_{nk})+h)=\psi(t_{nk},x_{nk})+Q_{b_{nk},p_{nk},A_{nk}}(h)+\rho_{nk}(h)$
for $h\in{\cal O}_{r^{nk}}(0,0)$, where $C^{1,2}$-function
$\rho_{nk}:{{\mathbb{R}}}^{2}\to{{\mathbb{R}}}$ with compact support is such
that

$\lim_{|h|\to 0}\frac{\rho_{nk}(h)}{|h_{1}|+|h_{2}|^{2}}=0,$ (15)

and $f^{nk}=\psi$ on the set $K\setminus{\cal O}_{r^{\prime}}(t_{nk},x_{nk})$
where $r^{\prime}_{nk}:=r_{nk}+\delta_{n}$.

Similarly, there are $\tilde{r}_{nk}>0$ and a function $\tilde{f}^{nk}\in
C_{b}^{1,2}(t^{\prime}_{nk},y_{nk})$ with the properties
$\tilde{\psi}^{\delta}\geq\tilde{f}^{nk}\geq\tilde{\psi}^{\delta}-{\varepsilon}_{nk}$,

$\displaystyle\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk})$ $\displaystyle=$
$\displaystyle\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk}),\quad\tilde{f}^{nk}_{t}(t^{\prime}_{nk},y_{nk})=b^{\prime}_{nk},\quad\tilde{f}^{nk}_{x}(t^{\prime}_{nk},y_{nk})=p^{\prime}_{nk},$
$\displaystyle\tilde{f}^{nk}_{xx}(t^{\prime}_{nk},y_{nk})=\tilde{A}^{\prime}_{nk},$

$\tilde{f}^{nk}((t^{\prime}_{nk},y_{nk})+h)=\tilde{\psi}(t^{\prime}_{nk},y_{nk})+Q_{b^{\prime}_{nk},p^{\prime}_{nk},\tilde{A}^{\prime}_{nk}}(h)+\tilde{\rho}_{nk}(h)$
for $h\in{\cal O}_{\tilde{r}^{nk}}(0,0)$, where $C^{1,2}$-function
$\tilde{\rho}_{nk}:{{\mathbb{R}}}^{2}\to{{\mathbb{R}}}$ with compact support
is such that

$\lim_{|h|\to 0}\frac{\tilde{\rho}_{nk}(h)}{|h_{1}|+|h_{2}|^{2}}=0,$ (16)

and $\tilde{f}^{nk}=\tilde{\psi}^{\delta}$ on the set $K\setminus{\cal
O}_{\tilde{r}^{\prime}_{nk}}(t^{\prime}_{nk},y_{nk})$ where
$\tilde{r}^{\prime}_{nk}:=\tilde{r}_{nk}+\tilde{\delta}_{n}$.

By definitions of sub- and supersolutions we have that

$\displaystyle f^{nk}_{t}(t_{nk},x_{nk})+{\cal L}f^{nk}(t_{nk},x_{nk})$
$\displaystyle\geq$ $\displaystyle 0$
$\displaystyle\tilde{f}^{nk}_{t}(t^{\prime}_{nk},y_{nk})+{\cal
L}\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk})$ $\displaystyle\leq$
$\displaystyle-\delta/T^{2}.$

Hence,

$B_{nk}:=f^{nk}_{t}(t_{nk},x_{nk})-\tilde{f}^{nk}_{t}(t^{\prime}_{nk},y_{nk})+{\cal
L}f^{nk}(t_{nk},x_{nk})-{\cal
L}\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk})\geq\delta/T^{2}.$ (17)

3. The third step consists in getting a contradiction with our hypothesis.

After regrouping terms we get that $B_{nk}=B^{(1)}_{nk}+B^{(2)}_{nk}$ where

$\displaystyle B^{(1)}_{nk}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\sigma^{2}(A_{nk}x_{nk}^{2}-\tilde{A}^{\prime}_{nk}y_{nk}^{2})+\frac{1}{2}\sigma^{2}_{P}(A_{nk}-\tilde{A}^{\prime}_{nk})+a(p_{nk}x_{nk}-p^{\prime}_{nk}y_{nk})$
$\displaystyle+a_{P}(p_{nk}-p^{\prime}_{nk})+b_{nk}-b^{\prime}_{nk},$
$\displaystyle B^{(2)}_{nk}$ $\displaystyle:=$ $\displaystyle{\cal
L}^{i}_{R}f^{nk}(t_{nk},x_{nk})-{\cal
L}^{i}_{R}\tilde{f}^{nk}(t^{\prime}_{nk},y_{n})+{\cal
L}^{i}_{P}f^{nk}(t_{nk},x_{nk})-{\cal
L}^{i}_{P}\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk})$

with ${\cal L}^{i}_{R}$ and ${\cal L}^{i}_{P}$ defined in (7) and (9).

Passing to the limit as $k\to\infty$ we get that

$\lim_{k}B^{(1)}_{nk}=\frac{1}{2}\sigma^{2}(A_{n}x_{n}^{2}-\tilde{A}_{n}y_{n}^{2})+\frac{1}{2}\sigma^{2}_{P}(A_{n}-\tilde{A}_{n})+an(x_{n}-y_{n})^{2}.$

By virtue of the Ishii lemma $A_{n}-\tilde{A}_{n}\leq 0$, the first term is
dominated by a constant multiplied by $n|x_{n}-y_{n}|^{2}$ and, therefore,
converges to zero as $n\to\infty$. It follows that
$\limsup_{n}\lim_{k}B^{(1)}_{nk}\leq 0$.

To get a contradiction with (17) it is sufficient to show that

$\displaystyle\limsup_{n}\limsup_{k}\big{(}{\cal
L}^{i}_{R}f^{nk}(t_{nk},x_{nk})-{\cal
L}^{i}_{R}\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk})\big{)}\leq 0,$ (18)
$\displaystyle\limsup_{n}\limsup_{k}\big{(}{\cal
L}^{i}_{P}f^{nk}(t_{nk},x_{nk})-{\cal
L}^{i}_{P}\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk})\big{)}\leq 0.$ (19)

Let

$\displaystyle H_{nk}(z)$ $\displaystyle:=$
$\displaystyle[f^{nk}(t_{nk},x_{nk}+x_{nk}z)-f^{nk}(t_{nk},x_{nk})-f^{nk}_{x}(t_{nk},x_{nk})x_{nk}zI_{\\{|z|\leq
1\\}}]$
$\displaystyle-[\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk}+y_{nk}z)-\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk})-\tilde{f}^{nk}_{x}(t^{\prime}_{nk},y_{nk})y_{nk}zI_{\\{|z|\leq
1\\}}],$ $\displaystyle G_{nk}(z)$ $\displaystyle:=$
$\displaystyle[f^{nk}(t_{nk},x_{nk}+z)-f^{nk}(t_{nk},x_{nk})-f^{nk}_{x}(t_{nk},x_{nk})zI_{\\{|z|\leq
1\\}}]$
$\displaystyle-[\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk}+z)-\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk})-\tilde{f}^{nk}_{x}(t^{\prime}_{nk},y_{nk})zI_{\\{|z|\leq
1\\}}].$

In this notation the claimed relations (18) and (19) have the form

$\displaystyle\limsup_{n}\limsup_{k}\int H_{nk}(z)\Pi(dz)\leq 0,$ (20)
$\displaystyle\limsup_{n}\limsup_{k}\int G_{nk}(z)\Pi_{P}(dz)\leq 0.$ (21)

In the sequel we work with $n$ and $k$ large enough to ensure that $\kappa\leq
x_{nk},y_{kn}\leq 1/\kappa$ for some (small) constant $\kappa>0$.

On the set $\\{|z|>1\\}$ the functions $H_{nk}$ and $G_{nk}$ admit a uniform
bounds

$\displaystyle H_{nk}(z)$ $\displaystyle\leq$
$\displaystyle\psi(t_{nk},x_{nk}+x_{nk}z)-\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk}+y_{nk}z)-\psi(t_{nk},x_{nk})+\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk})+2{\varepsilon}_{nk},$
$\displaystyle G_{nk}(z)$ $\displaystyle\leq$
$\displaystyle\psi(t_{nk},x_{nk}+z)-\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk}+z)-\psi(t_{nk},x_{nk})+\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk})+2{\varepsilon}_{nk}.$

Taking ${\varepsilon}_{nk}=1/n+1/k$ we get that

$\displaystyle\limsup_{k}H_{nk}(z)$ $\displaystyle\leq$
$\displaystyle\psi(t_{n},x_{n}+x_{n}z)-\tilde{\psi}^{\delta}(t_{n},y_{n}+y_{n}z)-\psi(t_{n},x_{n})+\tilde{\psi}^{\delta}(t_{n},y_{n})+1/n.$

Notice that on the set $\\{|z|>1\\}$

$\displaystyle\psi(t_{n},x_{n}+x_{n}z)-\tilde{\psi}^{\delta}(t_{n},y_{n}+y_{n}z)$
$\displaystyle=$
$\displaystyle\Delta_{n}(t_{n},x_{n}+x_{n}z,y_{n}+y_{n}z)+(1/2)nz(z+2)|x_{n}-y_{n}|^{2}$
$\displaystyle+(1/2)n|x_{n}-y_{n}|^{2}$ $\displaystyle\leq$
$\displaystyle\Delta_{n}(t_{n},x_{n}+x_{n}z,y_{n}+y_{n}z)+(3/2)nz^{2}|x_{n}-y_{n}|^{2}$
$\displaystyle+(1/2)n|x_{n}-y_{n}|^{2}.$

Hence, on the set $\\{|z|>1\\}$

$\displaystyle\limsup_{k}H_{nk}(z)$ $\displaystyle\leq$
$\displaystyle\Delta_{n}(t_{n},x_{n}+x_{n}z,y_{n}+y_{n}z)-\Delta_{n}(t_{n},x_{n},y_{n})+\frac{3}{2}nz^{2}|x_{n}-y_{n}|^{2}+\frac{1}{n}.$

The first term in the inequality above is dominated by the second term and the
last two terms converge to zero as $n\to\infty$.

Similarly, we get that

$\displaystyle\limsup_{k}G_{nk}(z)$ $\displaystyle\leq$
$\displaystyle\psi(t_{n},x_{n}+z)-\tilde{\psi}^{\delta}(t_{n},y_{n}+z)-\psi(t_{n},x_{n})+\tilde{\psi}^{\delta}(t_{n},y_{n})$
$\displaystyle\leq$
$\displaystyle\Delta_{n}(t_{n},x_{n}+z,y_{n}+z)-\Delta_{n}(t_{n},x_{n},y_{n})\leq
0.$

Using the boundedness of $H_{nk}(z)I_{\\{|z|>1\\}}$ and
$G_{nk}(z)I_{\\{|z|>1\\}}$ and applying the Fatou lemma we get the bounds

$\displaystyle\limsup_{n}\limsup_{k}\int H_{nk}(z)I_{\\{|z|>1\\}}\Pi(dz)\leq
0,$ (22) $\displaystyle\limsup_{n}\limsup_{k}\int
G_{nk}(z)I_{\\{|z|>1\\}}\Pi_{P}(dz)\leq 0.$ (23)

Take $r_{nk}\in]0,\kappa[$ small enough to guaranty that for
$\theta\in]0,r_{nk}]$ we have the representations

$\displaystyle f^{nk}(t_{nk},x_{nk}+\theta)$ $\displaystyle=$
$\displaystyle\psi(t_{nk},x_{nk})+p_{nk}\theta+(1/2)A_{nk}\theta^{2}+R_{nk}(\theta),$
$\displaystyle\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk}+\theta)$ $\displaystyle=$
$\displaystyle\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk})+p^{\prime}_{nk}\theta+(1/2)\tilde{A}^{\prime}_{nk}\theta^{2}+\tilde{R}_{nk}(\theta),$

where $R_{nk}(\theta)=\rho_{nk}(0,\theta)$,
$\tilde{R}_{nk}(\theta)=\tilde{\rho}_{nk}(0,\theta)$.

On the set $U_{nk}:=\\{z\colon|z|\leq\kappa r_{nk}\\}\cap\\{z\colon|z|\leq
1\\}$

$\displaystyle H_{nk}(z)$ $\displaystyle=$
$\displaystyle(1/2)(A_{n}x^{2}_{nk}-\tilde{A}_{n}y^{2}_{nk})z^{2}$
$\displaystyle+(1/2)\Big{(}(A_{nk}-A_{n})x^{2}_{nk}-(\tilde{A}^{\prime}_{nk}-\tilde{A}_{n})y^{2}_{nk}\Big{)}z^{2}+R_{nk}(z{x_{nk}})-\tilde{R}_{nk}(z{y_{nk}}),$
$\displaystyle G_{nk}(z)$ $\displaystyle=$
$\displaystyle(1/2)(A_{nk}-\tilde{A}^{\prime}_{nk})z^{2}+R_{nk}(z)-\tilde{R}_{nk}(z).$

and we have the bounds

$\displaystyle H_{nk}(z)$ $\displaystyle\leq$
$\displaystyle(3/2)n(x_{nk}-y_{nk})^{2}z^{2}+(1/(2\kappa^{2}))\big{(}|A_{nk}-A_{n}|+|\tilde{A}^{\prime}_{nk}-\tilde{A}_{n}|\big{)}z^{2}$
$\displaystyle+|R_{nk}(z{x_{nk}})|+|\tilde{R}_{nk}(z{y_{nk}})|,$
$\displaystyle G_{nk}(z)$ $\displaystyle\leq$
$\displaystyle(1/2)|A_{nk}-\tilde{A}^{\prime}_{nk}|z^{2}+|R_{nk}(z)|+|\tilde{R}_{nk}(z)|.$

The ratios $R_{nk}(zx_{nk})/z^{2}$, $\tilde{R}_{nk}(zy_{nk})/z^{2}$,
$R_{nk}(z)/z^{2}$ and $\tilde{R}_{nk}(z)/z^{2}$ are bounded and tend to zero
as $z\to 0$. Thus, for all sufficiently large $n,k$

$\displaystyle\int_{U_{nk}}H_{nk}(z)\Pi(dz)\leq\frac{1}{n},$
$\displaystyle\int_{U_{nk}}G_{nk}(z)\Pi_{P}(dz)\leq\frac{1}{n}.$

Also we can find $\delta_{n}>0$ small enough and $r^{\prime}_{nk}:=\kappa
r_{nk}+\delta_{n}$ (from Lemma 3) such that for sufficiently large $k,n$ on
the set $V_{nk}:=\\{z\colon\kappa r_{nk}\leq|z|\leq
r^{\prime}_{nk}\\}\cap\\{z\colon|z|\leq 1\\}$ we get

$\displaystyle\int_{V_{nk}}H_{nk}(z)\Pi(dz)\leq\frac{1}{n},$
$\displaystyle\int_{V_{nk}}G_{nk}(z)\Pi_{P}(dz)\leq\frac{1}{n}.$

On the set $\\{z\colon|z|\geq{r^{\prime}_{nk}}\\}$

$f^{nk}(t_{nk},x_{nk}+zx_{nk})=\psi(t_{nk},x_{nk}+zx_{nk}),\quad\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk}+zy_{nk})=\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk}+zy_{nk}),$
$f^{nk}(t_{nk},x_{nk}+z)=\psi(t_{nk},x_{nk}+z),\quad\tilde{f}^{nk}(t^{\prime}_{nk},y_{nk}+z)=\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk}+z).$

implying that on the set $Y_{nk}:=\\{z\colon r^{\prime}_{nk}\leq|z|\leq 1\\}$

$\displaystyle H_{nk}(z)$ $\displaystyle=$
$\displaystyle\psi(t_{nk},x_{nk}+x_{nk}z)-\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk}+y_{nk}z)-\psi(t_{nk},x_{nk})+\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk})$
$\displaystyle-(p_{nk}x_{nk}-p^{\prime}_{nk}y_{nk})z,$ $\displaystyle
G_{nk}(z)$ $\displaystyle=$
$\displaystyle\psi(t_{nk},x_{nk}+z)-\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk}+z)-\psi(t_{nk},x_{nk})+\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk})$
$\displaystyle-(p_{nk}-p^{\prime}_{nk})z.$

The functions $H_{nk}(z)$, $G_{nk}(z)$ are continuous on the compact set
$\\{z\colon\delta_{n}\leq|z|\leq 1\\}\supseteq Y_{nk}$, and hence uniformly
continuous. Together with the facts that

$p_{nk}x_{nk}-p^{\prime}_{nk}y_{nk}\to n(x_{n}-y_{n})^{2},\quad
p_{nk}-p^{\prime}_{nk}\to 0,\quad k\to\infty,$

we get that for sufficiently large $k$ and for small enough
$\eta_{n},\mu_{n}>0$

$\displaystyle H_{nk}(z)$ $\displaystyle\leq$
$\displaystyle\psi(t_{n},x_{n}+x_{n}z)-\tilde{\psi}^{\delta}(t^{\prime}_{n},y_{n}+y_{n}z)-\psi(t_{n},x_{n})+\tilde{\psi}^{\delta}(t^{\prime}_{n},y_{n})$
$\displaystyle-nz(x_{n}-y_{n})^{2}+\eta_{n}$ $\displaystyle\leq$
$\displaystyle\Delta_{n}(t_{n},x_{n}+x_{n}z,y_{n}+y_{n}z)-\Delta_{n}(t_{n},x_{n},y_{n})+\frac{1}{2}nz^{2}|x_{n}-y_{n}|^{2}+\eta_{n}.$

Similarly,

$\displaystyle G_{nk}(z)$ $\displaystyle\leq$
$\displaystyle\psi(t_{nk},x_{nk}+z)-\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk}+z)-\psi(t_{nk},x_{nk})+\tilde{\psi}^{\delta}(t^{\prime}_{nk},y_{nk})+\mu_{n}$
$\displaystyle\leq$
$\displaystyle\Delta_{n}(t_{n},x_{n}+x_{n}z,y_{n}+y_{n}z)-\Delta_{n}(t_{n},x_{n},y_{n})+\mu_{n}.$

The first terms in both inequalities are dominated by the second terms. The
third term in the first inequality is integrable and converges to zero as
$n\to\infty$. Hence,

$\displaystyle\limsup_{n}\frac{1}{2}n|x_{n}-y_{n}|^{2}\int z^{2}I_{\\{|z|\leq
1\\}}\Pi(dz)\leq 0.$

The last terms $\eta_{n}$ and $\mu_{n}$ can be chosen simultaneously with
$\delta_{n}$ to gurantee that for sufficiently large $n$

$\displaystyle\mu_{n}\int I_{\\{\delta_{n}\leq|z|\leq
1\\}}\Pi(dz)\leq\frac{1}{n},\quad\eta_{n}\int I_{\\{\delta_{n}\leq|z|\leq
1\\}}\Pi_{P}(dz)\leq\frac{1}{n}.$

Hence, applying the dominated convergence theorem,

$\displaystyle\limsup_{n}\limsup_{k}\int H_{nk}(z)I_{\\{|z|\leq
1\\}}\Pi(dz)\leq 0,$ (24) $\displaystyle\limsup_{n}\limsup_{k}\int
G_{nk}(z)I_{\\{|z|\leq 1\\}}\Pi_{P}(dz)\leq 0.$ (25)

$\Box$

As a corollary we get the following uniqueness result.

###### Theorem 5.2

Let $\Psi$ and $\tilde{\Psi}$ be two continuous functions on
$\bar{K}:=[0,T]\times\bar{\mathbb{R}}_{+}$ which are viscosity solutions of
(10) on $K$ such that

$\Psi(t,0)=\tilde{\Psi}(t,0),\quad\Psi(t,\infty)=\tilde{\Psi}(t,\infty),\quad\Psi(T,u)=\tilde{\Psi}(T,u).$

Then $\Psi=\tilde{\Psi}$.

###### Acknowledgements.

The research is funded by the grant of RSF $n^{\circ}$ 20-68-47030
“Econometric and probabilistic methods for the analysis of financial markets
with complex structure”.

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# Explanation of Y(4630) as hadronic resonant state

Xiao-Hui Mei1 Zhuo Yu2 Mao Song1<EMAIL_ADDRESS>Jian-You Guo1 Gang
Li1 Xuan Luo1 1School of Physics and Optoelectronic engineering, Anhui
University, Hefei 230601, China
2School of Physics, Southeast University, Nanjing 210094, China

###### Abstract

After Y(4630) is discovered, theorists have given various explanations. We
find that if Y(4630) is interpreted as the D-wave resonant state of
$\Lambda_{c}\bar{\Lambda}_{c}$ system, the particle mass, decay width and all
quantum numbers are consistent with experimental observations. We use the Bonn
approximation to get the interaction potential of one boson exchange model,
then extend the complex scaling method (CSM) to calculate the bound and
resonant states. The results indicate that the $\Lambda_{c}\bar{\Lambda}_{c}$
system can form not only the bound state of S wave, but also the resonant
state of the high angular momentum, and the ${}^{3}D_{1}$ wave resonant state
can explain the structure of Y(4630) very well.

###### pacs:

12.39.Pn, 14.40.Lb, 14.40.Pt, 25.80.-e

##  I introduction

Deuteron is a proton and neutron molecular state, which is well explained by
one boson exchange model Tornqvist:1993vu ; Tornqvist:1993ng . Along this
line, we may wonder whether the heavy baryon pair can also form a deuteron-
like bound state by exchanging virtual light mesons. Intuitionally, the larger
mass of the heavy baryons can reduce the kinetic of the systems and easier to
form bound states. Therefore, it is interesting to study whether one boson
exchange interactions are strong enough to bind the two heavy baryons
(dibaryon) or a heavy baryon and an anti-baryon (baryonium).

$\Lambda_{c}$ is the lightest charmed baryon, which contains a charm quark and
two light quarks, and the composition is similar to proton. Having more
knowledge about $\Lambda_{c}$ is helpful for studying the properties of other
charmed baryons. Until now, our understanding about the $\Lambda_{c}$ behavior
is very limited. The Belle Collaboration firstly reported a charmonium-like
state $Y(4630)$ in $\Lambda_{c}\bar{\Lambda}_{c}$ invariant mass spectrum from
the $e^{+}e^{-}\to\gamma_{ISR}\Lambda_{c}\bar{\Lambda}_{c}$ process, where
$\gamma_{ISR}$ is the emitted photon from the initial leptons, the related
parameters are mass $M=4634^{+8+5}_{-7-8}$ MeV, width
$\Gamma=92^{+40+10}_{-24-21}$ MeV, quantum number $J^{PC}=1^{--}$
Pakhlova:2008vn .

After the observation of $Y(4630)$, various theoretial interpretations were
proposed, such as conventional charmonium state Badalian:2008dv ;
Segovia:2008ta , tetraquark state Maiani:2014aja ; Cotugno:2009ys ;
Brodsky:2014xia , $\Lambda_{c}\bar{\Lambda}_{c}$ baryonium Lee:2011rka ;
Chen:2011cta  and threshold effect vanBeveren:2008rt . Simonov proposed a
mechanism to study baryon-antibaryon production, which can explain why the
$Y(4630)$ enhancement structure appears in the electroproduction of
$\Lambda_{c}\bar{\Lambda}_{c}$ Simonov:2011jc . Recently, a series of
investigations on the strong decay behaviors were proposed, which intended to
reveal the inner structure of $Y(4630)$ Liu:2016sip ; Liu:2016nbm ;
Guo:2016iej .

Resonance is one of most striking phenomenon in the whole range of scattering
experiments, which appear widely in atoms, molecules, nuclei and chemical
reactions. Based on conventional scattering theory, $R$-matrix method
Wigner:1947zz ; Hale:1987zz , $K$-matrix method Humblet:1991zz , scattering
phase shift method, continuous spectrum theory, $J$-matrix method Taylor  have
been developed. For the convenience of calculation, several bound-state-like
methods are developed, such as real stabilization method (RSM)Hazi , analytic
continuation method of coupling constant (ACCC) Kukulin  and complex scaling
method (CSM) csm1 ; csm2 , $etc$. The complex scaling method can describe the
bound state, resonant state and continuum in a consistent way, which is widely
used to exploring the resonance in atomic, molecular and nuclear physics. The
CSM has been extended from nonrelativistic to the relativistic framework csm3
; csm4 ; csm5 ; csm6 ; csm7 and from spherical nuclei to deformed nuclei csm9
, which has been applied in halo nuclei. In Ref.Yu:2021lmb , the authors
firstly extend CSM from atomic, molecular and nuclear physics to hadron
physics for explaining hadron molecular state. Recently, the CSM has been used
more and more in hadronic physics Wang:2022yes ; Cheng:2022qcm ; Cheng:2022vgy
.

Among various explanations, hadronic molecules gain more attention, since
Y(4630) is close to the thresholds of two hadrons. The bound state of S wave
is easier to form, but the higher excited states also have a certain
probability to form. For example, the heavy quarkonium not only found the
ground state $J/\psi$, but also the excited states $h_{c}(1P)$,
$\chi_{c2}(2P)$ and $\Upsilon(1^{3}D_{2})$ were observed in experiment. The
higher excited states of hadrons provide a unique way to study the structure
of hadron states. In the framework of one-boson-exchange model, if the hadrons
can bind to hadron molecular states, whether can form resonant states with
high angular momentum. In Ref.Yu:2021lmb , the authors have calculated the
resonant states for $DD(\bar{D})$, $\Lambda_{c}D(\bar{D})$ and
$\Lambda_{c}\Lambda_{c}(\bar{\Lambda}_{c})$ systems in heavy quark effective
theory. The bound state of $\Lambda_{c}\bar{\Lambda}_{c}$ system has been
investigated in several previous works Chen:2017jjn ; Meguro:2011nr ;
Li:2012bt ; Lee:2011rka . In this paper, we will consider the spin-orbit
coupling effect and further investigate whether $\Lambda_{c}\bar{\Lambda}_{c}$
system can form a resonant state consistent with the quantum number, mass and
width of Y(4630).

This paper is organized as follows. After the introduction, we present the
theoretical framework and calculation method in Section II. The numerical
results and discussion are given in Section III. A short summary is given in
Section IV.

##  II Theoretical framework

In our work, we calculate the effective interaction potential for the
$\Lambda_{c}\bar{\Lambda}_{c}$ system in the Bonn meson-exchange model. Due to
the spin and isospin conservation in hadron systems, the contributions of
$\pi$, $\eta$, $\rho$ meson exchanges are forbidden or suppressed heavily. The
interactions of $\Lambda_{c}\bar{\Lambda}_{c}$ system are mainly mediated by
$\sigma$ and $\omega$ mesons, and the effective Lagrangian densities for
one-$\sigma$-exchange and one-$\omega$-exchange are expressed,

$\displaystyle\mathcal{L}=g_{\sigma\Lambda_{c}\Lambda_{c}}\bar{\psi}\sigma\psi-
g_{\omega\Lambda_{c}\Lambda_{c}}\bar{\psi}\gamma_{\mu}\omega^{\mu}\psi+\frac{f_{\omega\Lambda_{c}\Lambda_{c}}}{2m_{\Lambda_{c}}}\bar{\psi}\sigma_{\mu\nu}\psi\partial^{\mu}\omega^{\nu},$
(1)

Here, $\psi$ is the Dirac-spinor for the spin-$\frac{1}{2}$ particle of
$\Lambda_{c}$. In the tensor coupling term, the constant
$f_{\omega\Lambda_{c}\Lambda_{c}}=-g_{\omega\Lambda_{c}\Lambda_{c}}$ in
Ref.Lee:2011rka , so the tensor coupling coefficient is $2m_{\Lambda_{c}}$
lower than vector coupling coefficient. Moreover the tensor term is
proportional to the relative momentum $q^{\nu}$, which is small, thus the
contribution of the tensor term can be ignored.

Although there are no definite values for the coupling strengths
$g_{\omega/\sigma\Lambda_{c}\Lambda_{c}}$ in experiment, they can estimated by
using the quark model. Since the exchange of $\sigma$ and $\omega$ mesons
occurs mainly between the light quarks in heavy hadrons, the interactions of
light quarks $(q=u,d)$ and $\sigma/\omega$ can be written as

$\displaystyle\mathcal{L}_{qq\sigma/\omega}$ $\displaystyle=$ $\displaystyle-
g_{\sigma}^{q}\bar{\psi}_{q}\sigma\psi_{q}-g_{\omega}^{q}\bar{\psi}_{q}\gamma^{\mu}\omega_{\mu}\psi_{q}.$
(2)

Compared with the vertices of $\bar{\Lambda}_{c}\Lambda_{c}\sigma/\omega$ and
$\bar{q}q\sigma/\omega$ in Eqs. (1)$-$(2), the coupling constants can be
related by,

$\displaystyle g_{\sigma\Lambda_{c}\Lambda_{c}}=2g_{\sigma}^{q},\quad\quad
g_{\omega\Lambda_{c}\Lambda_{c}}=2g_{\omega}^{q}.$ (3)

In a $\sigma$ model Riska:1999fn , the value of $g_{\sigma}^{q}$ is taken as
$g_{\sigma}^{q}=3.65$. For the $\omega$ coupling $g_{\omega}^{q}$, in the
Nijmegen model, $g_{\omega}^{q}=3.45$, whereas it is equal to 5.28 in the Bonn
model Rijken:1998yy . In Ref. Riska:2000gd , $g_{\omega}^{q}$ was roughly
assumed to be 3.00.

Figure 1: The Feynman diagrams at the tree level.

Based on the Lagrangins in Eqs (1), we can obtain the scattering Feynman
amplitudes for $\Lambda_{c}\bar{\Lambda}_{c}\to\Lambda_{c}\bar{\Lambda}_{c}$
in Fig. 1. The annihilation effect from S-channel has not been take into
account in the calculation. In the center-of-mass frame, the initial four-
momenta are $p_{1}(E_{1},\vec{p})$ and $p_{2}(E_{2},-\vec{p})$, the final
four-momenta are $k_{1}(E_{1},\vec{p^{\prime}})$ and
$k_{2}(E_{2},-\vec{p^{\prime}})$, as shown in Fig.1. Thus the four-momenta of
propagator is

$q=k_{1}-p_{1}=p_{2}-k_{2}=(0,\vec{p^{\prime}}-\vec{p})=(0,\vec{q})$ (4)

For convenience of calculations, we make the substitution for the following
four-momenta,

$\vec{q}=\vec{p^{\prime}}-\vec{p},~{}~{}~{}\vec{k}=\frac{1}{2}(\vec{p}+\vec{p^{\prime}}).$
(5)

In the nonrealistic approximation, we keep the terms up to order of
$\frac{1}{m_{\Lambda_{c}}^{2}}$. The scattering amplitudes are

$\displaystyle iM_{\sigma}=$ $\displaystyle-
g_{\sigma\Lambda_{c}\Lambda_{c}}^{2}\bar{u}(k_{1})u(p_{1})\frac{i}{q^{2}-m_{\sigma}^{2}}\bar{\upsilon}(p_{2})\upsilon(k_{2})$
$\displaystyle=$ $\displaystyle
i\frac{g_{\sigma\Lambda_{c}\Lambda_{c}}^{2}}{\vec{q}^{2}+{m_{\sigma}}^{2}}\left[1-\frac{\vec{k}^{2}}{2m_{\Lambda_{c}}^{2}}+\frac{\vec{q}^{2}}{8m_{\Lambda_{c}}^{2}}+i\frac{\vec{S}\cdot(\vec{k}\times\vec{q})}{2m_{\Lambda_{c}}^{2}}\right],$
(6)

and

$\displaystyle iM_{\omega}=$ $\displaystyle-
g_{\omega\Lambda_{c}\Lambda_{c}}^{2}\bar{u}(k_{1})\gamma^{\mu}u(p_{1})i\frac{-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{\omega}^{2}}}{q^{2}-m_{\omega}^{2}}\bar{\upsilon}(p_{2})\gamma^{\nu}\upsilon(k_{2})$
$\displaystyle=$ $\displaystyle
i\frac{g_{\omega\Lambda_{c}\Lambda_{c}}^{2}}{\vec{q}^{2}+m_{\omega}^{2}}\Bigg{[}1-\frac{\vec{q}^{2}}{8m_{\Lambda_{c}}^{2}}+\frac{3\vec{k}^{2}}{2m_{\Lambda_{c}}^{2}}+i\frac{3\vec{S}\cdot\left(\vec{k}\times\vec{q}\right)}{2m_{\Lambda_{c}}^{2}}$
(7)
$\displaystyle-\frac{(\vec{\sigma_{1}}\cdot\vec{\sigma_{2}})\cdot\vec{q}^{2}}{4m_{\Lambda_{c}}^{2}}+\frac{(\vec{\sigma}_{1}\cdot\vec{q})(\vec{\sigma}_{2}\cdot\vec{q})}{4m_{\Lambda_{c}}^{2}}\Bigg{]},$

where $\vec{S}=\frac{1}{2}(\vec{\sigma}_{1}+\vec{\sigma}_{2})$ is the total
spin of $\Lambda_{c}\bar{\Lambda}_{c}$ system.

In the Breit approximation, the relation between the effective potential in
momentum space $\mathcal{V}_{fi}$ and the scattering amplitude
$\mathcal{M}_{fi}$ in the momentum space are expressed,

$\displaystyle\mathcal{V}_{fi}(\bm{q})$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{M}_{fi}(\Lambda_{c}\bar{\Lambda}_{c}\to\Lambda_{c}\bar{\Lambda}_{c})}{\sqrt{\prod_{i}2m_{i}\prod_{f}2m_{f}}}.$
(8)

Here, $m_{i}$ and $m_{f}$ are the masses of the initial ($\Lambda_{c}$,
$\bar{\Lambda}_{c}$) and final particles ($\Lambda_{c}$, $\bar{\Lambda}_{c}$),
respectively.

In the above paper, hadron is directly treated as point particles without
considering the internal structure of the hadron. In order to regularize the
off shell effect of the exchanged meson, it is necessary to introduce a
monopole form factor $\mathcal{F}(q^{2})$ at every vertex, which has a form
as:

$\displaystyle\mathcal{F}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}-q^{2}},$ (9)

here, $\Lambda$ is the cutoff parameter, $m$ and $q$ correspond to the mass
and momentum of the exchanged meson, respectively. In Refs.Tornqvist:1993vu ;
Tornqvist:1993ng , $\Lambda$ is related to the root-mean-square radius of the
source hadron which propagate the interaction through the intermediate boson
($\sigma$ or $\omega$). According to the previous experience of the deuteron,
the cutoff $\Lambda$ is taken around 1.0 GeV.

After adding the monopole form factor $\mathcal{F}(q^{2})$, the effective
potential in the coordinate space $\mathcal{V}(r)$ is obtained by performing
the Fourier transformation as

$\displaystyle\mathcal{V}(\bm{r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\mathcal{V}(\bm{q})\mathcal{F}^{2}(q^{2}).$
(10)

The detailed Fourier transformations for different types of effective
potentials are expressed as Chen:2019uvv ; Zhao:2014gqa

$\displaystyle\mathcal{F}$
$\displaystyle\left\\{\frac{1}{\vec{q}^{2}+m^{2}}\left(\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}+\vec{q}^{2}}\right)^{2}\right\\}=Y\left(\Lambda,m,r\right),$
$\displaystyle\mathcal{F}$
$\displaystyle\left\\{\frac{\vec{q}^{2}}{\vec{q}^{2}+m^{2}}\left(\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}+\vec{q}^{2}}\right)^{2}\right\\}=-\nabla^{2}Y\left(\Lambda,m,r\right),$
$\displaystyle\mathcal{F}$
$\displaystyle\left\\{\frac{\vec{k}^{2}}{\vec{q^{2}}+m^{2}}\left(\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}+\vec{q}^{2}}\right)^{2}\right\\}=\frac{1}{4}\nabla^{2}Y\left(\Lambda,m,r\right)$
$\displaystyle-\frac{1}{2}\left\\{\nabla^{2},Y\left(\Lambda,m,r\right)\right\\},$
$\displaystyle\mathcal{F}$
$\displaystyle\left\\{\frac{\left(\vec{S}\cdot(\vec{q}\times\vec{k})\right)}{\vec{q}^{2}+m^{2}}\left(\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}+\vec{q}^{2}}\right)^{2}\right\\}=-i\vec{S}\cdot\vec{L}\frac{1}{r}\frac{\partial}{\partial
r}Y\left(\Lambda,m,r\right),$ $\displaystyle\mathcal{F}$
$\displaystyle\left\\{\frac{\left(\vec{\sigma}_{1}\cdot\vec{q}\right)\left(\vec{\sigma}_{2}\cdot\vec{q}\right)}{\vec{q}^{2}+m^{2}}\left(\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}+\vec{q}^{2}}\right)^{2}\right\\}$
$\displaystyle=$
$\displaystyle-\frac{1}{3}\left(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\right)\nabla^{2}Y\left(\Lambda,m,r\right)-\frac{1}{3}S\left(\vec{\hat{r}},\vec{\sigma}_{1},\vec{\sigma}_{2}\right)T\left(\Lambda,m,r\right),$

As shown in Ref.Lee:2011rka , due to the cancellations of the coupling
constants $f_{\omega\Lambda_{c}\Lambda_{c}}$ and
$g_{\omega\Lambda_{c}\Lambda_{c}}$ in the tensor terms, and there is no mixing
of S and D states. Therefore, $\Lambda_{c}\bar{\Lambda}_{c}$ systems do not
need to consider S-D coupling. The $\vec{k}^{2}$ term is named as the recoil
correction term, and the function $Y\left(\Lambda,m,r\right)$ is defined as

$\displaystyle Y\left(\Lambda,m,r\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi r}\left(e^{-mr}-e^{-\Lambda
r}\right)-\frac{\Lambda^{2}-m^{2}}{8\pi\Lambda}e^{-\Lambda r}.$ (12)

The one-$\sigma$-exchange and one-$\omega$-exchange interactions are
corresponding to intermediate- and short-range forces, therefore they are
suppressed when the radius $r$ reaches 1.0 fm or larger. For the ${}^{1}S_{0}$
state, both the $\omega$-exchange and $\sigma$-exchange provide attractive
force. For the ${}^{3}S_{1}$ state, the vector meson $\omega$ provides
repulsive force in the short range but attractive force in the medium range,
while the scalar meson $\sigma$ always provides attractive force.

After the interaction potential $\mathcal{V}(r)$ in coordinate space is
obtained, the eigenvalue and eigenfunction of the bound state for
$\Lambda_{c}\Lambda_{c}$ system can be obtained by solving the non-
relativistic Schrödinger equation. In this paper, we extend the CSM to solve
the Schrödinger equation in the complex energy plane. The Aguilar-Balslev-
Combes(ABC) theorem abc  proved that under the complex scaling transformation,
the energy spectrum has three parts: (i) the bound states are discrete set of
real points on the negative energy axis and remain unchanged under complex
scale transformation; (ii) the resonant states correspond to the discrete set
of points in the lower half of complex energy plane, which does not change
with the coordinate transformation; and (iii) The continuous spectrum is
rotated at $2\theta$ around the origin of the coordinates. In the CSM, bound
states, resonant states and continuous spectra can be described uniformly. As
the $\theta$ increases, the resonant states are exposed in the fourth quadrant
of the complex energy plane and do not change with the rotation of the
continuous spectrum. We solve the complex scaled Schrödinger equation by basis
expansion method, where the radial function use spherical harmonic oscillator
basis. The detailed calculation scheme can refer to our previous work
Yu:2021lmb .

##  III Numerical results

In this section, we discuss and analyze the effects of the
one-$\sigma$-exchange and one-$\omega$-exchange interactions for
$\Lambda_{c}\bar{\Lambda}_{c}$ system. The total boson exchange potentials are
used to calculate the numerical solution of the Schrödinger equation in the
CSM. The related parameters are given in Table 1.

Table 1: The related parameters are used in this work Olive:2016xmw . Hadron | $I(J^{P})$ | Mass (MeV) | Hadron | $I(J^{P})$ | Mass (MeV)
---|---|---|---|---|---
$\sigma$ | $0(0^{+})$ | 600 | $\omega$ | $0(1^{-})$ | 782.65
$\Lambda_{c}$ | $0(\frac{1}{2}^{+})$ | 2286.46 |  |  |

The corresponding eigenvalues can be obtained by diagonalizing the
Hamiltonian, then we can get the information about the resonant state.

Figure 2: (Color online) The resonant state is presented with
$\theta=60^{\circ}$. Here, the cutoff parameter $\Lambda=1.2$ GeV,
$g_{\sigma\Lambda_{c}\Lambda_{c}}$=7.3 and
$g_{\omega\Lambda_{c}\Lambda_{c}}$=10.57. The result is performed by expending
the basis function with N=120.

The eigenvalues of the transformed Hamiltonian $H_{\theta}$ are drawn in
Fig.2. We can clearly see that all the eigenvalues of $H_{\theta}$ have three
parts: the dark blue square, deep yellow triangle and green circle represent
the bound state, resonant state and continuum, respectively. The bound state
locates on the negative energy axis, while the continuous spectrum rotates
clockwise with the angle $2\theta$, and the resonant state in the lower half
of the complex energy plane, which is surrounded by the positive energy axis
and the rotated continuum line and become isolated.

Figure 3: (Color online) The resonant and continuous spectra varying with the
complex rotation angle in the complex energy plane. Except for the complex
rotation angle, other parameters are set same as those in Fig. 2.

In order to show how the resonant state is separated from the continuum by
complex rotation, the eigenvalues of $H_{\theta}$ with different complex scale
angle $\theta$ are plotted in Fig.3, and other parameters are the same as
those in Fig.2. In Fig.3(a), when $\theta=20^{\circ}$, we can only see the
continuous spectrum, it is quite difficult to observe the resonant state in
the complex energy plane. In Fig.3(b), when the rotation angel increases to
$\theta=30^{\circ}$, the resonant state begins to be separated from the
continuum spectrum gradually. When $\theta=40^{\circ}$, it is obviously that
the resonant state is completely separated from the continuum in Fig.3(c). On
the whole, from Fig.3(b) to Fig.3(d), no matter how the complex scale angle
rotates, the position of the resonant state is almost unchanged in the complex
energy plane. The above results suggest that the resonant state can be
determined as long as the selected rotation angle is large enough.

Figure 4: (Color online) The total potentials in one boson exchange model with
different orbit angular momentum L = 0, 1,2 for the
$\Lambda_{c}\bar{\Lambda}_{c}$ system dependence on the r, respectively. The
cutoff $\Lambda$ is set as 1.1 GeV, the value of coupling constant
$g_{\omega}^{q}$ are set as 3.45 in the Nijmegen model, and 5.28 in the Bonn
model Rijken:1998yy , $g_{\sigma}^{q}$ is set as $g_{\sigma}^{q}$= 3.65,
respectively. The relation between
$g_{\omega/\sigma\Lambda_{c}\Lambda_{c}}$and $g_{\omega/\sigma}^{q}$ is
$g_{\omega/\sigma\Lambda_{c}\Lambda_{c}}=2g_{\omega/\sigma}^{q}$.

The total potentials of the $\Lambda_{c}\bar{\Lambda}_{c}$ system in one boson
exchange model with the different orbit angular momentum L =0, 1, 2 are ploted
in Fig.4. The black and red line correspond to the potential with
$g_{\omega\Lambda_{c}\Lambda_{c}}=6.9$ and
$g_{\omega\Lambda_{c}\Lambda_{c}}=10.56$ in Fig.4, respectively. The
centrifugal force term, $L(L+1)/2\mu r^{2}$ is a repulsive force, the
potential energy term provides attractive force, these two part are
competitive. From Fig.4(a), we can see that under the reasonable parameters,
the potential becomes large enough to binding the two heavy baryons. In
Fig.4(b), The depth of the potential well becomes smaller, and a low potential
barrier appears, compared with Fig.4(a), which only has the center the
potentials. In Table 2, we find that when
$g_{\omega\Lambda_{c}\Lambda_{c}}=6.9$, the energy and width of P resonant
state is very small, and there is no P wave resonant for the
$\Lambda_{c}\bar{\Lambda}_{c}$ system when
$g_{\omega\Lambda_{c}\Lambda_{c}}=10.56$. In Fig.4(c), the total potentials
are larger than that in Fig.4(b), they are relatively easy to form resonant
states.

Table 2:  The energy and width of bound and resonant states for the $\Lambda_{c}\bar{\Lambda}_{c}$ system. $E$ and $\Gamma$ represent the energy and width of resonant states in units of MeV, respectively. The cutoff $\Lambda$ is set as 1.1 GeV. The value of coupling constant $g_{\omega}^{q}$ are set as 3.45 in the Nijmegen model, and 5.28 in the Bonn model Rijken:1998yy , $g_{\sigma}^{q}$ is set as $g_{\sigma}^{q}$= 3.65, respectively. The relation between $g_{\omega/\sigma\Lambda_{c}\Lambda_{c}}$ and $g_{\omega/\sigma}^{q}$ is $g_{\omega/\sigma\Lambda_{c}\Lambda_{c}}=2g_{\omega/\sigma}^{q}$. The notation $\ldots$ stands for no bound or resonant state solutions. $g_{\omega\Lambda_{c}\Lambda_{c}}$ | $L$ | $E$ | $\Gamma$ | $L$ | $E$ | $\Gamma$
---|---|---|---|---|---|---
| ${}^{1}S_{0}$ | -131.66 | … | ${}^{3}D_{1}$ | 38.86 | 183.82
| ${}^{3}S_{1}$ | -126.13 | … | ${}^{1}D_{2}$ | 32.71 | 192.12
6.9 | ${}^{3}P_{0}$ | 7.15 | 8.28 | ${}^{3}D_{2}$ | 34.94 | 188.42
| ${}^{1}P_{1}$ | 9.16 | 13.82 | ${}^{3}D_{3}$ | 28.78 | 193.7
| ${}^{3}P_{2}$ | 9.90 | 17.0 |  |  |
| ${}^{1}S_{0}$ | -240.84 | … | ${}^{3}D_{1}$ | 54.61 | 131.8
| ${}^{3}S_{1}$ | -224.53 | … | ${}^{1}D_{2}$ | 49.66 | 152.64
10.56 | ${}^{3}P_{0}$ | … | … | ${}^{3}D_{2}$ | 51.23 | 144.34
| ${}^{1}P_{1}$ | … | … | ${}^{3}D_{3}$ | 44.79 | 160.1
| ${}^{3}P_{2}$ | … | … |  |  |

We extend the complex scale method to solve the Schrödinger equation and
obtain the bound state and the resonant state for
$\Lambda_{c}\bar{\Lambda}_{c}$ system numerically. The solution of resonant
state has the form $E-i\Gamma/2$ , where $E$ is the resonance energy and
$\Gamma$ is its decay width. The coupling strength
$g_{\omega\Lambda_{c}\Lambda_{c}}$ is twice as much as $g_{\omega}^{q}$ in
heavy quark effective theory. The coupling constant $g_{\omega}^{q}$ are set
as 3.45 in the Nijmegen model, and 5.28 in the Bonn model as input benchmark
parameters. We can obtain the energies of bound states, the energies and
widths of the resonant states for the $\Lambda_{c}\bar{\Lambda}_{c}$ system
with different angular momentum $L$, which are listed in Table 2.

In Table 2, we find that there exists S wave bound state in each case with
$\Lambda=1.1$ GeV, whose binding energy is about hundreds MeV, and the
difference of energies between the ${}^{1}S_{0}$ and ${}^{3}S_{1}$ state is
about ten MeV. For the case $g_{\omega\Lambda_{c}\Lambda_{c}}=6.9$, the
$\Lambda_{c}\bar{\Lambda}_{c}$ system can form P wave resonant states, the
energies and widths are about several to more than a dozen MeV. In addition,
the $\Lambda_{c}\bar{\Lambda}_{c}$ system can also form the D wave resonant
states, the energies are dozens of MeV and widths are more than 100 MeV. The
resonance widths increases with the increasing of angular momentum L for
$\Lambda_{c}\bar{\Lambda}_{c}$ system, which indicates that the resonant
states become more and more unstable. We find that the quantum number of the
resonant state ${}^{3}D_{1}$ agrees with the quantum number $J^{PC}=1^{--}$ of
Y(4630). Then, We want to know whether there are any suitable parameter
consistent with the energy and width of Y(4630).

Figure 5: (Color online) Radial density distributions in the coordinate space
for the bound state ${}^{3}S_{1}$, and the resonant states ${}^{1}P_{1}$,
${}^{3}D_{1}$ with $g_{\sigma\Lambda_{c}\Lambda_{c}}=7.3$,
$g_{\omega\Lambda_{c}\Lambda_{c}}=10.56$, and $\Lambda=1.1$ GeV.

In Fig.5, we show the radial density distributions for the bound state
${}^{3}S_{1}$, and the resonant states ${}^{1}P_{1}$, and ${}^{3}D_{1}$ with
the coupling constants $g_{\sigma\Lambda_{c}\Lambda_{c}}=7.3$,
$g_{\omega\Lambda_{c}\Lambda_{c}}=10.56$, the cutoff parameter $\Lambda=1.1$
GeV. The black, red and blue lines represent bound state ${}^{3}S_{1}$ ,and
resonant states ${}^{1}P_{1}$, ${}^{3}D_{1}$, respectively. It can been seen
that the black bound state converges when the radius around 0.25 fm, but the
red and blue resonant states begin to converge when the radius about 0.75 and
1.25 fm, which indicate that compared with the bound state, the resonant
states are more dispersed.

Figure 6: (Color online) The values of $g_{\omega\Lambda_{c}\Lambda_{c}}$ and
$g_{\sigma\Lambda_{c}\Lambda_{c}}$ for the decay width of
$\Lambda_{c}\bar{\Lambda}_{c}$ system varying from 60 to 133 MeV, with
$\Lambda=1.25$ GeV.

The coupling constants $g_{\sigma\Lambda_{c}\Lambda_{c}}$ and
$g_{\omega\Lambda_{c}\Lambda_{c}}$ in the Lagrangians are difficult to extract
from experiments. In Ref.Lee:2011rka , the coupling constants of the heavy
charmed baryons and light mesons can be approximately determined by nucleon-
meson coupling the obtained the numerical values
$g_{\sigma\Lambda_{c}\Lambda_{c}}$=5.64 and
$g_{\omega\Lambda_{c}\Lambda_{c}}$=10.57. However, the coupling constant
$g_{\sigma\Lambda_{c}\Lambda_{c}}$ is 7.3 in a $\sigma$ model Riska:1999fn ,
the coupling constant $g_{\omega\Lambda_{c}\Lambda_{c}}=6.9$ in the Nijmegen
model, and $g_{\omega\Lambda_{c}\Lambda_{c}}=10.56$ in the Bonn model
Rijken:1998yy . Therefore, the value of these two coupling strengths have
large uncertainties. Taking into account the width uncertainties from 60 to
133 MeV of Y(4630), we show the different values of
$g_{\omega\Lambda_{c}\Lambda_{c}}$ and $g_{\sigma\Lambda_{c}\Lambda_{c}}$
which satisfy the mass of Y(4630) in Fig.6. When
$g_{\sigma\Lambda_{c}\Lambda_{c}}=7.2$, and
$g_{\omega\Lambda_{c}\Lambda_{c}}=9.13$, there is a resonant state for
$\Lambda_{c}\bar{\Lambda}_{c}$ system, with the energy 61.08 MeV, and width
91.24 MeV. The width of X(4630) reported by Belle Collaboration is its total
width, the $\Lambda_{c}\bar{\Lambda}_{c}$ is just one of its partial decay
channels, thus the decay width of the $\Lambda_{c}\bar{\Lambda}_{c}$ resonance
state should be less than the width of X(4630). In Fig.6, the orange dots
represents the width less than the width 92 MeV of Y(4630), while the gray
dots represent the width greater than 92 MeV. We can see that the range
covered by the orange dots can satisfy the requirements of the experiment.

Figure 7: (Color online) The variation of energy and width of ${}^{3}D_{1}$
wave resonant state with the change of coupling constant
$g_{\omega\Lambda_{c}\Lambda_{c}}$ for $\Lambda_{c}\bar{\Lambda}_{c}$ system
with $\Lambda=1.25$ GeV .

In order to clear the dependence of energy and width on the coupling strength
$g_{\omega\Lambda_{c}\Lambda_{c}}$, we present the energy and width as a
function of coupling strength $g_{\omega\Lambda_{c}\Lambda_{c}}$ for resonant
state ${}^{3}D_{1}$ in Fig.7. As we can see, in Fig.7(a), the energy of the
resonant state increases slowly and then gradually decreases with the
increasing of $g_{\omega\Lambda_{c}\Lambda_{c}}$, and reaches a maximum value
when $g_{\omega\Lambda_{c}\Lambda_{c}}$ is about 9.0. Unlike the change of the
energy, in Fig. 7(b), the width decreases significantly with the change of the
coupling constant $g_{\omega\Lambda_{c}\Lambda_{c}}$.

Figure 8: (Color online) The energy and width of ${}^{3}D_{1}$ wave resonant
state as a function of the cutoff parameter $\Lambda$ for
$\Lambda_{c}\bar{\Lambda}_{c}$ system with
$g_{\omega\Lambda_{c}\Lambda_{c}}=9.13$.
Figure 9: (Color online) The total potentials in one boson exchange model with
different cutoff parameter $\Lambda$ for ${}^{3}D_{1}$ wave resonant state
dependence on the r. The coupling constants are set as
$g_{\sigma\Lambda_{c}\Lambda_{c}}=7.2$, and
$g_{\omega\Lambda_{c}\Lambda_{c}}=9.13$.

The cutoff parameter $\Lambda$ is related to the size of hadrons, and has a
significant impact on the results of energy and width. For nucleon-nucleon
interaction, the cutoff parameter $\Lambda$ is usually from 0.8 to 1.5 GeV,
for the heavy hadron state, this value should be slightly larger. In Fig.8(a)
and (b), we present the energy and width of resonant state ${}^{3}D_{1}$ for
$\Lambda_{c}\bar{\Lambda}_{c}$ system as a function of the $\Lambda$. From
Fig.8(a), we can see that the energy of resonant state ${}^{3}D_{1}$ increases
from 25 MeV to 60 MeV, and then decreases to 40 MeV, when the cutoff parameter
$\Lambda$ varies from 1.0 to 1.4. The corresponding width decreases from 200
MeV down to 20 MeV, as shown in Fig.8(b). The energies and widths are related
to the potential functions, which depend on the coupling constants and cutoff
parameter. The dependence of potential function on coupling constants and
cutoff is similar, thus we present the potential functions for the different
cutoff parameters in Fig.9. We can see that when the cutoff parameter becomes
larger, the depth of potential well becomes deeper, the barrier effect is more
obvious, the formed resonance state is more stable, and the decay width is
smaller.

Table 3:  The energy and width of bound and resonant states for $\Lambda_{c}\bar{\Lambda}_{c}$ systems, when the cutoff $\Lambda$ is 1.25 GeV, the coupling constant are set as $g_{\sigma\Lambda_{c}\Lambda_{c}}=7.2$, and $g_{\omega\Lambda_{c}\Lambda_{c}}=9.13$. $E$ and $\Gamma$ represent the energy and width of resonant states in units of MeV, respectively. The notation $\ldots$ stands for no bound or resonant state solutions. $L$ | $E$ | $\Gamma$ | $L$ | $E$ | $\Gamma$
---|---|---|---|---|---
${}^{1}S_{0}$ | -408.9 | … | ${}^{3}D_{1}$ | 61.08 | 91.24
${}^{3}S_{1}$ | -376.27 | … | ${}^{1}D_{2}$ | 60.41 | 125.06
${}^{3}P_{0}$ | … | … | ${}^{3}D_{2}$ | 60.69 | 112.68
${}^{1}P_{1}$ | … | … | ${}^{3}D_{3}$ | 55.99 | 140.98
${}^{3}P_{2}$ | … | … |  |  |

In Table 3, we list the bound and resonant states for the
$\Lambda_{c}\bar{\Lambda}_{c}$ systems, when the cutoff $\Lambda$ is 1.25 GeV,
the coupling constants are set as $g_{\sigma\Lambda_{c}\Lambda_{c}}=7.2$, and
$g_{\omega\Lambda_{c}\Lambda_{c}}=9.13$. From the Table 3, we can see that if
${}^{3}D_{1}$ resonant state is a reasonable explanation for Y(4630), there
may also exist ${}^{1}D_{2}$, ${}^{3}D_{2}$ and ${}^{3}D_{2}$ resonant states
around Y(4630), ${}^{1}S_{0}$ and ${}^{3}S_{1}$ deeper bound states, but
cannot form P resonant state. These states can be investigated in future
experiments. Although the angular momentum of the P state is lower than that
of the D state, the height of potential barrier is smaller, and it is not
easier to form a resonance state than the D state. It can be seen that the P
state can be formed when $g_{\omega\Lambda_{c}\Lambda_{c}}$=6.9 in Table 2,
and there are no resonance state when $g_{\omega\Lambda_{c}\Lambda_{c}}$=10.56
in Table 2 and in Table 3. Therefore, the P-wave resonance state can be formed
within a certain parameter range. The P-wave resonance state is not observed
in the experiment, probably because it not form a resonance state, or the
researchers need further to analyze more data.

##  IV Summary

Recent year, many new exotic hadrons have been discovered, some of them can be
explained by hadronic molecular states in one-boson-exchange model. In this
paper, we have investigated the $\Lambda_{c}\bar{\Lambda}_{c}$ system using
complex scaling method in one boson exchange model. The numerical results
indicated that the $\Lambda_{c}\bar{\Lambda}_{c}$ system can form not only S
wave bound state, but also higher angular momentum L resonant states. When the
coupling constants are taken as $g_{\sigma\Lambda_{c}\Lambda_{c}}=7.2$ and
$g_{\omega\Lambda_{c}\Lambda_{c}}=9.13$, there exists a ${}^{3}D_{1}$ wave
resonant state with energy $E=61.08$ MeV and decay width $\Gamma=91.24$ MeV,
which is consistent with the exotic hadron state Y(4630) with the quantum
number $J^{PC}=1^{--}$, mass 4634 MeV and width 91.24 MeV. If ${}^{3}D_{1}$
resonant state is a reasonable explanation for Y(4630), there may also exist
other bound and resonant states around Y(4630), which remains to be verified
experimentally in future.

## ACKNOWLEDGMENTS

This work was supported in part by the National Natural Science Foundation of
China (No.11935001), the Natural Science Foundation of Anhui Province
(No.2108085MA20, No.2208085MA10), and the Key Research Foundation of Education
Ministry of Anhui Province of China (No.KJ2021A0061).

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