#  Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic
cylinder-kernel determinant

Jun Xia  Department of Mathematics, Jiaying University, MeiZhou 514015, China.
Email<EMAIL_ADDRESS>Department of Mathematics, Sun Yat-sen
University, GuangZhou 510275, China. Email<EMAIL_ADDRESS>Shuai-Xia
Xu  Institut Franco-Chinois de l’Energie Nucléaire, Sun Yat-sen University,
GuangZhou 510275, China. Email<EMAIL_ADDRESS>Yu-Qiu Zhao
Department of Mathematics, Sun Yat-sen University, GuangZhou 510275, China.
Email<EMAIL_ADDRESS>
###### Abstract

The Clarkson-McLeod solutions of the fourth Painlevé equation behave like
$\kappa D_{\alpha-\frac{1}{2}}^{2}(\sqrt{2}x)$ as $x\rightarrow+\infty$, where
$\kappa$ is some real constant and $D_{\alpha-\frac{1}{2}}(x)$ is the
parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent
method, we derive the asymptotic behaviors for this class of solutions as
$x\to-\infty$. This completes a proof of Clarkson and McLeod’s conjecture on
the asymptotics of this family of solutions. The total integrals of the
Clarkson-McLeod solutions and the asymptotic approximations of the
$\sigma$-form of this family of solutions are also derived. Furthermore, we
find a determinantal representation of the $\sigma$-form of the Clarkson-
McLeod solutions via an integrable operator with the parabolic cylinder
kernel.

2010 mathematics subject classification: 30E15; 33E17; 34E05; 41A60;
Keywords and phrases: The fourth Painlevé equation, asymptotic expansion,
Riemann-Hilbert problem, Deift-Zhou method, Hamiltonian, total integral,
Fredholm determinant.

###### Contents

  1. 1 Introduction and statement of results
    1. 1.1 Statement of results
  2. 2 Riemann-Hilbert problem for the Painlevé IV equation
  3. 3 RH analysis as $x\to-\infty$ with $0\leq|s_{*}|<1$
    1. 3.1 Deformations of the jump contour
    2. 3.2 Global parametrix
    3. 3.3 Local parametrices near the saddle points $z_{2,\pm}$
    4. 3.4 Local parametrices near the saddle points $z_{1,\pm}$
    5. 3.5 Local parametrix near the origin
    6. 3.6 Final transformation
    7. 3.7 Case $|s_{*}|=0$
  4. 4 RH analysis as $x\to-\infty$ with $|s_{*}|>1$
    1. 4.1 Modified global parametrix
    2. 4.2 Modified local parametrices at $z_{1,\pm}$, $z_{2,\pm}$
    3. 4.3 Modified local parametrix near the origin
    4. 4.4 Final transformation
  5. 5 RH analysis as $x\to-\infty$ with $|s_{*}|=1$ but $s_{*}\neq 1$
    1. 5.1 Global parametrix
    2. 5.2 Local parametrices at $z_{\pm}=\pm\sqrt{2}$
    3. 5.3 Local parametrix near the origin
    4. 5.4 Final transformation
  6. 6 Proof of Theorems 1.1 \- 1.4
    1. 6.1 Derivations of (1.18) and (1.25)
    2. 6.2 Derivations of (1.20) and (1.27)
    3. 6.3 Derivations of (1.19) and (1.26)
    4. 6.4 Derivation of (1.24)
    5. 6.5 Derivations of () and ()
      1. 6.5.1 Derivation of ()
      2. 6.5.2 Derivation of ()
  7. 7 Proof of Theorem 1.6
  8. A Local parametrix models
    1. A.1 Airy parametrix
    2. A.2 Parabolic cylinder parametrix
    3. A.3 Bessel parametrix

##  1 Introduction and statement of results

This paper is concerned with the fourth Painlevé (PIV) equation

$\frac{\mathrm{d}^{2}q}{\mathrm{d}x^{2}}=\frac{1}{2q}\left(\frac{\mathrm{d}q}{\mathrm{d}x}\right)^{2}+\frac{3}{2}q^{3}+4xq^{2}+2(x^{2}-2\alpha)q,$
(1.1)

where the parameter $\alpha\in\mathbb{R}$. We are interested in the real
solutions of (1.1) satisfying the asymptotic condition

$q(x)\rightarrow 0\quad\mathrm{as}\quad x\rightarrow+\infty.$ (1.2)

A result due to Bassom _et al._ [6] shows that any real solution of the PIV
equation (1.1), fulfilling the boundary condition (1.2), possesses the
following asymptotic behavior

$q(x)\sim\kappa D_{\alpha-\frac{1}{2}}^{2}(\sqrt{2}x)\quad\mathrm{as}\quad
x\to+\infty$ (1.3)

for some constant $\kappa\in\mathbb{R}$. Here, $D_{\mu}(x)$ denotes the
parabolic cylinder function, which is the solution of Weber’s equation (cf.
[24, Chapter 12])

$\displaystyle\frac{\mathrm{d}^{2}D_{\mu}(x)}{\mathrm{d}x^{2}}=\left(\frac{1}{4}x^{2}-\mu-\frac{1}{2}\right)D_{\mu}(x),$

uniquely characterized by the asymptotic property

$D_{\mu}(x)=x^{\mu}e^{-\frac{1}{4}x^{2}}\left(1+O\left(x^{-2}\right)\right),\quad\mathrm{as}\quad
x\to+\infty.$ (1.4)

Conversely, for any constant $\kappa\in\mathbb{R}$, there exists a unique
solution to the PIV equation (1.1) asymptotic to $\kappa
D_{\alpha-\frac{1}{2}}^{2}(\sqrt{2}x)$ as $x\to+\infty$. For convenience, we
denote such solutions by $q(x)=q(x;\alpha,\kappa)$.

Concerning the asymptotic behaviors of $q(x;\alpha,\kappa)$ as
$x\rightarrow-\infty$, there has been the following conjecture.

### Conjecture (Clarkson-McLeod [9])

There exists a constant $\kappa^{*}>0$ such that:

(a) When $0<\kappa<\kappa^{*}$ and as $x\rightarrow-\infty$, we have

$q(x;\alpha,\kappa)\sim c_{n}\,2^{\alpha-\frac{1}{2}}x^{2\alpha-1}e^{-x^{2}}$
(1.5)

if $\alpha+\frac{1}{2}=n+1\in\mathbb{N}$ with $\mathbb{N}$ being the set of
all positive integers; and

$q(x;\alpha,\kappa)\sim-\frac{2x}{3}+\frac{4d_{1}}{\sqrt{3}}\sin\left(\frac{x^{2}}{\sqrt{3}}-\frac{4d_{1}^{2}}{\sqrt{3}}\ln(\sqrt{2}|x|)+d_{2}+O\left(x^{-2}\right)\right)+O\left(x^{-1}\right)$
(1.6)

if $\alpha-\frac{1}{2}\notin\mathbb{Z}$, where the constants $c_{n}$, $d_{1}$,
$d_{2}$ are dependent on $\kappa$.

(b) When $\kappa=\kappa^{*}$, $q(x;\alpha,\kappa)$ is asymptotic to $-2x$ as
$x\rightarrow-\infty$.

(c) When $\kappa>\kappa^{*}$, $q(x;\alpha,\kappa)$ has a pole at some point
$x$ on the real axis.

The solutions $q(x;\alpha,\kappa)$ in this class are now called the _Clarkson-
McLeod solutions_ of the PIV equation (1.1). In the case
$\alpha+\frac{1}{2}\in\mathbb{N}$ of part (a), the asymptotic formula (1.5)
has been proven in [6, 9]. Therein, the values of $c_{n}$ and $\kappa^{*}$
were explicitly evaluated by

$c_{n}=\frac{\kappa}{1-\sqrt{\pi}\,n!\,\kappa},\qquad\kappa^{*}=\frac{1}{\sqrt{\pi}\,n!}.$
(1.7)

While $\alpha-\frac{1}{2}\notin\mathbb{Z}$, the value of $\kappa^{*}$ in [9]
was conjectured to be

$\kappa^{*}=\frac{1}{\sqrt{\pi}\,\Gamma\left(\alpha+\frac{1}{2}\right)}.$
(1.8)

For $\alpha+\frac{1}{2}\in\mathbb{N}$, each solution $q(x)=q(x;\alpha,\kappa)$
of the PIV equation (1.1) can be explicitly expressed in terms of the
classical special functions. The behavior of the solution as $x\to-\infty$ can
then be determined in a straightforward manner. For example, when
$\alpha=\frac{1}{2}$, the exact solution is given by (see [6, Equation
(4.13)])

$q\left(x;{1}/{2},\kappa\right)=\frac{2\kappa\,e^{-x^{2}}}{2-\kappa\sqrt{\pi}\,\mathrm{erfc}(x)},\quad
x\in\mathbb{R},$ (1.9)

where $\mathrm{erfc}(x)=\frac{2}{\sqrt{\pi}}\int^{\infty}_{x}e^{-t^{2}}dt$ is
the complementary error function; cf. [24, Chapter 7]. It is readily verified
that $\kappa^{*}=1/\sqrt{\pi}$. The approximation (1.5) is valid since
$\mathrm{erfc}(x)$ is strictly monotone decreasing on $\mathbb{R}$ with range
$(0,2)$. Similarly, it holds that
$q\left(x;{1}/{2},1/\sqrt{\pi}\right)\sim-2x$ as $x\to-\infty$. For
$\kappa\sqrt{\pi}>1$, the solution has a real pole which is the zero of the
denominator in (1.9). For general $\alpha+\frac{1}{2}\in\mathbb{N}$, the
solutions and their asymptotic approximations as given in (1.5) can be
obtained by using (1.9) and the Bäcklund transformation; see [6, Equation
(5.9)], or [4, Equation (3.24)].

The solutions $q(x;\alpha,\kappa)$ with positive half-integers $\alpha$ find
prominent applications in the theory of orthogonal polynomials with the
discontinuous Hermite weight [8], and in random matrix theory [15, 25]. It is
well known that for the Gaussian unitary ensemble of $n\times n$ Hermitian
matrices, the probability of having no eigenvalues in the interval
$(x,+\infty)$ can be expressed in terms of the Fredholm determinant

$\det\left(\mathbf{I}-K_{n,x}\right).$ (1.10)

Here, $K_{n,x}$ is the integrable operator acting on $L^{2}(0,+\infty)$ with
the classical Hermite kernel

$K_{n,x}(\lambda,\mu)=e^{-\frac{(x+\lambda)^{2}+(\mu+x)^{2}}{2}}\gamma_{n-1}^{2}\frac{\pi_{n}(\lambda+x)\pi_{n-1}(\mu+x)-\pi_{n-1}(\lambda+x)\pi_{n}(\mu+x)}{\lambda-\mu},$
(1.11)

where $\pi_{n}(\lambda)$ is the $n$-th monic Hermite polynomial determined by
the orthogonal relation

$\int_{\mathbb{R}}\pi_{n}(x)\pi_{m}(x)e^{-x^{2}}dx=\gamma_{n}^{-2}\delta_{n,m},$
(1.12)

with the normalization constant

$\gamma_{n-1}^{2}=\frac{2^{n-1}}{\sqrt{\pi}\,\Gamma(n)};$ (1.13)

see [24, Table 18.3.1]. It is found by Tracy and Widom in [25] that

$\frac{d}{dx}\ln\det(\mathbf{I}-K_{n,x})=\sigma_{n}(x)$ (1.14)

where $\sigma_{n}(x)$ is the unique solution of the $\sigma$-form of the PIV
equation [20] with the parameter $\nu=n$

$\left(\sigma_{\nu}^{\prime\prime}\right)^{2}+4\left(\sigma_{\nu}^{\prime}\right)^{2}\left(\sigma_{\nu}^{\prime}+2\nu\right)-4\left(x\sigma_{\nu}^{\prime}-\sigma_{\nu}\right)^{2}=0,$
(1.15)

characterized by the boundary condition

$\sigma_{\nu}(x)\sim\frac{2^{\nu-1}x^{2\nu-2}e^{-x^{2}}}{\sqrt{\pi}(\nu-1)!},\quad
x\to+\infty.$ (1.16)

It is worth mentioning that $\sigma_{\nu}(x)$ is closely related to the
Hamiltonian for the PIV equation (1.1); see Remark 1.7 below and [20].

For general $\alpha-\frac{1}{2}\notin\mathbb{Z}$, the asymptotic formula (1.6)
in part (a) was proved by Abdullayev [1] using the integral equation method.
The connection formulas, that is, the explicit expressions of the parameters
$d_{1}$ and $d_{2}$ in (1.6), in terms of the parameter $\kappa$, were derived
later by Its and Kapaev [18], and by Wong and Zhang [27], using respectively
the isomonodromy method and the uniform asymptotic approach. Moreover,
according to the numerical investigations performed in [5],
$q(x;\alpha,\kappa)$ might blow up at finite $x$ if $\alpha<-1/2$. While
$\alpha>-1/2$, the same numerical results allow us to expect the absence of
the real poles of $q(x;\alpha,\kappa)$. The asymptotic behavior of
$q(x;\alpha,\kappa)$ and the connection formulas in part (c) were recently
derived by the current authors in [28]. While, to the best of our knowledge,
the asymptotic result in part (b) of the Clarkson-McLeod conjecture has not
been confirmed. It is also desirable to know whether there exists a
determinantal representation of the $\sigma$-form of the Clarkson-McLeod
solutions similar to (1.14), for general $\alpha$ not being a half-integer.

In the present paper, we derive the asymptotic approximations and the
connection formulas for the Clarkson-McLeod solutions as $x\to-\infty$ by
using the Deift-Zhou nonlinear steepest descent method [11, 12, 13].
Particularly, we prove part (b) and revisit part (a), (c) of the Clarkson-
McLeod conjecture. We also show that the $\sigma$-form of the Clarkson-McLeod
solutions of the PIV equation (1.1) with general parameter $\alpha$ can be
represented by the Fredholm determinant of an integrable operator whose kernel
is expressed in terms of the classical parabolic cylinder functions, thus
generalizing the result (1.14) of Tracy and Widom. Furthermore, the
asymptotics of the Hamiltonian of the Clarkson-McLeod solutions and the
evaluations of total integrals of the Clarkson-McLeod solutions are also
obtained.

###  1.1 Statement of results

#### Asymptotics of the Clarkson-McLeod solutions

Our first result is the following complete description of the asymptotic
behaviors of the Clarkson-McLeod solutions to the PIV equation (1.1) when the
parameter $\alpha\in\mathbb{R}$ with $\alpha+\frac{1}{2}\notin\mathbb{N}$.

######  Theorem 1.1.

Let $\kappa\neq 0$ be a given real number and $\kappa^{*}$ be the constant
defined by (1.8). For any $\alpha\in\mathbb{R}$ with
$\alpha+\frac{1}{2}\notin\mathbb{N}$, there exists a unique real solution
$q(x;\alpha,\kappa)$ to the PIV equation (1.1) satisfying the following
asymptotic behavior

$q(x;\alpha,\kappa)=\kappa\,2^{\alpha-\frac{1}{2}}x^{2\alpha-1}e^{-x^{2}}\left(1+O\left(x^{-2}\right)\right)\quad\mathrm{as}\quad
x\to+\infty.$ (1.17)

For $\alpha\in\mathbb{R}$ with $\alpha-\frac{1}{2}\notin\mathbb{Z}$, the
solution $q(x;\alpha,\kappa)$ possesses the following asymptotic behaviors as
$x\rightarrow-\infty$.

(1) If $\kappa(\kappa-\kappa^{*})<0$, then

$q(x;\alpha,\kappa)=-\frac{2}{3}x+\frac{2\sqrt{6}\,b_{1}}{3}\sin\left(\frac{x^{2}}{\sqrt{3}}-\frac{b_{1}^{2}}{\sqrt{3}}\ln(2\sqrt{3}x^{2})+\psi_{1}\right)+O\left(\frac{1}{x}\right).$
(1.18)

(2) If $\kappa=\kappa^{*}$, then

$q(x;\alpha,\kappa)=-2x+O\left(\frac{1}{x}\right).$ (1.19)

(3) If $\kappa(\kappa-\kappa^{*})>0$, then

$q(x;\alpha,\kappa)=-\frac{2}{3}x+\frac{2x}{2\cos\left(\frac{x^{2}}{\sqrt{3}}-\frac{b_{2}}{\sqrt{3}}\ln\left(2\sqrt{3}x^{2}\right)+\psi_{2}\right)+1}+O\left(\frac{1}{x}\right).$
(1.20)

The error term in the asymptotic expansion (1.20) is uniform for $x$ bounded
away from the singularities appearing on the right-hand side of the asymptotic
expansion. Moreover, the corresponding connection formulas are respectively
given by

$\left\\{\begin{aligned}
b_{1}^{2}&=-\frac{\sqrt{3}}{2\pi}\ln(1-\left|\rho\right|^{2}),\quad b_{1}\geq
0,\\\
\psi_{1}&=-\frac{\pi}{4}-\frac{2\pi}{3}\alpha-\arg\Gamma\left(-\frac{b_{1}^{2}}{\sqrt{3}}i\right)-\arg\rho,\end{aligned}\right.$
(1.21)

and

$\left\\{\begin{aligned} b_{2}&=-\frac{\sqrt{3}}{2\pi}\ln(|\rho|^{2}-1),\\\
\psi_{2}&=-\frac{2\pi}{3}\alpha-\arg\Gamma\left(-\frac{b_{2}}{\sqrt{3}}i+\frac{1}{2}\right)-\arg\rho,\end{aligned}\right.$
(1.22)

where

$\rho=1-\frac{2\pi^{\frac{3}{2}}}{e^{\pi
i\alpha}\Gamma\left(\frac{1}{2}-\alpha\right)}\kappa.$ (1.23)

When $\frac{1}{2}-\alpha\in\mathbb{N}$, it holds $\kappa^{*}=0$, and
$q(x;\alpha,\kappa)$ possesses the asymptotic behavior (1.20) for
$\kappa\in\mathbb{R}\setminus\\{0\\}$ as $x\rightarrow-\infty$.

######  Remark 1.2.

For all $\alpha\in\mathbb{R}$, the remaining case $\kappa=0$ corresponds to
the trivial solution $q(x;\alpha,0)=0$. As aforementioned, the asymptotic
formula (1.18) and connection formulas (1.21) have been derived in [18, 27].
In this paper, we give alternative proofs of these formulas by performing
asymptotic analysis of the Riemann-Hilbert (RH, for short) problem for PIV
equation. Furthermore, we accomplish the case $b_{1}=0$ in (1.18) and (1.21)
which is not covered in [18, Equation (1.10)] and [27, Equation (1.5)]. We
also show that (1.20) is true for negative half-integer $\alpha$, a case not
considered in our previous work [28]. We also provide a novel proof of (1.19).
Theorem 1.1, along with the case with positive half-integer $\alpha$ solved in
terms of the special functions in [6, 9], fully confirms the conjecture of
Clarkson and McLeod. Minor extension allows $\kappa^{*}$ to be negative or
zero.

The asymptotic analysis of the RH problem for the PIV equation enables us to
derive simultaneously the asymptotics of the Hamiltonian associated with the
Clarkson-McLeod solutions and the total integrals of Clarkson-McLeod solutions
along the real axis, which are of independent interests.

######  Theorem 1.3.

Under the conditions in Theorem 1.1, for any $\alpha\in\mathbb{R}$ with
$\alpha+\frac{1}{2}\notin\mathbb{N}$, the Hamiltonian
$\mathcal{H}(x;\alpha,\kappa)$ of the Clarkson-McLeod solutions, defined by
(2.14) below, has the following asymptotic behavior as $x\rightarrow+\infty$:

$\mathcal{H}(x;\alpha,\kappa)=-\kappa\,2^{\alpha-\frac{1}{2}}x^{2\alpha-1}e^{-x^{2}}\left(1+O\left(x^{-2}\right)\right).$
(1.24)

While for $\alpha\in\mathbb{R}$ with $\alpha-\frac{1}{2}\notin\mathbb{Z}$,
$\mathcal{H}(x;\alpha,\kappa)$ satisfies the following asymptotic behaviors as
$x\rightarrow-\infty$.

(1) If $\kappa(\kappa-\kappa^{*})<0$, then

$\mathcal{H}(x;\alpha,\kappa)=-\frac{8x^{3}}{27}+\frac{4}{3}\left(\alpha+b_{1}^{2}\right)x-\frac{2\sqrt{2}\,b_{1}}{3}\cos\left(\frac{x^{2}}{\sqrt{3}}-\frac{b_{1}^{2}}{\sqrt{3}}\ln(2\sqrt{3}x^{2})+\psi_{1}\right)+O\left(\frac{1}{x}\right),$
(1.25)

(2) If $\kappa=\kappa^{*}$, then

$\mathcal{H}(x;\alpha,\kappa)=4\alpha x+O\left(\frac{1}{x}\right),$ (1.26)

(3) If $\kappa(\kappa-\kappa^{*})>0$, then

$\mathcal{H}(x;\alpha,\kappa)=-\frac{8x^{3}}{27}+\frac{4}{3}(\alpha+b_{2})x+\frac{\frac{4}{\sqrt{3}}x\sin\left(\frac{x^{2}}{\sqrt{3}}-\frac{b_{2}}{\sqrt{3}}\ln\left(2\sqrt{3}x^{2}\right)+\psi_{2}\right)}{2\cos\left(\frac{x^{2}}{\sqrt{3}}-\frac{b_{2}}{\sqrt{3}}\ln\left(2\sqrt{3}x^{2}\right)+\psi_{2}\right)+1}+O\left(\frac{1}{x}\right).$
(1.27)

The parameters $b_{1}$, $\psi_{1}$ and $b_{2}$, $\psi_{2}$ are the same as in
(1.21) and (1.22), respectively. The error term in the asymptotic expansion
(1.27) is uniform for $x$ bounded away from the singularities appearing on the
right-hand side of the asymptotic expansion.

Moreover, when $\frac{1}{2}-\alpha\in\mathbb{N}$, for any non-vanishing
$\kappa\in\mathbb{R}$, $\mathcal{H}(x;\alpha,\kappa)$ possesses the same
asymptotic behavior (1.27) as $x\rightarrow-\infty$.

Our next result is the evaluation of the total integrals of the Clarkson-
McLeod solutions $q(x;\alpha,\kappa)$. Similar results for the integrals of
the Painlevé II transcendents have been derived in [2, 3, 10, 22, 23].

######  Theorem 1.4.

Let $\mathrm{P.V.}$ denote the Cauchy principal value. For
$\alpha\in\mathbb{R}$ with $\alpha-\frac{1}{2}\notin\mathbb{Z}$, we evaluate
the total integrals of $q(x;\alpha,\kappa)$ as follows.

(1) If $\kappa(\kappa-\kappa^{*})<0$, take $c<0<d$ such that all real poles of
$q(x;\alpha,\kappa)$ lie in the interval $(c,d)$,

$\displaystyle\exp\Bigg{\\{}\int^{c}_{-\infty}\left(q(t;\alpha,\kappa)+\frac{2t}{3}-\frac{2\alpha}{t}\right)dt+\mathrm{P.V.}\int_{c}^{d}q(t;\alpha,\kappa)\,dt+\int^{+\infty}_{d}q(t;\alpha,\kappa)\,dt\Bigg{\\}}$
$\displaystyle\qquad=\frac{(-1)^{N_{+}-N_{-}}\sqrt{\pi}e^{\frac{c^{2}}{3}}|c|^{-2\alpha}3^{2\alpha}(1-\rho)e^{\pi
i\alpha}}{2^{\frac{1}{2}-\alpha}\Gamma(\frac{1}{2}-\alpha)(1-|\rho|^{2})^{\frac{2}{3}}},$
(1.28)

where $\rho$ is related to $\kappa$ by (1.23) and $N_{\pm}$ denote the numbers
of real poles of $q(x;\alpha,\kappa)$ in the interval $(c,d)$ of residues $\pm
1$, respectively.

(2) If $\kappa=\kappa^{*}$, take $c<0<d$ such that all real poles of
$q(x;\alpha,\kappa)$ lie in the interval $(c,d)$,

$\displaystyle\exp\Bigg{\\{}\int^{c}_{-\infty}\left(q(t;\alpha,\kappa)+2t+\frac{2\alpha}{t}\right)dt+\mathrm{P.V.}\int_{c}^{d}q(t;\alpha,\kappa)\,dt+\int^{+\infty}_{d}q(t;\alpha,\kappa)\,dt\Bigg{\\}}$
$\displaystyle\qquad=\frac{(-1)^{N_{+}-N_{-}}\sqrt{\pi}e^{-c^{2}}|c|^{-2\alpha}2^{\frac{1}{2}+\alpha}}{\Gamma(\frac{1}{2}+\alpha)},$
(1.29)

where $N_{\pm}$ denote the numbers of real poles of $q(x;\alpha,\kappa)$ of
residues $\pm 1$ in the interval $(c,d)$.

######  Remark 1.5.

According to the numerical analysis performed in [5], it is expected that
$q(x;\alpha,\kappa)$ is pole free on the real axis when
$0<\kappa\leq\kappa^{*}$. However, to the best of our knowledge, there is no
rigorous proof of the numerical evidence. The Cauchy principal values in (1.4)
and (1.4) may be removed if one can prove that $q(x;\alpha,\kappa)$ is pole
free on the real axis in these cases.

#### Determinantal representation of the Clarkson-McLeod solutions

Let $\gamma K_{\nu,x}$ be the integrable operator acting on $L^{2}(0,+\infty)$
with the parabolic cylinder kernel

$\gamma
K_{\nu,x}(\lambda,\mu)=\gamma\frac{D_{\nu}(\sqrt{2}(\lambda+x))D_{\nu-1}(\sqrt{2}(\mu+x))-D_{\nu-1}(\sqrt{2}(\lambda+x))D_{\nu}(\sqrt{2}(\mu+x))}{\lambda-\mu},$
(1.30)

where $D_{\nu}$ is the parabolic cylinder function with the parameter
$\nu\in\mathbb{R}$ and $\gamma$ is a real parameter. Then, the Fredholm
determinant of the operator is related to the $\sigma$-form of the Clarkson-
McLeod solutions as stated in the following theorem.

######  Theorem 1.6.

Suppose that $\nu\in\mathbb{R}$, $\gamma\in\mathbb{R}$ and $\gamma K_{\nu,x}$
is the integrable operator acting on $L^{2}(0,+\infty)$ with the kernel
(1.30). We have

$\frac{d}{dx}\ln\det(\mathbf{I}-\gamma K_{\nu,x})=\sigma_{\nu}(x;\gamma),$
(1.31)

where $\sigma_{\nu}(x;\gamma)$ is the unique solution of the $\sigma$-form
(1.15) of the PIV equation, determined by the boundary condition near positive
infinity

$\sigma_{\nu}(x;\gamma)\sim\sqrt{2}\gamma D_{\nu-1}^{2}(\sqrt{2}x).$ (1.32)

######  Remark 1.7.

It is seen from [20, Equations (C.34)-(C.36)] and (2.14) below that the
$\sigma$-form of the PIV equation is related to $q(x;\alpha,\kappa)$ and the
Hamiltonian $\mathcal{H}(x;\alpha,\kappa)$ by

$\sigma_{\alpha+\frac{1}{2}}(x;\gamma)=\frac{1}{2}\left(q(x;\alpha,\kappa)-\mathcal{H}(x;\alpha,\kappa)\right),$
(1.33)

and conversely

$q(x;\alpha,\kappa)=-\frac{\sigma^{\prime\prime}_{\alpha+\frac{1}{2}}(x;\gamma)+2x\sigma_{\alpha+\frac{1}{2}}^{\prime}(x;\gamma)-2\sigma_{\alpha+\frac{1}{2}}(x;\gamma)}{2\sigma_{\alpha+\frac{1}{2}}^{\prime}(x;\gamma)+4\alpha+2}.$
(1.34)

It follows from the approximation (1.32) and the relation (1.34) that
$q(x;\alpha,\kappa)$ actually satisfies the boundary condition (1.17), and the
parameters $\kappa$ and $\gamma$ are related by

$\kappa=\sqrt{2}\gamma.$ (1.35)

In view of (1.33), the asymptotic behavior near negative infinity of the
solution of the $\sigma$-form (1.15) of the PIV equation subject to the
boundary condition (1.32) can be obtained by using Theorems 1.1 and 1.3.

######  Remark 1.8.

For $\nu=n\in\mathbb{N}$, the parabolic cylinder function is reduced to the
Hermite polynomial [24, (12.7.2)]

$D_{n}(\sqrt{2}x)=e^{-\frac{1}{2}x^{2}}2^{\frac{n}{2}}\pi_{n}(x),$ (1.36)

where $\pi_{n}(x)$ is the monic Hermite polynomial defined through (1.12).
Therefore, for given $\nu\in\mathbb{N}$, the kernel
$\gamma^{*}K_{\nu,x}(\lambda,\mu)$ with

$\gamma^{*}=\frac{1}{\sqrt{2\pi}\Gamma(\nu)}$ (1.37)

is reduced to the classical Hermite kernel defined in (1.11). Applying Theorem
1.6, we recover (1.14)-(1.16), which was obtained first by Tracy and Widom.

The rest of the present paper is arranged as follows. In Section 2, we state
the RH problem for the PIV equation (1.1) and express the solutions of the PIV
equation and the associated Hamiltonian in terms of the solution to this RH
problem. Subsequently, in Sections 3-5, we apply the Deift-Zhou nonlinear
steepest descent method to the mentioned RH problem as $x\to-\infty$, for the
parameter $\kappa$ respectively in three different regimes. Using the
asymptotic analysis of the RH problems we performed, Theorems 1.1-1.4 are then
proved in Section 6. The final Section 7 is devoted to the proof of Theorem
1.6. For the convenience of the reader, we collect in the Appendix three local
parametrix models used in the asymptotic analysis of the RH problems.

##  2 Riemann-Hilbert problem for the Painlevé IV equation

In this section, we review the RH problem for the PIV equation (1.1), a
detailed description can be found in [18, Section 2] and [14, Chapter 5.1].

Denote $\Sigma=\bigcup^{8}_{k=1}\gamma_{k}$, where
$\gamma_{k}=\\{\xi\in\mathbb{C}\mid\arg\xi=k\pi/4\\}$; see Figure 1. Then, the
$2\times 2$ matrix-valued function $\Psi(\xi,x)$ solves the following RH
problem.

### RH problem for $\Psi(\xi,x)$

(1) $\Psi(\xi,x)$ is analytic for all $\xi\in\mathbb{C}\setminus\Sigma$.

(2) $\Psi(\xi,x)$ satisfies the jump relations

$\Psi_{+}(\xi,x)=\Psi_{-}(\xi,x)\left\\{\begin{aligned}
&S_{k},\quad&\xi&\in\gamma_{k},\ k=1,\cdots,7,\\\ &S_{8}e^{-2\pi
i\alpha\sigma_{3}},\quad&\xi&\in\gamma_{8},\end{aligned}\right.$

where the Stokes matrices

$S_{2i-1}=\begin{pmatrix}1&s_{2i-1}\\\ 0&1\end{pmatrix},\quad
S_{2i}=\begin{pmatrix}1&0\\\ s_{2i}&1\end{pmatrix},\quad\ i=1,2,3,4.$ (2.1)

The Stokes multipliers $s_{k}$ satisfy the following restrictions

$s_{k+4}=-s_{k}e^{(-1)^{k}2\pi i\alpha},\quad k=0,1,2,3,4,$ (2.2)

and

$[(1+s_{3}s_{4})(1+s_{1}s_{2})+s_{1}s_{4}]e^{-i\pi\alpha}-(1+s_{2}s_{3})e^{i\pi\alpha}=-2i\sin(\pi\alpha).$
(2.3)

(3) As $\xi\to\infty$, $\Psi(\xi,x)$ satisfies the following asymptotic
condition

$\Psi(\xi,x)=\Psi^{(\infty)}(\xi,x)e^{\Theta(\xi,x)\sigma_{3}},$ (2.4)

where

$\Psi^{(\infty)}(\xi,x)=\mathbf{I}+\frac{\Psi_{1}}{\xi}+\frac{\Psi_{2}}{\xi^{2}}+O\left(\frac{1}{\xi^{3}}\right),\quad\Theta(\xi,x)=\frac{1}{8}\xi^{4}+\frac{x}{2}\xi^{2}+\alpha\ln\xi,$
(2.5)

with the branch of $\ln\xi$ chosen so that $\arg\xi\in(0,2\pi)$.

(4) As $\xi\to 0$, $\Psi(\xi,x)$ has the asymptotic behavior of the form

$\Psi(\xi,x)=\left\\{\begin{aligned}
&\Psi^{(0)}(\xi,x)\xi^{\alpha\sigma_{3}}E,&&\mathrm{for}\quad\alpha-\frac{1}{2}\notin\mathbb{Z},\\\
&\Psi^{(0)}(\xi,x)\xi^{\alpha\sigma_{3}}\begin{pmatrix}1&\frac{s_{0}}{\pi
i}\ln\xi\\\
0&1\end{pmatrix}E,&&\mathrm{for}\quad\alpha+\frac{1}{2}\in\mathbb{N},\\\
&\Psi^{(0)}(\xi,x)\xi^{\alpha\sigma_{3}}\begin{pmatrix}1&0\\\
-\frac{s_{0}}{\pi
i}\ln\xi&1\end{pmatrix}E,&&\mathrm{for}\quad\frac{1}{2}-\alpha\in\mathbb{N},\end{aligned}\right.$
(2.6)

where $\Psi^{(0)}(\xi,x)$ is analytic in the neighborhood of $\xi=0$. The
functions $\xi^{\alpha}$ and $\ln\xi$ take principal values. The connection
matrix $E$ is given by

$E=E_{0}S_{0}\cdots S_{k-1},\quad\xi\in\Omega_{k},\quad k=1,\cdots,8,$ (2.7)

where $S_{0}:=\mathbf{I}$ and the regions
$\Omega_{k}=\left\\{\xi\in\mathbb{C}\mid\arg\xi\in(\frac{(k-1)\pi}{4},\frac{k\pi}{4})\right\\}$
are depicted in Figure 1. Moreover, the connection matrix $E_{0}$ takes the
form

$E_{0}=\left\\{\begin{aligned} &\begin{pmatrix}1&0\\\ \frac{s_{0}e^{2\pi
i\alpha}}{e^{2\pi
i\alpha}+1}&1\end{pmatrix},&&\mathrm{for}\quad\alpha-\frac{1}{2}\notin\mathbb{Z},\\\
&\begin{pmatrix}p_{1}&-1\\\
1&0\end{pmatrix},&&\mathrm{for}\quad\alpha+\frac{1}{2}\in\mathbb{N},\\\
&\begin{pmatrix}1&0\\\
p_{2}&1\end{pmatrix},&&\mathrm{for}\quad\frac{1}{2}-\alpha\in\mathbb{N},\end{aligned}\right.$
(2.8)

where $p_{1}$, $p_{2}$ are two arbitrary nonzero constants.

In the RH formulation, we denote the Pauli matrices by $\sigma_{k}$, $k=1,2,3$

$\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$
(2.9)

Figure 1: The jump contour $\Sigma$ and the regions $\Omega_{k}$

A significant fact is that the PIV transcendents and the associated
Hamiltonian can be expressed in terms of the solution of the RH problem for
$\Psi(\xi,x)$.

######  Proposition 2.1.

The solution of the PIV equation (1.1) and the associated Hamiltonian are
related to the solution to above RH problem for $\Psi(\xi,x)$ by

$q(x)=(\Psi_{1})_{12}(\Psi_{1})_{21},$ (2.10)

and

$\mathcal{H}(x)=\left(\Psi_{2}\right)_{11}-\left(\Psi_{2}\right)_{22},$ (2.11)

where $\Psi_{1}$ and $\Psi_{2}$ are given in (2.5). Let $\Psi^{(0)}(\xi,x)$ be
defined as (2.6), we have

$F(x):=\Psi^{(0)}(0,x)=\exp\bigg{[}\int^{x}q(x)dx\;\sigma_{3}\bigg{]}.$ (2.12)

###### Proof.

The relations (2.10) and (2.12) have been derived in [18, Equations (2.11) and
(2.40)]. According to Jimbo, Miwa and Ueno [19], it follows from (2.4) and
(2.5) that

$\mathcal{H}(x)=-\mathop{\mathrm{Res}}\limits_{z=\infty}\mathrm{Tr}\left(\Psi^{(\infty)}(\xi,x)^{-1}\frac{\mathrm{d}\Psi^{(\infty)}(\xi,x)}{\mathrm{d}\xi}\frac{\mathrm{d}\Theta(\xi,x)}{\mathrm{d}x}\sigma_{3}\right)=\left(\Psi_{2}\right)_{11}-\left(\Psi_{2}\right)_{22}.$
(2.13)

∎

We mention that using the Lax pair for $\Psi(\xi,x)$ [18, Equations
(2.1)-(2.2)], the Hamiltonian $\mathcal{H}(x)$ can be expressed in terms of
$q$ as follows

$\mathcal{H}(x)=\frac{q^{3}}{4}+xq^{2}+(x^{2}-2\alpha)q-\frac{(q^{\prime})^{2}}{4q}.$
(2.14)

Let $p=\frac{q^{\prime}}{q}$, then

$H(x):=-2\mathcal{H}(x)=-\frac{q^{3}}{2}-2xq^{2}-2(x^{2}-2\alpha)q+\frac{1}{2}p^{2}q.$
(2.15)

The PIV equation (1.1) can be derived by eliminating $p$ from the Hamilton
equations

$\frac{dq}{dx}=\frac{\partial H}{\partial
p},\quad\frac{dp}{dx}=-\frac{\partial H}{\partial q}.$ (2.16)

Applying (2.14)-(2.16), we also find the relation

$\frac{d}{dx}\mathcal{H}(x)=\frac{\partial}{\partial
x}\mathcal{H}(x)=q^{2}+2xq.$ (2.17)

It was observed by Its and Kapaev [18, Equation (2.23)] that, for any real
solution of the PIV equation (1.1), the Stokes multipliers satisfy the
conditions

$\overline{s}_{0}=s_{0},\quad\overline{s}_{1}=-e^{2\pi i\alpha}s_{3}.$ (2.18)

When $\alpha\in\mathbb{R}$ with $\alpha-\frac{1}{2}\notin\mathbb{Z}$, Its and
Kapaev proved that for solutions of the PIV equation (1.1) determined by the
asymptotic behavior (1.17) with real parameter $\kappa$, the Stokes
multipliers fulfill the following conditions (see [18, Theorem 3.1])

$s_{2}=0,\quad s_{1}+s_{3}=0,\quad s_{*}\neq 1,\quad(1-s_{*})e^{\pi
i\alpha}\in\mathbb{R},$ (2.19)

where $s_{*}$ is constituted by the Stokes multipliers $s_{0}$, $s_{1}$
through

$s_{*}=s_{0}s_{1}+1.$ (2.20)

The connection formula between the Stokes multipliers and the parameter
$\kappa$ in the asymptotic behavior (1.17) was also derived therein

$\kappa=-\frac{(s_{*}-1)e^{\pi
i\alpha}}{2\pi^{\frac{3}{2}}}\Gamma\left(\frac{1}{2}-\alpha\right).$ (2.21)

Actually, the relation (2.21) is also true for
$\frac{1}{2}-\alpha\in\mathbb{N}$.

For the case $\alpha-\frac{1}{2}\notin\mathbb{Z}$, combining (1.8) with
(2.21), we find the following correspondence between the conditions on the
parameter $\kappa$ and the Stokes multiplier $s_{*}$ as shown in Table 1.

$\kappa$ | $s_{*}$
---|---
$\kappa(\kappa-\kappa^{*})<0$ | $|s_{*}|<1$
$\kappa=\kappa^{*}$ | $|s_{*}|=1,\ s_{*}\neq 1$
$\kappa(\kappa-\kappa^{*})>0$ | $|s_{*}|>1$
Table 1: The correspondence between $\kappa$ and $s_{*}$

We point out that the exceptional case $s_{*}=1$ in the table is equivalent to
$\kappa=0$ and the solution determined by (1.17) is trivial; see Remark 1.2.
While, for the case $\frac{1}{2}-\alpha\in\mathbb{N}$, it follows from (2.18),
(2.20) and (2.21) that we always have

$|s_{*}|^{2}=1+\frac{4\pi^{3}\kappa^{2}}{\Gamma(\frac{1}{2}-\alpha)^{2}}>1.$

For $\alpha+\frac{1}{2}\in\mathbb{N}$, as mentioned in the introduction, the
solution of the PIV equation (1.1) determined by (1.17) can be expressed in
terms of the complementary error function. These special solutions correspond
to the specified values of Stokes multipliers (see [21])

$s_{0}=s_{2}=0,\quad s_{1}+s_{3}=0.$ (2.22)

##  3 RH analysis as $x\to-\infty$ with $0\leq|s_{*}|<1$

In this section, we start by carrying out the Deift-Zhou nonlinear steepest
descent analysis of the RH problem for $\Psi$ as $x\to-\infty$ under the
condition $0<|s_{*}|<1$. Then, the reduced case $|s_{*}|=0$ is considered at
the end of this section.

Assume now that $x<0$. We start with the first transformation

$\Phi(z)=(-x)^{-\frac{\alpha}{2}\sigma_{3}}\Psi\left((-x)^{\frac{1}{2}}z,x\right).$
(3.1)

Immediately, $\Phi(z)$ solves the following RH problem.

### RH problem for $\Phi(z)$

(1) $\Phi(z)$ is analytic for all $z\in\mathbb{C}\setminus\widetilde{\Sigma}$,
where $\widetilde{\Sigma}=\Sigma\setminus(\gamma_{2}\cup\gamma_{6})$; see
Figure 1.

(2) $\Phi(z)$ satisfies the same jump conditions as $\Psi(\xi,x)$ on
$\widetilde{\Sigma}$.

(3) As $z\to\infty$,

$\Phi(z)=\left(\mathbf{I}+\frac{\Phi_{1}}{z}+\frac{\Phi_{2}}{z^{2}}+O(z^{-3})\right)z^{\alpha\sigma_{3}}e^{x^{2}(\frac{1}{8}z^{4}-\frac{1}{2}z^{2})\sigma_{3}},$
(3.2)

where the branch of $z^{\alpha}$ is chosen such that $\arg z\in(0,2\pi)$.

(4) $\Phi(z)$ has the same asymptotic behaviors as $\Psi(\xi,x)$ as $z\to 0$;
see (2.6) and (2.7).

Simultaneously, by (2.10), (2.11) and (3.1), we get

$q(x;\alpha,\kappa)=-x(\Phi_{1})_{12}(\Phi_{1})_{21},$ (3.3)

and

$\mathcal{H}(x;\alpha,\kappa)=-x\left[(\Phi_{2})_{11}-(\Phi_{2})_{22}\right],$
(3.4)

where $\Phi_{1}=\Phi_{1}(x)$ and $\Phi_{2}=\Phi_{2}(x)$ are the coefficients
in (3.2).

Introduce the following $g$-function

$g(z)=\frac{1}{8}z\left(z^{2}-\frac{8}{3}\right)^{\frac{3}{2}},$ (3.5)

where $\arg\left(z\pm\textstyle\sqrt{{8}/{3}}\right)\in(-\pi,\pi)$. By
straightforward computation,

$g(z)=\frac{1}{8}z^{4}-\frac{1}{2}z^{2}+\frac{1}{3}+\frac{4}{27}z^{-2}+O(z^{-4})$
(3.6)

as $z\rightarrow\infty$. Moreover, $g(z)$ has four saddle points

$z_{1,\pm}=\pm\sqrt{\frac{2}{3}},\quad z_{2,\pm}=\pm\sqrt{\frac{8}{3}}.$

We then make the second transformation

$\mathbf{U}(z)=e^{\frac{x^{2}}{3}\sigma_{3}}\Phi(z)e^{-x^{2}g(z)\sigma_{3}}.$
(3.7)

It is direct to check that $\mathbf{U}(z)$ satisfies the following RH problem.

### RH problem for $\mathbf{U}(z)$

(1) $\mathbf{U}(z)$ is analytic for
$z\in\mathbb{C}\setminus\widetilde{\Sigma}$.

(2) We have the jump relations
$\mathbf{U}_{+}(z)=\mathbf{U}_{-}(z)J_{\mathbf{U}}(z)$, where

$J_{\mathbf{U}}(z)=\left\\{\begin{aligned}
&\begin{pmatrix}1&s_{k}e^{2x^{2}g(z)}\\\
0&1\end{pmatrix},\quad&z&\in\gamma_{k},\ k=1,3,5,7,\\\
&\begin{pmatrix}e^{x^{2}(g_{-}(z)-g_{+}(z))}&0\\\ -s_{0}e^{2\pi
i\alpha}e^{-x^{2}(g_{+}(z)+g_{-}(z))}&e^{x^{2}(g_{+}(z)-g_{-}(z))}\end{pmatrix},\quad&z&\in\gamma_{4},\\\
&\begin{pmatrix}e^{-2\pi i\alpha}e^{x^{2}(g_{-}(z)-g_{+}(z))}&0\\\
s_{0}e^{2\pi i\alpha}e^{-x^{2}(g_{+}(z)+g_{-}(z))}&e^{2\pi
i\alpha}e^{x^{2}(g_{+}(z)-g_{-}(z))}\end{pmatrix},\quad&z&\in\gamma_{8}.\end{aligned}\right.$

(3) $\mathbf{U}(z)$ satisfies the asymptotic condition

$\mathbf{U}(z)=\left(\mathbf{I}+O(z^{-1})\right)z^{\alpha\sigma_{3}},\quad\mathrm{as}\quad
z\rightarrow\infty,$

where $\arg z\in(0,2\pi)$.

(4) $\mathbf{U}(z)$ satisfies the following asymptotic behavior as $z\to 0$

$\mathbf{U}(z)=\mathbf{U}_{0}(z)z^{\alpha\sigma_{3}}E_{0}S_{1}e^{-x^{2}g(z)\sigma_{3}},\qquad\arg
z\in(\pi/4,{\pi}/{2}),$ (3.8)

where $\mathbf{U}_{0}(z)$ is analytic in the neighborhood of $z=0$. The
asymptotic behaviors of $\mathbf{U}(z)$ in other regions are determined by
(3.8) and the jump relations satisfied by $\mathbf{U}(z)$.

Figure 2: The anti-Stokes curves of the exponent $g(z)$
Figure 3: The first deformation of the jump contour

###  3.1 Deformations of the jump contour

Next, we transform the RH problem for $\mathbf{U}(z)$ to a RH problem
formulated on the anti-Stokes curves of $g(z)$, as illustrated in Figure 2.

To this aim, we first notice that the RH problem for $\mathbf{U}(z)$ is
equivalent to the one posed on the contour shown in Figure 3, where we use the
notations $\widetilde{S}_{k}$ to denote the analytically extended jump
matrices $J_{\mathbf{U}}(z)$.

Secondly, since the jump matrices on $(z_{2,-},z_{2,+})$ are now oscillating
for large $|x|$, we should deform the segment $(z_{2,-},z_{2,+})$ to the anti-
Stokes curves of $g(z)$. Thus, we introduce the third transformation
$\mathbf{U}\rightarrow\mathbf{T}$. This transformation is based on the
following factorizations of the jump matrices on $(z_{2,-},z_{2,+})$.

$\displaystyle\widetilde{S}_{4}$
$\displaystyle=\begin{pmatrix}e^{x^{2}(g_{-}(z)-g_{+}(z))}&0\\\ -s_{0}e^{2\pi
i\alpha}&e^{x^{2}(g_{+}(z)-g_{-}(z))}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&-s_{0}^{-1}e^{-2\pi
i\alpha}e^{2x^{2}g_{-}(z)}\\\
0&1\end{pmatrix}\begin{pmatrix}0&s_{0}^{-1}e^{-2\pi i\alpha}\\\ -s_{0}e^{2\pi
i\alpha}&0\end{pmatrix}\begin{pmatrix}1&-s^{-1}_{0}e^{-2\pi
i\alpha}e^{2x^{2}g_{+}(z)}\\\ 0&1\end{pmatrix}$
$\displaystyle=:\widetilde{S}_{U_{1}}\widetilde{S}_{P_{-}}\widetilde{S}_{U_{2}},$
(3.9)
$\displaystyle(\widetilde{S}_{3}\widetilde{S}_{4}\widetilde{S}_{5})^{-1}$
$\displaystyle=\begin{pmatrix}s_{*}e^{x^{2}(g_{-}(z)-g_{+}(z))}&s_{1}(e^{-2\pi
i\alpha}+s_{*})\\\ s_{0}e^{2\pi
i\alpha}&\overline{s}_{*}e^{x^{2}(g_{+}(z)-g_{-}(z))}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&0\\\
\frac{\overline{s}_{*}e^{-2x^{2}g_{-}(z)}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix}\begin{pmatrix}0&(|s_{*}|^{2}-1)s_{0}^{-1}e^{-2\pi
i\alpha}\\\ s_{0}e^{2\pi i\alpha}(1-|s_{*}|^{2})^{-1}&0\end{pmatrix}$
$\displaystyle~{}~{}~{}~{}\times\begin{pmatrix}1&0\\\
\frac{s_{*}e^{-2x^{2}g_{+}(z)}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix}$
$\displaystyle=:\widetilde{S}_{L_{1}}\widetilde{S}_{P}\widetilde{S}_{L_{2}},$
(3.10) $\displaystyle\widetilde{S}_{7}\widetilde{S}_{8}\widetilde{S}_{1}$
$\displaystyle=\begin{pmatrix}\overline{s}_{*}e^{-2\pi
i\alpha}e^{x^{2}(g_{-}(z)-g_{+}(z))}&s_{1}(e^{-2\pi i\alpha}+s_{*})\\\
s_{0}e^{2\pi i\alpha}&s_{*}e^{2\pi
i\alpha}e^{x^{2}(g_{+}(z)-g_{-}(z))}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&0\\\ \frac{s_{*}e^{2\pi
i\alpha}e^{-2x^{2}g_{-}(z)}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix}\begin{pmatrix}0&(|s_{*}|^{2}-1)s_{0}^{-1}e^{-2\pi
i\alpha}\\\ s_{0}e^{2\pi i\alpha}(1-|s_{*}|^{2})^{-1}&0\end{pmatrix}$
$\displaystyle~{}~{}~{}~{}\times\begin{pmatrix}1&0\\\
\frac{\overline{s}_{*}e^{-2\pi i\alpha}e^{-2x^{2}g_{+}(z)}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix}$
$\displaystyle=:\widetilde{S}_{L_{3}}\widetilde{S}_{P}\widetilde{S}_{L_{4}},$
(3.11)

and

$\displaystyle\widetilde{S}_{8}$ $\displaystyle=\begin{pmatrix}e^{-2\pi
i\alpha}e^{x^{2}(g_{-}(z)-g_{+}(z))}&0\\\ s_{0}e^{2\pi i\alpha}&e^{2\pi
i\alpha}e^{x^{2}(g_{+}(z)-g_{-}(z))}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&s_{0}^{-1}e^{-4\pi
i\alpha}e^{2x^{2}g_{-}(z)}\\\
0&1\end{pmatrix}\begin{pmatrix}0&-s_{0}^{-1}e^{-2\pi i\alpha}\\\ s_{0}e^{2\pi
i\alpha}&0\end{pmatrix}\begin{pmatrix}1&s^{-1}_{0}e^{2x^{2}g_{+}(z)}\\\
0&1\end{pmatrix}$
$\displaystyle=:\widetilde{S}_{U_{3}}\widetilde{S}_{P_{+}}\widetilde{S}_{U_{4}}.$
(3.12)

In the above factorizations, we have utilized the property

$g_{+}(z)+g_{-}(z)=0,\quad\mathrm{for}\quad z\in(z_{2,-},z_{2,+}),$

and the complex conjugate relation

$\overline{s}_{*}=s_{0}s_{1}e^{2\alpha\pi i}+1.$ (3.13)

Now, we define the third transformation $\mathbf{U}\mapsto\mathbf{T}$ as

$\mathbf{T}(z)=\left\\{\begin{aligned} &\mathbf{U}(z),&\mathrm{for}\ &z\
\mathrm{outside\ the\ four\ lens\ regions},\\\
&\mathbf{U}(z)\widetilde{S}_{U_{2}},&\mathrm{for}\ &z\ \mathrm{in\ the\ upper\
part\ of\ the\ first\ lens\ region},\\\
&\mathbf{U}(z)\widetilde{S}_{U_{1}}^{-1},&\mathrm{for}\ &z\ \mathrm{in\ the\
lower\ part\ of\ the\ first\ lens\ region},\\\
&\mathbf{U}(z)\widetilde{S}_{L_{2}}^{-1},&\mathrm{for}\ &z\ \mathrm{in\ the\
upper\ part\ of\ the\ second\ lens\ region},\\\
&\mathbf{U}(z)\widetilde{S}_{L_{1}},&\mathrm{for}\ &z\ \mathrm{in\ the\ lower\
part\ of\ the\ second\ lens\ region},\\\
&\mathbf{U}(z)\widetilde{S}_{L_{4}}^{-1},&\mathrm{for}\ &z\ \mathrm{in\ the\
upper\ part\ of\ the\ third\ lens\ region},\\\
&\mathbf{U}(z)\widetilde{S}_{L_{3}},&\mathrm{for}\ &z\ \mathrm{in\ the\ lower\
part\ of\ the\ third\ lens\ region}\\\
&\mathbf{U}(z)\widetilde{S}_{U_{4}}^{-1},&\mathrm{for}\ &z\ \mathrm{in\ the\
upper\ part\ of\ the\ fourth\ lens\ region},\\\
&\mathbf{U}(z)\widetilde{S}_{U_{3}},&\mathrm{for}\ &z\ \mathrm{in\ the\ lower\
part\ of\ the\ fourth\ lens\ region},\end{aligned}\right.$ (3.14)

where the lens regions are illustrated in Figure 4 and the same notations are
used to denote the analytic extensions of the corresponding jump matrices.

Figure 4: The second deformation of the jump contour

Immediately, $\mathbf{T}(z)$ solves a RH problem whose jump contour and jump
matrices are shown in Figure 4. We proceed to blow up the four lens in Figure
4. Consequently, we obtain the following equivalent RH problem for
$\mathbf{T}(z)$.

Figure 5: The jump contour $\Sigma_{\mathbf{T}}$

### RH problem for $\mathbf{T}(z)$

(1) $\mathbf{T}(z)$ is analytic for
$z\in\mathbb{C}\setminus\Sigma_{\mathbf{T}}$, where $\Sigma_{\mathbf{T}}$ is
shown in Figure 5.

(2) We have $\mathbf{T}_{+}(z)=\mathbf{T}_{-}(z)J_{k}(z)$ for $z\in\pi_{k}$,
$k=1,\cdots,20$, where

$\displaystyle J_{1}(z)=\begin{pmatrix}1&-s_{0}^{-1}e^{2x^{2}g(z)}\\\
0&1\end{pmatrix},$ $\displaystyle
J_{2}(z)=\begin{pmatrix}1&s_{*}s_{0}^{-1}e^{2x^{2}g(z)}\\\ 0&1\end{pmatrix},$
$\displaystyle J_{3}(z)=\begin{pmatrix}1&0\\\ -\frac{\overline{s}_{*}e^{-2\pi
i\alpha}e^{-2x^{2}g(z)}}{s_{1}(e^{-2\pi i\alpha}+s_{*})}&1\end{pmatrix},$
$\displaystyle J_{4}(z)=\begin{pmatrix}1&0\\\ \frac{(e^{-2\pi
i\alpha}-1)e^{-2x^{2}g(z)}}{s_{1}(e^{-2\pi i\alpha}+s_{*})}&1\end{pmatrix},$
$\displaystyle J_{5}(z)=\begin{pmatrix}1&0\\\
\frac{s_{*}e^{-2x^{2}g(z)}}{s_{1}(e^{-2\pi i\alpha}+s_{*})}&1\end{pmatrix},$
$\displaystyle J_{6}(z)=\begin{pmatrix}1&-\overline{s}_{*}s_{0}^{-1}e^{-2\pi
i\alpha}e^{2x^{2}g(z)}\\\ 0&1\end{pmatrix},$ $\displaystyle
J_{7}(z)=J_{9}(z)=\begin{pmatrix}1&\frac{e^{2x^{2}g(z)}}{s_{0}e^{2\pi
i\alpha}}\\\ 0&1\end{pmatrix},$ $\displaystyle J_{8}(z)=\begin{pmatrix}1&0\\\
-s_{0}e^{2\pi i\alpha}e^{-2x^{2}g(z)}&1\end{pmatrix},$ $\displaystyle
J_{10}(z)=\begin{pmatrix}1&-\frac{s_{*}e^{2x^{2}g(z)}}{s_{0}e^{2\pi
i\alpha}}\\\ 0&1\end{pmatrix},$ $\displaystyle J_{11}(z)=\begin{pmatrix}1&0\\\
\frac{\overline{s}_{*}e^{-2x^{2}g(z)}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix},$ $\displaystyle
J_{12}(z)=\begin{pmatrix}1&0\\\ \frac{(e^{2\pi
i\alpha}-1)e^{-2x^{2}g(z)}}{s_{1}(e^{-2\pi i\alpha}+s_{*})}&1\end{pmatrix},$
$\displaystyle J_{13}(z)=\begin{pmatrix}1&0\\\ -\frac{s_{*}e^{2\pi
i\alpha}e^{-2x^{2}g(z)}}{s_{1}(e^{-2\pi i\alpha}+s_{*})}&1\end{pmatrix},$
$\displaystyle J_{14}(z)=\begin{pmatrix}1&\overline{s}_{*}s_{0}^{-1}e^{-4\pi
i\alpha}e^{2x^{2}g(z)}\\\ 0&1\end{pmatrix},$ $\displaystyle
J_{15}(z)=\begin{pmatrix}1&-s_{0}^{-1}e^{-4\pi i\alpha}e^{2x^{2}g(z)}\\\
0&1\end{pmatrix},$ $\displaystyle J_{16}(z)=\begin{pmatrix}e^{-2\pi
i\alpha}&0\\\ s_{0}e^{2\pi i\alpha}e^{-2x^{2}g(z)}&e^{2\pi
i\alpha}\end{pmatrix},$ $\displaystyle
J_{17}(z)=J_{20}(z)^{-1}=\begin{pmatrix}0&s_{0}^{-1}e^{-2\pi i\alpha}\\\
-s_{0}e^{2\pi i\alpha}&0\end{pmatrix},$

and

$J_{19}(z)=J_{18}(z)^{-1}=\begin{pmatrix}0&-s_{0}^{-1}e^{-2\pi
i\alpha}(1-|s_{*}|^{2})\\\ s_{0}e^{2\pi
i\alpha}(1-|s_{*}|^{2})^{-1}&0\end{pmatrix}.$

(3) As $z\rightarrow\infty$,
$\mathbf{T}(z)=\left(\mathbf{I}+O(z^{-1})\right)z^{\alpha\sigma_{3}}$, where
$\arg z\in(0,2\pi)$.

(4) As $z\to 0$, $\mathbf{T}(z)$ has the following asymptotic behavior

$\mathbf{T}(z)=\mathbf{T}_{0}(z)z^{\alpha\sigma_{3}}E_{0}S_{1}e^{-x^{2}g(z)\sigma_{3}}J_{3}(z),\quad\arg
z\in(\pi/4,{\pi}/{2}),$ (3.15)

where $\mathbf{T}_{0}(z)$ is analytic in the neighborhood of $z=0$. The
behaviors of $\mathbf{T}(z)$ in other regions are determined by (3.15) and the
jump relations satisfied by $\mathbf{T}(z)$.

Taking into account the lower and upper triangular structures of the jump
matrices, the properties of $\mathop{\rm Re}\nolimits g(z)$ on the anti-Stokes
curves (see Figure 2), and the fact that $\mathop{\rm Re}\nolimits g(z)>0$ on
the imaginary axis, we see that the jump matrices on $\pi_{k}$,
$k=1,\cdots,15$ of the RH problem for $\mathbf{T}(z)$ tend to the identity
matrix exponentially fast when $x\rightarrow-\infty$. Therefore, it is
expected that the dominant contribution to the asymptotics of $\mathbf{T}(z)$
as $x\rightarrow-\infty$ will come from the remaining jumps along the line
$(z_{2,-},+\infty)$ and the neighborhoods of the origin $z=0$ and the four
saddle points $z_{1,\pm}=\pm\sqrt{2/3}$, $z_{2,\pm}=\pm\sqrt{8/3}$. We are in
a position to construct the global parametrix solving the constant jumps along
$(z_{2,-},+\infty)$ and local parametrices near the origin and the four saddle
points.

###  3.2 Global parametrix

We need to solve the following RH problem.

### RH problem for $\mathbf{P}^{(\infty)}(z)$

(1) $\mathbf{P}^{(\infty)}(z)$ is analytic for
$z\in\mathbb{C}\setminus[z_{2,-},+\infty)$.

(2) $\mathbf{P}^{(\infty)}(z)$ satisfies the following jump conditions

$\mathbf{P}^{(\infty)}_{+}(z)=\mathbf{P}^{(\infty)}_{-}(z)J^{(\infty)}(z),$
(3.16)

where

$J^{(\infty)}(z)=\left\\{\begin{aligned} &\begin{pmatrix}0&-e^{-2\pi
i\alpha}s^{-1}_{0}\\\ e^{2\pi
i\alpha}s_{0}&0\end{pmatrix},&z&\in(z_{2,-},z_{1,-}),\\\
&\begin{pmatrix}0&-e^{-2\pi i\alpha}s_{0}^{-1}(1-|s_{*}|^{2})\\\ e^{2\pi
i\alpha}s_{0}(1-|s_{*}|^{2})^{-1}&0\end{pmatrix},&z&\in(z_{1,-},z_{1,+}),\\\
&\begin{pmatrix}0&-e^{-2\pi i\alpha}s^{-1}_{0}\\\ e^{2\pi
i\alpha}s_{0}&0\end{pmatrix},&z&\in(z_{1,+},z_{2,+}),\\\
&\begin{pmatrix}e^{-2\pi i\alpha}&0\\\ 0&e^{2\pi
i\alpha}\end{pmatrix},&z&\in(z_{2,+},+\infty).\end{aligned}\right.$

(3) As $z\rightarrow\infty$, we have

$\mathbf{P}^{(\infty)}(z)=\left(\mathbf{I}+O(z^{-1})\right)z^{\alpha\sigma_{3}},$
(3.17)

where the branch cut of $z^{\alpha}$ is taken along $\mathbb{R}_{+}$ so that
$\arg z\in(0,2\pi)$.

The solution of the above RH problem can be constructed as follows

$\mathbf{P}^{(\infty)}(z)=s_{0}^{-\frac{\sigma_{3}}{2}}f^{-\sigma_{3}}_{\infty}\mathbf{X}(z)f(z)^{\sigma_{3}}s_{0}^{\frac{\sigma_{3}}{2}},$
(3.18)

where $f(z)$ is the Szegő function

$f(z)=\left(\frac{3}{8}\right)^{\frac{\alpha}{2}}\left(z+\sqrt{z^{2}-\frac{8}{3}}\right)^{\alpha}\left(\frac{\left((2-\sqrt{3})z-i\sqrt{z^{2}-\frac{8}{3}}\right)^{2}-\frac{8}{3}}{\left((2-\sqrt{3})z+i\sqrt{z^{2}-\frac{8}{3}}\right)^{2}-\frac{8}{3}}\right)^{{\beta}},$
(3.19)

$\mathbf{X}(z)$ is given by

$\mathbf{X}(z)=\begin{pmatrix}1&1\\\
-i&i\end{pmatrix}\omega(z)^{\sigma_{3}}\begin{pmatrix}1&1\\\
-i&i\end{pmatrix}^{-1},\quad\omega(z)=\left(\frac{z-\sqrt{\frac{8}{3}}}{z+\sqrt{\frac{8}{3}}}\right)^{\frac{1}{4}},$
(3.20)

and $f_{\infty}$ is defined by

$f_{\infty}=2^{-\frac{\alpha}{2}}3^{\frac{\alpha}{2}}e^{\frac{\pi
i{\beta}}{3}}.$ (3.21)

The branches of the multi-valued functions in (3.18)-(3.20) are chosen by
specifying

$\arg
z\in(-\pi,\pi),\quad\arg\left(z\pm\textstyle\sqrt{\frac{8}{3}}\right)\in(-\pi,\pi),\qquad\arg\left(\textstyle
z+\sqrt{z^{2}-\frac{8}{3}}\right)\in(0,2\pi),$

and

$\arg\left(\frac{\left((2-\sqrt{3})z-i\sqrt{z^{2}-\frac{8}{3}}\right)^{2}-\frac{8}{3}}{\left((2-\sqrt{3})z+i\sqrt{z^{2}-\frac{8}{3}}\right)^{2}-\frac{8}{3}}\right)\in(-\pi,\pi).$

The constant ${\beta}$ in (3.19) and (3.21) is given by

${\beta}=-\frac{1}{2\pi i}\ln(1-|s_{*}|^{2}),$ (3.22)

where the logarithm takes the principal value. In the case $0<|s_{*}|<1$, it
is easily seen that

${\beta}\neq 0\quad\mathrm{and}\quad{\beta}\in i\mathbb{R}.$ (3.23)

From (3.19) and (3.20), as $z\rightarrow\infty$, straightforward computation
yields

$\displaystyle\mathbf{X}(z)$
$\displaystyle=\mathbf{I}+\sqrt{\frac{2}{3}}\sigma_{2}z^{-1}+\frac{\mathbf{I}}{3}z^{-2}+O(z^{-3}),$
(3.24) $\displaystyle f(z)$
$\displaystyle=2^{-\frac{\alpha}{2}}3^{\frac{\alpha}{2}}e^{\frac{\pi
i{\beta}}{3}}\left[1-\frac{2}{3}\left(\alpha+\sqrt{3}i{\beta}\right)z^{-2}+O(z^{-4})\right]z^{\alpha}.$
(3.25)

###  3.3 Local parametrices near the saddle points $z_{2,\pm}$

We will construct two parametrices $\mathbf{P}^{(2,\pm)}(z)$ satisfying the
same jumps as $\mathbf{T}(z)$ on $\Sigma_{\mathbf{T}}$ (see Figure 5),
respectively in the neighborhoods
$U(z_{2,\pm},\delta)=\\{z\in\mathbb{C}\mid|z-z_{2,\pm}|<\delta\\}$ of the
saddle points $z_{2,\pm}$, and matching with $\mathbf{P}^{(\infty)}(z)$ on the
boundaries $\partial
U(z_{2,\pm},\delta)=\\{z\in\mathbb{C}\mid|z-z_{2,\pm}|=\delta\\}$.

### RH problem for $\mathbf{P}^{(2,+)}(z)$

(1) $\mathbf{P}^{(2,+)}(z)$ is analytic for $z\in
U(z_{2,+},\delta)\setminus\Sigma_{\mathbf{T}}$.

(2) $\mathbf{P}^{(2,+)}(z)$ satisfies the same jumps as $\mathbf{T}(z)$ on
$U(z_{2,+},\delta)\cap\Sigma_{\mathbf{T}}$.

(3) On the circular boundary $\partial U(z_{2,+},\delta)$, it holds that

$\mathbf{P}^{(2,+)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\mathbf{P}^{(\infty)}(z),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (3.26)

To solve the above RH problem, we define a conformal mapping

$\varphi_{1}(z)=\left(\frac{3}{2}g(z)\right)^{\frac{2}{3}},$ (3.27)

where the branch is specified by the asymptotic condition

$\varphi_{1}(z)=2^{\frac{5}{6}}3^{-\frac{1}{6}}\textstyle\left(z-\sqrt{\frac{8}{3}}\right)\left(1+o(1)\right),\quad\mathrm{as}\quad
z\rightarrow\sqrt{\frac{8}{3}}.$ (3.28)

Then, the solution to the above RH problem can be explicitly constructed in
terms of the Airy function:

$\mathbf{P}^{(2,+)}(z)=\mathbf{E}^{(2,+)}(z)\Phi^{\mathrm{(Ai)}}\left(|x|^{\frac{4}{3}}\varphi_{1}(z)\right)\left(s_{0}e^{2\pi
i\alpha}\right)^{-\frac{\sigma_{3}}{2}}\sigma_{1}e^{\mp\pi
i\alpha\sigma_{3}}e^{-x^{2}g(z)\sigma_{3}},\quad\pm\mathop{\rm Im}\nolimits
z>0,$ (3.29)

where $\Phi^{\mathrm{(Ai)}}$ is the Airy model parametrix (see Appendix A.1),
and $\mathbf{E}^{(2,+)}(z)$ is given by

$\mathbf{E}^{(2,+)}(z)=\mathbf{P}^{(\infty)}(z)e^{\pm\pi
i\alpha\sigma_{3}}\sigma_{1}\left(s_{0}e^{2\pi
i\alpha}\right)^{\frac{\sigma_{3}}{2}}\frac{1}{\sqrt{2}}\begin{pmatrix}1&-i\\\
-i&1\end{pmatrix}|x|^{\frac{\sigma_{3}}{3}}\varphi_{1}(z)^{\frac{\sigma_{3}}{4}},\
\pm\mathop{\rm Im}\nolimits z>0.$ (3.30)

Here, the branch of $\varphi_{1}(z)^{\frac{1}{4}}$ is chosen such that
$\arg\varphi_{1}(z)\in(-\pi,\pi)$. This means that

$\left(\varphi_{1}(z)^{\frac{\sigma_{3}}{4}}\right)_{+}=\left(\varphi_{1}(z)^{\frac{\sigma_{3}}{4}}\right)_{-}e^{\frac{\pi
i}{2}\sigma_{3}},\quad z\in(z_{2,-},z_{2,+}).$ (3.31)

From (3.16) and (3.31), it is readily verified that $\mathbf{E}^{(2,+)}(z)$ is
analytic in the neighborhood $U(z_{2,+},\delta)$. Finally, combining (3.29),
(3.30) with the asymptotic behavior (A.2) gives us the matching condition
(3.26).

### RH problem for $\mathbf{P}^{(2,-)}(z)$

(1) $\mathbf{P}^{(2,-)}(z)$ is analytic for $z\in
U(z_{2,-},\delta)\setminus\Sigma_{\mathbf{T}}$.

(2) On $U(z_{2,-},\delta)\cap\Sigma_{\mathbf{T}}$, $\mathbf{P}^{(2,-)}(z)$
shares the same jumps as $\mathbf{T}(z)$.

(3) On the boundary $\partial U(z_{2,-},\delta)$, we have

$\mathbf{P}^{(2,-)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\mathbf{P}^{(\infty)}(z),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (3.32)

By symmetry, the solution to the above RH problem is given by

$\mathbf{P}^{(2,-)}(z)=\mathbf{E}^{(2,-)}(z)\Phi^{(\mathrm{Ai})}\left(|x|^{\frac{4}{3}}\varphi_{1}(-z)\right)\left(s_{0}e^{2\pi
i\alpha}\right)^{-\frac{\sigma_{3}}{2}}\sigma_{2}e^{-x^{2}g(z)\sigma_{3}},$
(3.33)

where $\varphi_{1}(z)$ is the conformal mapping defined by (3.27) and
$\mathbf{E}^{(2,-)}(z)$ is given by

$\mathbf{E}^{(2,-)}(z)=\mathbf{P}^{(\infty)}(z)\sigma_{2}\left(s_{0}e^{2\pi
i\alpha}\right)^{\frac{\sigma_{3}}{2}}\frac{1}{\sqrt{2}}\begin{pmatrix}1&-i\\\
-i&1\end{pmatrix}|x|^{\frac{\sigma_{3}}{3}}\varphi_{1}(-z)^{\frac{\sigma_{3}}{4}}.$
(3.34)

###  3.4 Local parametrices near the saddle points $z_{1,\pm}$

In this subsection, we seek two parametrices $\mathbf{P}^{(1,\pm)}(z)$
satisfying the same jumps as $\mathbf{T}(z)$ on $\Sigma_{\mathbf{T}}$ (see
Figure 5) in the neighborhoods
$U(z_{1,\pm},\delta)=\\{z\in\mathbb{C}\mid|z-z_{1,\pm}|<\delta\\}$ of the
saddle points $z_{1,\pm}$, and matching with $\mathbf{P}^{(\infty)}(z)$ on the
boundaries $\partial
U(z_{1,\pm},\delta)=\\{z\in\mathbb{C}\mid|z-z_{1,\pm}|=\delta\\}$.

### RH problem for $\mathbf{P}^{(1,+)}(z)$

(1) $\mathbf{P}^{(1,+)}(z)$ is analytic for $z\in
U(z_{1,+},\delta)\setminus\Sigma_{\mathbf{T}}$.

(2) $\mathbf{P}^{(1,+)}(z)$ shares the same jumps as $\mathbf{T}(z)$ on
$U(z_{1,+},\delta)\cap\Sigma_{\mathbf{T}}$.

(3) On the circle $\partial U(z_{1,+},\delta)$, $\mathbf{P}^{(1,+)}(z)$
satisfies

$\mathbf{P}^{(1,+)}(z)=\left(\mathbf{I}+O(|x|^{-1})\right)\mathbf{P}^{(\infty)}(z),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (3.35)

Let us define the conformal mapping

$\varphi_{2}(z)=\left\\{\begin{aligned}
&2\sqrt{-\textstyle\frac{\sqrt{3}i}{6}-g(z)},\quad&\mathop{\rm Im}\nolimits
z>0,\\\ &2\sqrt{-\textstyle\frac{\sqrt{3}i}{6}+g(z)},\quad&\mathop{\rm
Im}\nolimits z<0,\end{aligned}\right.$ (3.36)

where the branches of the square roots are specified by choosing

$\varphi_{2}(z)=e^{-\frac{\pi i}{4}}2\cdot
3^{-\frac{1}{4}}\textstyle\left(z-\sqrt{\frac{2}{3}}\right)\left(1+o(1)\right),\quad\mathrm{as}\quad
z\rightarrow\sqrt{\frac{2}{3}}.$ (3.37)

Let $\Phi^{\mathrm{(PC)}}$ be the parabolic cylinder model parametrix with
parameter ${\beta}$ given by (3.22); see Appendix A.2. The solution to the
above RH problem can be constructed as follows:

$\displaystyle\mathbf{P}^{(1,+)}(z)=\mathbf{E}^{(1,+)}(z)\Phi^{\mathrm{(PC)}}\left(|x|\varphi_{2}(z)\right)\left(\frac{s_{*}}{h_{0}}\right)^{\frac{\sigma_{3}}{2}}\left\\{\begin{aligned}
&s_{0}^{-\frac{\sigma_{3}}{2}}(-\sigma_{1})e^{-x^{2}g(z)\sigma_{3}},&\mathop{\rm
Im}\nolimits z>0,\\\ &s_{0}^{\frac{\sigma_{3}}{2}}\sigma_{3}e^{2\pi
i(\alpha+{\beta})\sigma_{3}}e^{-x^{2}g(z)\sigma_{3}},&\mathop{\rm Im}\nolimits
z<0,\end{aligned}\right.$ (3.38)

where $h_{0}$ is the Stokes multiplier in (A.4) and $\mathbf{E}^{(1,+)}(z)$ is
given by

$\mathbf{E}^{(1,+)}(z)=\mathbf{W}^{(+)}(z)\left(\frac{s_{*}}{h_{0}}\right)^{-\frac{\sigma_{3}}{2}}|x|^{{\beta}\sigma_{3}}e^{i\frac{\sqrt{3}}{6}x^{2}\sigma_{3}}2^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}|x|\varphi_{2}(z)&1\\\
1&0\end{pmatrix}$ (3.39)

with

$\mathbf{W}^{(+)}(z)=\mathbf{P}^{(\infty)}(z)\left\\{\begin{aligned}
&(-\sigma_{1})s_{0}^{\frac{1}{2}\sigma_{3}}\varphi_{2}(z)^{{\beta}\sigma_{3}},&\mathop{\rm
Im}\nolimits z&>0,\\\ &e^{-2\pi
i(\alpha+{\beta})\sigma_{3}}\sigma_{3}s_{0}^{-\frac{1}{2}\sigma_{3}}\varphi_{2}(z)^{{\beta}\sigma_{3}},&\mathop{\rm
Im}\nolimits z&<0.\end{aligned}\right.$ (3.40)

The branch of $\varphi_{2}(z)^{{\beta}}$ is chosen by specifying
$\arg\varphi_{2}(z)\in(0,2\pi)$, this implies the jump relations

$\left\\{\begin{aligned}
&\left(\varphi_{2}(z)^{{\beta}}\right)_{+}=\left(\varphi_{2}(z)^{{\beta}}\right)_{-}e^{2\pi
i{\beta}},\quad&z&\in(z_{1,+},z_{2,+}),\\\
&\left(\varphi_{2}(z)^{{\beta}}\right)_{+}=\left(\varphi_{2}(z)^{{\beta}}\right)_{-},\quad&z&\in(z_{2,-},z_{1,+}).\end{aligned}\right.$
(3.41)

From the jump conditions (3.16) and (3.41), it is readily seen that
$\mathbf{W}^{(+)}(z)$ is analytic in the neighborhood $U(z_{1,+},\delta)$. In
particular, it follows from (3.18), (3.19), (3.20), (3.37) and (3.40) that

$\mathbf{W}^{(+)}(\textstyle\sqrt{\frac{2}{3}})=-s_{0}^{-\frac{\sigma_{3}}{2}}f_{\infty}^{-\sigma_{3}}\begin{pmatrix}1&1\\\
-i&i\end{pmatrix}3^{-\frac{\sigma_{3}}{4}}e^{\frac{\pi
i}{4}\sigma_{3}}\begin{pmatrix}1&1\\\
-i&i\end{pmatrix}^{-1}2^{-\frac{{\beta}}{2}\sigma_{3}}3^{-\frac{{\beta}}{4}\sigma_{3}}e^{\frac{\pi
i{\beta}}{4}\sigma_{3}}e^{\frac{\pi i\alpha}{3}\sigma_{3}}\sigma_{1}.$ (3.42)

Finally, a combination of (3.18), (3.23), (A.5) and (3.38) gives

$\displaystyle\mathbf{P}^{(1,+)}(z)\mathbf{P}^{(\infty)}(z)^{-1}$
$\displaystyle=\mathbf{W}^{(+)}(z)\begin{pmatrix}1+\frac{{\beta}({\beta}+1)}{2|x|^{2}\varphi_{2}^{2}}+O\left(|x|^{-4}\right)&\frac{e^{i\frac{\sqrt{3}}{3}x^{2}}|x|^{2{\beta}}}{h_{0}^{-1}s_{*}}\left(\frac{{\beta}}{|x|\varphi_{2}}+O\left(|x|^{-3}\right)\right)\\\
\frac{h_{0}^{-1}s_{*}}{e^{i\frac{\sqrt{3}}{3}x^{2}}|x|^{2{\beta}}}\left(\frac{1}{|x|\varphi_{2}}+O\left(|x|^{-3}\right)\right)&1-\frac{{\beta}({\beta}-1)}{2|x|^{2}\varphi_{2}^{2}}+O\left(|x|^{-4}\right)\end{pmatrix}\mathbf{W}^{(+)}(z)^{-1}$
$\displaystyle=\mathbf{I}+\mathbf{G}^{(+)}(z)|x|^{-1}+O\left(|x|^{-2}\right),$
(3.43)

where

$\mathbf{G}^{(+)}(z)=\mathbf{W}^{(+)}(z)\begin{pmatrix}0&\frac{\beta
h_{0}e^{i\frac{\sqrt{3}}{3}x^{2}}|x|^{2\beta}}{s_{*}\varphi_{2}(z)}\\\
\frac{s_{*}|x|^{-2\beta}}{h_{0}e^{i\frac{\sqrt{3}}{3}x^{2}}\varphi_{2}(z)}&0\end{pmatrix}\mathbf{W}^{(+)}(z)^{-1}.$
(3.44)

### RH problem for $\mathbf{P}^{(1,-)}(z)$

(1) $\mathbf{P}^{(1,-)}(z)$ is analytic for all $z\in
U(z_{1,-},\delta)\setminus\Sigma_{\mathbf{T}}$.

(2) $\mathbf{P}^{(1,-)}(z)$ satisfies the same jump conditions as
$\mathbf{T}(z)$ on $U(z_{1,-},\delta)\cap\Sigma_{\mathbf{T}}$.

(3) $\mathbf{P}^{(1,-)}(z)$ fulfills the matching condition on the boundary
$\partial U(z_{1,-},\delta)$:

$\mathbf{P}^{(1,-)}(z)=\left(\mathbf{I}+O(|x|^{-1})\right)\mathbf{P}^{(\infty)}(z),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (3.45)

Similarly, we introduce the conformal mapping

$\varphi_{3}(z)=\left\\{\begin{aligned}
&2\sqrt{-\textstyle\frac{\sqrt{3}i}{6}+g(z)},\quad&\mathop{\rm Im}\nolimits
z>0,\\\ &2\sqrt{-\textstyle\frac{\sqrt{3}i}{6}-g(z)},\quad&\mathop{\rm
Im}\nolimits z<0,\end{aligned}\right.$ (3.46)

where the branches of square roots are taken so that

$\varphi_{3}(z)=e^{\frac{3\pi i}{4}}2\cdot
3^{-\frac{1}{4}}\textstyle\left(z+\sqrt{\frac{2}{3}}\right)\left(1+o(1)\right),\quad\mathrm{as}\quad
z\rightarrow-\sqrt{\frac{2}{3}}.$ (3.47)

The solution to the above RH problem can be also constructed in terms of the
parabolic cylinder functions as follows:

$\displaystyle\mathbf{P}^{(1,-)}(z)=\mathbf{E}^{(1,-)}(z)\Phi^{\mathrm{(PC)}}\left(|x|\varphi_{3}(z)\right)\left(\frac{s_{*}}{h_{0}}\right)^{\frac{\sigma_{3}}{2}}\left\\{\begin{aligned}
&e^{2\pi i{\beta}\sigma_{3}}\left(s_{0}e^{2\pi
i\alpha}\right)^{\frac{\sigma_{3}}{2}}e^{-x^{2}g(z)\sigma_{3}},&\mathop{\rm
Im}\nolimits z>0,\\\ &\left(s_{0}e^{2\pi
i\alpha}\right)^{-\frac{\sigma_{3}}{2}}\sigma_{3}\sigma_{1}e^{-x^{2}g(z)\sigma_{3}},&\mathop{\rm
Im}\nolimits z<0,\end{aligned}\right.$ (3.48)

where $\Phi^{\mathrm{(PC)}}$ is the parabolic cylinder model parametrix,
$h_{0}$ is given in (A.4), ${\beta}$ is defined by (3.22) and
$\mathbf{E}^{(1,-)}(z)$ is given by

$\mathbf{E}^{(1,-)}(z)=\mathbf{W}^{(-)}(z)\left(\frac{s_{*}}{h_{0}}\right)^{-\frac{\sigma_{3}}{2}}|x|^{{\beta}\sigma_{3}}e^{\frac{i\sqrt{3}x^{2}}{6}\sigma_{3}}2^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}|x|\varphi_{3}(z)&1\\\
1&0\end{pmatrix}$ (3.49)

with

$\mathbf{W}^{(-)}(z)=\mathbf{P}^{(\infty)}(z)\left\\{\begin{aligned}
&\left(s_{0}e^{2\pi i\alpha}\right)^{-\frac{\sigma_{3}}{2}}e^{-2\pi
i{\beta}\sigma_{3}}\varphi_{3}(z)^{{\beta}\sigma_{3}},\ &\mathop{\rm
Im}\nolimits z&>0,\\\ &\sigma_{1}\sigma_{3}\left(s_{0}e^{2\pi
i\alpha}\right)^{\frac{\sigma_{3}}{2}}\varphi_{3}(z)^{{\beta}\sigma_{3}},\
&\mathop{\rm Im}\nolimits z&<0.\end{aligned}\right.$ (3.50)

We point out that the branch of the function $\varphi_{3}(z)^{{\beta}}$ is
chosen such that $\arg\varphi_{3}(z)\in(-\pi,\pi)$. This implies that

$\left\\{\begin{aligned}
&\left(\varphi_{3}(z)^{{\beta}}\right)_{+}=\left(\varphi_{3}(z)^{{\beta}}\right)_{-}e^{2\pi
i{\beta}},\quad&z&\in(z_{2,-},z_{1,-}),\\\
&\left(\varphi_{3}(z)^{{\beta}}\right)_{+}=\left(\varphi_{3}(z)^{{\beta}}\right)_{-},\quad&z&\in(z_{1,-},z_{2,+}).\end{aligned}\right.$
(3.51)

Using the jump relations (3.16) and (3.51), it is straightforward to check
that $\mathbf{W}^{(-)}(z)$ is analytic in the neighborhood
$U(z_{1,-},\delta)$. Particularly, by substituting (3.18), (3.19), (3.20) and
(3.47) into (3.50), we have

$\mathbf{W}^{(-)}(-\textstyle\sqrt{\frac{2}{3}})=s_{0}^{-\frac{\sigma_{3}}{2}}f_{\infty}^{-\sigma_{3}}\begin{pmatrix}1&1\\\
-i&i\end{pmatrix}3^{\frac{1}{4}\sigma_{3}}e^{\frac{\pi
i}{4}\sigma_{3}}\begin{pmatrix}1&1\\\
-i&i\end{pmatrix}^{-1}2^{\frac{{\beta}}{2}\sigma_{3}}3^{\frac{{\beta}}{4}\sigma_{3}}e^{-\frac{\pi
i\alpha}{3}\sigma_{3}}e^{-\frac{\pi i{\beta}}{4}\sigma_{3}}.$ (3.52)

Furthermore, using (3.18), (3.23), (3.48) and the asymptotic behavior (A.5),
we obtain the matching condition

$\displaystyle\mathbf{P}^{(1,-)}(z)\mathbf{P}^{(\infty)}(z)^{-1}$
$\displaystyle=\mathbf{W}^{(-)}(z)\begin{pmatrix}1+\frac{{\beta}({\beta}+1)}{2|x|^{2}\varphi_{3}^{2}}+O\left(|x|^{-4}\right)&\frac{e^{i\frac{\sqrt{3}}{3}x^{2}}|x|^{2{\beta}}}{h_{0}^{-1}s_{*}}\left(\frac{{\beta}}{|x|\varphi_{3}}+O\left(|x|^{-3}\right)\right)\\\
\frac{h_{0}^{-1}s_{*}}{e^{i\frac{\sqrt{3}}{3}x^{2}}|x|^{2{\beta}}}\left(\frac{1}{|x|\varphi_{3}}+O\left(|x|^{-3}\right)\right)&1-\frac{{\beta}({\beta}-1)}{2|x|^{2}\varphi_{3}^{2}}+O\left(|x|^{-4}\right)\end{pmatrix}\mathbf{W}^{(-)}(z)^{-1}$
$\displaystyle=\mathbf{I}+\mathbf{G}^{(-)}(z)|x|^{-1}+O\left(|x|^{-2}\right)$
(3.53)

where

$\mathbf{G}^{(-)}(z)=\mathbf{W}^{(-)}(z)\begin{pmatrix}0&\frac{{\beta}h_{0}e^{i\frac{\sqrt{3}}{3}x^{2}}|x|^{2{\beta}}}{s_{*}\varphi_{3}(z)}\\\
\frac{s_{*}|x|^{-2{\beta}}}{h_{0}e^{i\frac{\sqrt{3}}{3}x^{2}}\varphi_{3}(z)}&0\end{pmatrix}\mathbf{W}^{(-)}(z)^{-1}.$
(3.54)

###  3.5 Local parametrix near the origin

In this subsection, we intend to find a function $\mathbf{P}^{(0)}(z)$ having
the same jumps as $\mathbf{T}(z)$ on $\Sigma_{\mathbf{T}}$ (see Figure 5) in
the neighborhood $U(0,\delta)=\\{z\in\mathbb{C}\mid|z|<\delta\\}$ of the
origin, and matching with $\mathbf{P}^{(\infty)}(z)$ on the boundary $\partial
U(0,\delta)=\\{z\in\mathbb{C}\mid|z|=\delta\\}$.

### RH problem for $\mathbf{P}^{(0)}(z)$

(1) $\mathbf{P}^{(0)}(z)$ is analytic for $z\in
U(0,\delta)\setminus\Sigma_{\mathbf{T}}$.

(2) $\mathbf{P}^{(0)}(z)$ satisfies the same jump conditions as
$\mathbf{T}(z)$ on $U(0,\delta)\cap\Sigma_{\mathbf{T}}$.

(3) On the boundary $\partial U(0,\delta)$, we have

$\mathbf{P}^{(0)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\mathbf{P}^{(\infty)}(z),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (3.55)

(4) $\mathbf{P}^{(0)}(z)$ has the same asymptotic behavior as $\mathbf{T}(z)$
near the origin; see (3.15).

To begin with, we construct a conformal mapping

$\varphi_{4}(z)=\pm\frac{1}{8}iz\left(z^{2}-\frac{8}{3}\right)^{\frac{3}{2}},\quad\pm\mathop{\rm
Im}\nolimits z>0,$ (3.56)

which behaves like

$\varphi_{4}(z)=2^{\frac{3}{2}}3^{-\frac{3}{2}}z(1+o(1)),\quad\mathrm{as}\quad
z\rightarrow 0.$ (3.57)

The solution of the above RH problem can be built explicitly in term of the
modified Bessel functions. To see this, let $\Phi^{(\mathrm{Bes})}$ be the
Bessel model parametrix with parameter $\alpha$ as given in Appendix A.3. The
parametrix $\mathbf{P}^{(0)}(z)$ is then constructed as follows:

$\mathbf{P}^{(0)}(z)=\mathbf{E}^{(0)}(z)\Phi^{(\mathrm{Bes})}\left(x^{2}\varphi_{4}(z)\right)\mathbf{K}(z)\left[s_{1}(e^{-2\pi
i\alpha}+s_{*})\right]^{-\frac{\sigma_{3}}{2}}e^{-x^{2}g(z)\sigma_{3}},$
(3.58)

where $\mathbf{K}(z)$ is a piecewise constant matrix defined in regions
$\Lambda_{k}$ described in Figure 11

$\mathbf{K}(z)=\left\\{\begin{aligned}
&\mathbf{I},\quad&z&\in\Lambda_{1}\cup\Lambda_{8},\\\ &\begin{pmatrix}1&0\\\
-e^{-2\pi i\alpha}&1\end{pmatrix},\quad&z&\in\Lambda_{2},\\\
&\begin{pmatrix}e^{-\pi i\alpha}&0\\\ -e^{\pi i\alpha}&e^{\pi
i\alpha}\end{pmatrix},\quad&z&\in\Lambda_{3},\\\ &e^{-\pi
i\alpha\sigma_{3}},\quad&z&\in\Lambda_{4},\\\ &e^{\pi
i\alpha\sigma_{3}},\quad&z&\in\Lambda_{5},\\\ &\begin{pmatrix}e^{\pi
i\alpha}&0\\\ e^{-\pi i\alpha}&e^{-\pi
i\alpha}\end{pmatrix},\quad&z&\in\Lambda_{6},\\\ &\begin{pmatrix}1&0\\\
e^{2\pi i\alpha}&1\end{pmatrix},\quad&z&\in\Lambda_{7},\end{aligned}\right.$
(3.59)

and $\mathbf{E}^{(0)}(z)$ is given by

$\mathbf{E}^{(0)}(z)=\mathbf{P}^{(\infty)}(z)\left[s_{1}(e^{-2\pi
i\alpha}+s_{*})\right]^{\frac{\sigma_{3}}{2}}\mathbf{Q}(z)e^{-\frac{1}{4}\pi
i\sigma_{3}}\frac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\ i&1\end{pmatrix}$ (3.60)

with

$\mathbf{Q}(z)=\left\\{\begin{aligned} &e^{\frac{1}{2}\pi
i\alpha\sigma_{3}},\quad&\mathop{\rm Im}\nolimits z>0,~{}~{}|z|<\delta,\\\
&e^{-\frac{1}{2}\pi i\alpha\sigma_{3}}\sigma_{3}\sigma_{1},\quad&\mathop{\rm
Im}\nolimits z<0,~{}~{}|z|<\delta.\end{aligned}\right.$ (3.61)

First, using the jump conditions (A.6), it can be easily checked that the
function $\mathbf{P}^{(0)}(z)$ constructed by (3.58)-(3.61) satisfies the same
jump relations as $\mathbf{T}(z)$ on $\Sigma_{\mathbf{T}}\cap U(0,\delta)$.
Applying (3.16), (3.60) and (3.61), one can check that $\mathbf{E}^{(0)}(z)$
is also analytic in the neighborhood $U(0,\delta)$. Moreover, the matching
condition (3.55) follows from (3.58), (3.60) and (A.7). Next, we verify the
behavior of $\mathbf{P}^{(0)}(z)$ as $z\to 0$. Recalling the definition of the
connection matrix $E_{0}$ as given in (2.8), we may rewrite the asymptotic
behavior (3.15) in the form

$\mathbf{T}(z)=\mathbf{T}_{1}(z)z^{\alpha\sigma_{3}}\begin{pmatrix}1&\frac{1}{1+e^{-2\pi
i\alpha}}\\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\\ -e^{-2\pi
i\alpha}&1\end{pmatrix}\left[s_{1}(e^{-2\pi
i\alpha}+s_{*})\right]^{-\frac{\sigma_{3}}{2}}e^{-x^{2}g(z)\sigma_{3}},$
(3.62)

where $\mathbf{T}_{1}(z)$ is analytic in a neighborhood of $z=0$. Comparing
(3.62) with (3.58) and (A.9), one can see that $\mathbf{P}^{(0)}(z)$ satisfies
the asymptotic behavior (3.15) as $z\to 0$.

###  3.6 Final transformation

The final transformation is defined as

$\mathbf{R}(z)=\left\\{\begin{aligned}
&\mathbf{T}(z)\left[\mathbf{P}^{(1,\pm)}(z)\right]^{-1},\quad&z&\in
U(z_{1,\pm},\delta)\setminus\Sigma_{\mathbf{T}},\\\
&\mathbf{T}(z)\left[\mathbf{P}^{(2,\pm)}(z)\right]^{-1},\quad&z&\in
U(z_{2,\pm},\delta)\setminus\Sigma_{\mathbf{T}},\\\
&\mathbf{T}(z)\left[\mathbf{P}^{(0)}(z)\right]^{-1},\quad&z&\in
U(0,\delta)\setminus\Sigma_{\mathbf{T}},\\\
&\mathbf{T}(z)\left[\mathbf{P}^{(\infty)}(z)\right]^{-1},\quad&\mathrm{e}&\mathrm{lsewhere}.\\\
\end{aligned}\right.$ (3.63)

Immediately, $\mathbf{R}(z)$ fulfills the following RH problem.

### RH problem for $\mathbf{R}(z)$

Figure 6: The jump contour $\Sigma_{\mathbf{R}}$

(1) $\mathbf{R}(z)$ is analytic for
$z\in\mathbb{C}\setminus\Sigma_{\mathbf{R}}$, where the contour
$\Sigma_{\mathbf{R}}$ is illustrated in Figure 6.

(2) On the contour $\Sigma_{\mathbf{R}}$, we have
$\mathbf{R}_{+}(z)=\mathbf{R}_{-}(z)J_{\mathbf{R}}(z)$, where

$J_{\mathbf{R}}(z)=\left\\{\begin{aligned}
&\mathbf{P}^{(1,\pm)}(z)\mathbf{P}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(z_{1,\pm},\delta),\\\
&\mathbf{P}^{(2,\pm)}(z)\mathbf{P}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(z_{2,\pm},\delta),\\\
&\mathbf{P}^{(0)}(z)\mathbf{P}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(0,\delta),\\\
&\mathbf{P}^{(\infty)}_{-}(z)J_{\mathbf{T}}(z)\mathbf{P}^{(\infty)}_{+}(z)^{-1},\quad&\mathrm{e}&\mathrm{lsewhere}.\\\
\end{aligned}\right.$ (3.64)

(3) As $z\rightarrow\infty$, we have

$\mathbf{R}(z)=\mathbf{I}+\frac{\mathbf{R}_{1}}{z}+\frac{\mathbf{R}_{2}}{z^{2}}+O(z^{-3}).$
(3.65)

In view of the matching conditions (3.26), (3.32), (3.35), (3.45) and (3.55),
it is now readily seen that as $x\rightarrow-\infty$,

$J_{\mathbf{R}}(z)=\left\\{\begin{aligned}
&\mathbf{I}+O(|x|^{-1}),&z&\in\partial U(z_{1,\pm},\delta),\\\
&\mathbf{I}+O(|x|^{-2}),&z&\in\partial U(0,\delta)\cup\partial
U(z_{2,\pm},\delta),\\\
&\mathbf{I}+O(e^{-c_{1}|x|^{2}}),&\mathrm{e}&\mathrm{lsewhere},\end{aligned}\right.$
(3.66)

where $c_{1}$ is some positive constant. We thus have

$\mathbf{R}(z)=\mathbf{I}+O(|x|^{-1})\quad\mathrm{as}\quad
x\rightarrow-\infty,$ (3.67)

where the error bound is uniform for
$z\in\mathbb{C}\setminus\Sigma_{\mathbf{R}}$.

###  3.7 Case $|s_{*}|=0$

In this section, we consider the special case $|s_{*}|=0$. It follows from the
definitions (2.20), (3.13) of $s_{*}$, $\bar{s}_{*}$ that

$\left\\{\begin{aligned} &1+s_{0}s_{1}=0,\\\ &1+s_{0}s_{1}e^{2\pi
i\alpha}=0.\end{aligned}\right.$

Substituting the first equation into the second one gives $e^{2\pi
i\alpha}=1$. Therefore, $\alpha$ is an integer. Recalling the jump matrices on
the contour $\Sigma_{\mathbf{T}}$ (see Figure 5), it is readily seen that in
this case, there are no jumps on the infinite branches $\pi_{k}$,
$k=2,\cdots,6,10,\cdots,14$. As a result, the RH problem for $\mathbf{T}(z)$
given in Section 3.1 is reduced to the following RH problem for
$\widehat{\mathbf{T}}(z)$.

### RH problem for $\widehat{\mathbf{T}}(z)$

(1) $\widehat{\mathbf{T}}(z)$ is analytic for
$z\in\mathbb{C}\setminus(\Sigma_{\mathbf{T}}\setminus\bigcup\pi_{k})$, where
$k=2,\cdots,6,10,\cdots,14$.

(2) $\widehat{\mathbf{T}}(z)$ satisfies the jump relations

$\widehat{\mathbf{T}}_{+}(z)=\widehat{\mathbf{T}}_{-}(z)\left\\{\begin{aligned}
&\begin{pmatrix}1&-s_{0}^{-1}e^{2x^{2}g(z)}\\\
0&1\end{pmatrix},\quad&z&\in\pi_{1},\pi_{15}\\\
&\begin{pmatrix}1&s_{0}^{-1}e^{2x^{2}g(z)}\\\
0&1\end{pmatrix},\quad&z&\in\pi_{7},\pi_{9},\\\ &\begin{pmatrix}1&0\\\
-s_{0}e^{-2x^{2}g(z)}&1\end{pmatrix},\quad&z&\in\pi_{8},\pi^{-}_{16},\\\
&\begin{pmatrix}0&-s_{0}^{-1}\\\
s_{0}&0\end{pmatrix},\quad&z&\in(z_{2,-},z_{2,+}),\end{aligned}\right.$

(3)
$\widehat{\mathbf{T}}(z)=\left(\mathbf{I}+O(z^{-1})\right)z^{\alpha\sigma_{3}}$
as $z\rightarrow\infty$.

(4) $\widehat{\mathbf{T}}(z)$ has the following behavior at the origin

$\widehat{\mathbf{T}}(z)=\widehat{\mathbf{T}}_{0}(z)z^{\alpha\sigma_{3}}E_{0}S_{1}e^{-x^{2}g(z)\sigma_{3}},\qquad\arg
z\in(\pi/4,\pi/2),$ (3.68)

where $\widehat{\mathbf{T}}_{0}(z)$ is analytic in a neighborhood of the
origin. The asymptotic behaviors of $\widehat{\mathbf{T}}(z)$ in other regions
are determined by (3.68) and the jump relations satisfied by
$\widehat{\mathbf{T}}(z)$.

The following RH analysis is similar to what we have done in the case
$0<|s_{*}|<1$. Firstly, neglecting the exponentially small entries in the jump
matrices for $\widehat{\mathbf{T}}(z)$, we arrive at an approximate RH problem
with constant jump on $(z_{2,-},z_{2,+})$. The solution to this RH problem,
namely the global paramatrix, can be constructed as (3.18) by taking the
parameter $\beta=0$ therein. Next, we build two local parametrices near
$z_{2,\pm}=\pm\sqrt{8/3}$, satisfying exactly the same RH problems for
$\mathbf{P}^{(2,\pm)}(z)$, of which the solutions have been constructed in
Section 3.3. Finally, by defining $\mathbf{R}(z)$ as the ratio of the solution
of the RH problem for $\widehat{\mathbf{T}}(z)$ and the parametrices similarly
to (3.63), we show that $\mathbf{R}(z)$ satisfies the asymptotic
$\mathbf{R}(z)=\mathbf{I}+O(|x|^{-2})$ as $x\to-\infty$ and thus the
parametrices are good approximations of $\widehat{\mathbf{T}}(z)$ as
$x\to-\infty$.

##  4 RH analysis as $x\to-\infty$ with $|s_{*}|>1$

When $|s_{*}|>1$ and as $x\to-\infty$, a detailed RH analysis has been carried
out by the current authors in [28] for $\alpha\in\mathbb{R}$ but
$\alpha-\frac{1}{2}\notin\mathbb{Z}$. In this section, we will briefly review
the RH analysis therein, and discuss the exceptional case
$\frac{1}{2}-\alpha\in\mathbb{N}$.

In the case $|s_{*}|>1$, we have $1-|s_{*}|^{2}<0$. From the definition (3.22)
of ${\beta}$, it follows that

${\beta}=-\frac{1}{2\pi i}\ln(1-|s_{*}|^{2})=-\frac{1}{2\pi
i}\ln(|s_{*}|^{2}-1)-\frac{1}{2}:={\beta}_{0}-\frac{1}{2},\quad{\beta}_{0}\in
i\mathbb{R}.$ (4.1)

This, together with (3.44) and (3.54), implies that
$\mathbf{P}^{(1,\pm)}(z)\mathbf{P}^{\infty}(z)^{-1}\nrightarrow\mathbf{I}$ as
$x\rightarrow-\infty$. Therefore, the matching conditions (3.35) and (3.45)
are no longer fulfilled.

Similar problems have also occurred in deriving singular asymptotics for the
PII transcendents [7, 16]. In [7], Bothner and Its develop a certain
“dressing” technique to tackle this problem. However, we find their method
does not apply to our case. Instead, our idea is to modify the global
parametrix $\mathbf{P}^{(\infty)}(z)$, so as to satisfy the matching
condition. Similar technique was first used in [30] to derive the uniform
asymptotics of the Pollaczek polynomials and subsequently to derive the
uniform asymptotics of a system of Szegő class polynomials [29].

###  4.1 Modified global parametrix

We define

$\widetilde{\mathbf{P}}^{(\infty)}(z)=\mathbf{H}(z)\mathbf{P}^{(\infty)}(z),$
(4.2)

where $\mathbf{P}^{(\infty)}(z)$ is introduced in (3.18). The meromorphic
function $\mathbf{H}(z)$ with $\det{\mathbf{H}}(z)=1$ is given by

$\mathbf{H}(z)=\mathbf{I}+\frac{\widetilde{A}}{z-\sqrt{\frac{2}{3}}}+\frac{\widetilde{B}}{z+\sqrt{\frac{2}{3}}}.$
(4.3)

The explicit expressions of $\widetilde{A}$ and $\widetilde{B}$ are

$\widetilde{A}=\sqrt{\frac{2}{3}}\begin{pmatrix}\frac{l}{l+\sqrt{2}}&e^{\frac{5\pi
i}{6}}\frac{l\,s_{0}^{-1}f^{-2}_{\infty}}{\sqrt{2}+l}\\\\[8.5359pt]
e^{\frac{\pi
i}{6}}\frac{l\,s_{0}f^{2}_{\infty}}{\sqrt{2}-l}&\frac{l}{l-\sqrt{2}}\end{pmatrix},\quad\widetilde{B}=\sqrt{\frac{2}{3}}\begin{pmatrix}-\frac{l}{l+\sqrt{2}}&e^{\frac{5\pi
i}{6}}\frac{l\,s_{0}^{-1}f^{-2}_{\infty}}{\sqrt{2}+l}\\\\[8.5359pt]
e^{\frac{\pi
i}{6}}\frac{l\,s_{0}f^{2}_{\infty}}{\sqrt{2}-l}&-\frac{l}{l-\sqrt{2}}\end{pmatrix},$
(4.4)

where $f_{\infty}$ is given by (3.21) and $l=l(x)$ is defined as

$l=-\frac{i\sqrt{6}e^{i\phi}}{2+e^{i\phi}},\quad\phi=-\frac{\sqrt{3}}{3}x^{2}+i{\beta}_{0}\ln\left(2\sqrt{3}x^{2}\right)+\frac{2\pi\alpha}{3}+\arg\Gamma\left({\beta}_{0}+\frac{1}{2}\right)+\arg
s_{*},$ (4.5)

with ${\beta}_{0}$ and $s_{*}$ given in (4.1) and (2.20), respectively.

We shall assume in (4.4) that $x$ does not take the zeros of the functions
$\sqrt{2}\pm l(x)$. The set of zeros consists of two sequences $\\{x_{n}\\}$
and $\\{y_{n}\\}$ for $n\in\mathbb{N}$, defined respectively by the equations

$\frac{\sqrt{3}}{3}x_{n}^{2}-i{\beta}_{0}\ln\left(2\sqrt{3}x_{n}^{2}\right)-\frac{2\pi\alpha}{3}-\arg\Gamma\left({\beta}_{0}-\frac{1}{2}\right)-\arg
s_{*}-\frac{2\pi}{3}-2n\pi=0,$ (4.6)

and

$\frac{\sqrt{3}}{3}y_{n}^{2}-i{\beta}_{0}\ln\left(2\sqrt{3}y_{n}^{2}\right)-\frac{2\pi\alpha}{3}-\arg\Gamma\left({\beta}_{0}+\frac{1}{2}\right)-\arg
s_{*}+\frac{2\pi}{3}-2n\pi=0.$ (4.7)

For detailed derivations of $\widetilde{A}$ and $\widetilde{B}$, we refer to
[28, Section 3.3].

###  4.2 Modified local parametrices at $z_{1,\pm}$, $z_{2,\pm}$

We first consider the local parametrix at $z_{1,+}=\sqrt{\frac{2}{3}}$.
Instead of (3.38), we define

$\displaystyle\widetilde{\mathbf{P}}^{(1,+)}(z)=\widetilde{\mathbf{E}}^{(1,+)}(z)\Phi^{\mathrm{(PC)}}\left(|x|\varphi_{2}(z)\right)\left(\frac{s_{*}}{h_{0}}\right)^{\frac{\sigma_{3}}{2}}\left\\{\begin{aligned}
&s_{0}^{-\frac{\sigma_{3}}{2}}(-\sigma_{1})e^{-x^{2}g(z)\sigma_{3}},&\mathop{\rm
Im}\nolimits z>0,\\\ &s_{0}^{\frac{\sigma_{3}}{2}}\sigma_{3}e^{2\pi
i(\alpha+{\beta})\sigma_{3}}e^{-x^{2}g(z)\sigma_{3}},&\mathop{\rm Im}\nolimits
z<0,\end{aligned}\right.$

where $\Phi^{\mathrm{(PC)}}$ is the parabolic cylinder model parametrix,
$\varphi_{2}(z)$ is the conformal mapping (3.36), ${\beta}$ is given by (3.22)
and $\widetilde{\mathbf{E}}^{(1,+)}(z)$ is defined as

$\widetilde{\mathbf{E}}^{(1,+)}(z)=\mathbf{H}(z)\mathbf{W}^{(+)}(z)\left(\frac{s_{*}}{h_{0}}\right)^{-\frac{\sigma_{3}}{2}}|x|^{{\beta}\sigma_{3}}e^{i\frac{\sqrt{3}}{6}x^{2}\sigma_{3}}\begin{pmatrix}1&0\\\
-\frac{1}{|x|\varphi_{2}(z)}&1\end{pmatrix}2^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}|x|\varphi_{2}(z)&1\\\
1&0\end{pmatrix}$

with $\mathbf{H}(z)$ and $\mathbf{W}^{(+)}(z)$ given by (4.3) and (3.40),
respectively.

It is straightforward to check that $\widetilde{\mathbf{P}}^{(1,+)}(z)$ solves
the following RH problem.

### RH problem for $\widetilde{\mathbf{P}}^{(1,+)}(z)$

(1) $\widetilde{\mathbf{P}}^{(1,+)}(z)$ is analytic for all $z\in
U(z_{1,+},\delta)\setminus\Sigma_{\mathbf{T}}$.

(2) $\widetilde{\mathbf{P}}^{(1,+)}(z)$ fulfills the same jump relations as
$\mathbf{T}(z)$ on $U(z_{1,+},\delta)\cap\Sigma_{\mathbf{T}}$.

(3) On the circle $\partial U(z_{1,+},\delta)$, we have the matching condition

$\widetilde{\mathbf{P}}^{(1,+)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\widetilde{\mathbf{P}}^{(\infty)}(z),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (4.8)

Similarly, the local parametrix at $z_{1,-}=-\sqrt{\frac{2}{3}}$ is
constructed by

$\displaystyle\widetilde{\mathbf{P}}^{(1,-)}(z)=\widetilde{\mathbf{E}}^{(1,-)}(z)\Phi^{\mathrm{(PC)}}\left(|x|\varphi_{3}(z)\right)\left(\frac{s_{*}}{h_{0}}\right)^{\frac{\sigma_{3}}{2}}\left\\{\begin{aligned}
&e^{2\pi i{\beta}\sigma_{3}}\left(s_{0}e^{2\pi
i\alpha}\right)^{\frac{\sigma_{3}}{2}}e^{-x^{2}g(z)\sigma_{3}},&\mathop{\rm
Im}\nolimits z>0,\\\ &\left(s_{0}e^{2\pi
i\alpha}\right)^{-\frac{\sigma_{3}}{2}}\sigma_{3}\sigma_{1}e^{-x^{2}g(z)\sigma_{3}},&\mathop{\rm
Im}\nolimits z<0,\end{aligned}\right.$

where $\varphi_{3}(z)$ is the conformal mapping (3.46) and
$\widetilde{\mathbf{E}}^{(1,-)}(z)$ is given by

$\widetilde{\mathbf{E}}^{(1,-)}(z)=\mathbf{H}(z)\mathbf{W}^{(-)}(z)\left(\frac{s_{*}}{h_{0}}\right)^{-\frac{\sigma_{3}}{2}}|x|^{{\beta}\sigma_{3}}e^{\frac{i\sqrt{3}x^{2}}{6}\sigma_{3}}\begin{pmatrix}1&0\\\
-\frac{1}{|x|\varphi_{3}(z)}&1\end{pmatrix}2^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}|x|\varphi_{3}(z)&1\\\
1&0\end{pmatrix}$

with $\mathbf{H}(z)$ and $\mathbf{W}^{(-)}(z)$ given by (4.3) and (3.50),
respectively.

Then, $\widetilde{\mathbf{P}}^{(1,-)}(z)$ satisfies the following RH problem.

### RH problem for $\widetilde{\mathbf{P}}^{(1,-)}(z)$

(1) $\widetilde{\mathbf{P}}^{(1,-)}(z)$ is analytic for all $z\in
U(z_{1,-},\delta)\setminus\Sigma_{\mathbf{T}}$.

(2) $\widetilde{\mathbf{P}}^{(1,-)}(z)$ satisfies the same jump relations as
$\mathbf{T}(z)$ on $U(z_{1,-},\delta)\cap\Sigma_{\mathbf{T}}$.

(3) On the boundary $\partial U(z_{1,-},\delta)$, we have

$\widetilde{\mathbf{P}}^{(1,-)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\widetilde{\mathbf{P}}^{(\infty)}(z),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (4.9)

Since the function $\mathbf{H}(z)$ defined by (4.3) is analytic at
$z_{2,\pm}=\pm\sqrt{8/3}$, in order to construct the local parametrices
$\widetilde{\mathbf{P}}^{(2,\pm)}(z)$ at these points, we only need to replace
$\mathbf{P}^{(\infty)}(z)$ by $\widetilde{\mathbf{P}}^{(\infty)}(z)$ in the
constructions of local parametrices at these points in the case $0<|s_{*}|<1$;
see (3.29) and (3.33). Moreover, we have the following matching conditions on
the circular boundaries:

$\displaystyle\widetilde{\mathbf{P}}^{(2,+)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\widetilde{\mathbf{P}}^{(\infty)}(z),\quad\mathrm{as}\quad
x\to-\infty$ (4.10)

and

$\widetilde{\mathbf{P}}^{(2,-)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\widetilde{\mathbf{P}}^{(\infty)}(z),\quad\mathrm{as}\quad
x\to-\infty.$ (4.11)

###  4.3 Modified local parametrix near the origin

Accordingly, the parametrix $\widetilde{\mathbf{P}}^{(0)}(z)$ is defined as

$\widetilde{\mathbf{P}}^{(0)}(z)=\widetilde{\mathbf{E}}^{(0)}(z)\Phi^{(\mathrm{Bes})}\left(x^{2}\varphi_{4}(z)\right)\mathbf{K}(z)\left[s_{1}\left(e^{-2\pi
i\alpha}+s_{*}\right)\right]^{-\frac{\sigma_{3}}{2}}e^{-x^{2}g(z)\sigma_{3}},$
(4.12)

where $\varphi_{4}(z)$ is the conformal mapping (3.56), $\mathbf{K}(z)$ is
defined by (3.59) and $\widetilde{\mathbf{E}}^{(0)}(z)$ is given by

$\widetilde{\mathbf{E}}^{(0)}(z)=\widetilde{\mathbf{P}}^{(\infty)}(z)\left[s_{1}(e^{-2\pi
i\alpha}+s_{*})\right]^{\frac{\sigma_{3}}{2}}\mathbf{Q}(z)e^{-\frac{1}{4}\pi
i\sigma_{3}}\frac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\ i&1\end{pmatrix}$ (4.13)

with $\mathbf{Q}(z)$ defined by (3.61).

It is direct to verify that $\widetilde{\mathbf{P}}^{(0)}(z)$ satisfies the
same jump conditions as $\mathbf{T}(z)$ and $\widetilde{\mathbf{P}}^{(0)}(z)$
possesses the same asymptotic behavior near the origin as $\mathbf{T}(z)$ for
the case $\alpha-\frac{1}{2}\notin\mathbb{Z}$. Moreover,
$\widetilde{\mathbf{P}}^{(0)}(z)$ fulfills the following matching condition

$\widetilde{\mathbf{P}}^{(0)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\widetilde{\mathbf{P}}^{(\infty)}(z)\quad\mathrm{as}\quad
x\to-\infty.$ (4.14)

While, in the case $\frac{1}{2}-\alpha\in\mathbb{N}$ and $|s_{*}|>1$, we need
to check that $\widetilde{\mathbf{P}}^{(0)}(z)$ shares the same asymptotic
behaviors as $\mathbf{T}(z)$ near the origin. From (2.8) and (3.14), it
follows that as $z\to 0$ with $\arg z\in(0,\frac{\pi}{2})$

$\mathbf{T}(z)=\widetilde{\mathbf{T}}_{0}(z)z^{\alpha\sigma_{3}}S_{1}e^{-x^{2}g(z)\sigma_{3}}\begin{pmatrix}1&0\\\
-\frac{\overline{s}_{*}e^{-2\pi i\alpha}e^{-2x^{2}g(z)}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix},$ (4.15)

where $\widetilde{\mathbf{T}}_{0}(z)$ is analytic in a neighborhood of $z=0$.
Using the following factorization

$\begin{pmatrix}1&s_{1}\\\ 0&1\end{pmatrix}=\begin{pmatrix}1&0\\\
s_{1}^{-1}&1\end{pmatrix}\begin{pmatrix}0&s_{1}\\\
-s_{1}^{-1}&0\end{pmatrix}\begin{pmatrix}1&0\\\ s_{1}^{-1}&1\end{pmatrix},$
(4.16)

we can rewrite (4.15) as

$\mathbf{T}(z)=\widetilde{\mathbf{T}}_{1}(z)z^{-\alpha\sigma_{3}}\begin{pmatrix}1&0\\\
1&1\end{pmatrix}\left[s_{1}(e^{-2\pi
i\alpha}+s_{*})\right]^{-\frac{\sigma_{3}}{2}}e^{-x^{2}g(z)\sigma_{3}},$
(4.17)

where $\widetilde{\mathbf{T}}_{1}(z)$ is analytic in a neighborhood of $z=0$.
Comparing (4.17) with (4.12) and (A.11), it is readily seen that
$\widetilde{\mathbf{P}}^{(0)}(z)$ satisfies the asymptotic behavior (4.15) as
$z\to 0$ for $\frac{1}{2}-\alpha\in\mathbb{N}$.

###  4.4 Final transformation

The final transformation is defined as

$\widetilde{\mathbf{R}}(z)=\left\\{\begin{aligned}
&\mathbf{T}(z)\left[\widetilde{\mathbf{P}}^{(1,\pm)}(z)\right]^{-1},\quad&z&\in
U(z_{1,\pm},\delta)\setminus\Sigma_{\mathbf{T}},\\\
&\mathbf{T}(z)\left[\widetilde{\mathbf{P}}^{(2,\pm)}(z)\right]^{-1},\quad&z&\in
U(z_{2,\pm},\delta)\setminus\Sigma_{\mathbf{T}},\\\
&\mathbf{T}(z)\left[\widetilde{\mathbf{P}}^{(0)}(z)\right]^{-1},\quad&z&\in
U(0,\delta)\setminus\Sigma_{\mathbf{T}},\\\
&\mathbf{T}(z)\left[\widetilde{\mathbf{P}}^{(\infty)}(z)\right]^{-1},\quad&\mathrm{e}&\mathrm{lsewhere}.\\\
\end{aligned}\right.$ (4.18)

As a consequence, $\widetilde{\mathbf{R}}(z)$ solves the following RH problem.

### RH problem for $\widetilde{\mathbf{R}}(z)$

(1) $\widetilde{\mathbf{R}}(z)$ is analytic for
$z\in\mathbb{C}\setminus\Sigma_{\mathbf{R}}$, where $\Sigma_{\mathbf{R}}$ is
illustrated in Figure 6.

(2) On $\Sigma_{R}$, we have
$\widetilde{\mathbf{R}}_{+}(z)=\widetilde{\mathbf{R}}_{-}(z)J_{\widetilde{\mathbf{R}}}(z)$,
where

$J_{\widetilde{\mathbf{R}}}(z)=\left\\{\begin{aligned}
&\widetilde{\mathbf{P}}^{(1,\pm)}(z)\widetilde{\mathbf{P}}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(z_{1,\pm},\delta),\\\
&\widetilde{\mathbf{P}}^{(2,\pm)}(z)\widetilde{\mathbf{P}}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(z_{2,\pm},\delta),\\\
&\widetilde{\mathbf{P}}^{(0)}(z)\widetilde{\mathbf{P}}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(0,\delta),\\\
&\widetilde{\mathbf{P}}^{(\infty)}_{-}(z)J_{\mathbf{T}}(z)\widetilde{\mathbf{P}}^{(\infty)}_{+}(z)^{-1},\quad&\mathrm{e}&\mathrm{lsewhere}.\\\
\end{aligned}\right.$ (4.19)

(3) As $z\rightarrow\infty$, we have

$\widetilde{\mathbf{R}}(z)=\mathbf{I}+\frac{\widetilde{\mathbf{R}}_{1}}{z}+\frac{\widetilde{\mathbf{R}}_{2}}{z^{2}}+O(z^{-3}).$
(4.20)

Using the matching conditions (4.8), (4.9), (4.10), (4.11) and (4.14), we have
the following estimates for the jump matrices (4.19) as $x\rightarrow-\infty$

$J_{\widetilde{\mathbf{R}}}(z)=\left\\{\begin{aligned}
&\mathbf{I}+O(|x|^{-2}),&z&\in\partial U(0,\delta)\cup\partial
U(z_{1,\pm},\delta)\cup\partial U(z_{2,\pm},\delta),\\\
&\mathbf{I}+O(e^{-c_{2}|x|^{2}}),&z&\in\pi_{k},\
k=1,\cdots,16.\end{aligned}\right.$ (4.21)

where $c_{2}$ is some positive constant. Consequently, we have that for any
$z\in\mathbb{C}\setminus\Sigma_{\mathbf{R}}$

$\widetilde{\mathbf{R}}(z)=\mathbf{I}+O(|x|^{-2}),\qquad\mathrm{as}\qquad
x\rightarrow-\infty.$ (4.22)

##  5 RH analysis as $x\to-\infty$ with $|s_{*}|=1$ but $s_{*}\neq 1$

In the case when $|s_{*}|=1$ with $s_{*}\neq 1$, it follows from (2.20) and
(3.13) that

$s_{*}=-e^{-2\pi i\alpha}.$ (5.1)

We need another $g$-function

$\widehat{g}(z)=\frac{1}{8}(z^{2}-2)^{2}=\frac{1}{8}z^{4}-\frac{1}{2}z^{2}+\frac{1}{2}.$
(5.2)

It is direct to see that $\widehat{g}(z)$ has three saddle points $z_{0}=0$,
$z_{\pm}=\pm\sqrt{2}$. As shown in Figure 7, the topology of the anti-Stokes
curves of $\widehat{g}(z)$ is quite different from $g(z)$ depicted in Figure
2.

As before, we begin with the re-scaling transformation (3.1). To proceed, we
introduce

$\widehat{\mathbf{U}}(z)=e^{\frac{x^{2}}{2}\sigma_{3}}|x|^{\alpha\sigma_{3}}\Phi(z)|x|^{-\alpha\sigma_{3}}e^{-x^{2}\widehat{g}(z)\sigma_{3}}.$
(5.3)

Then, $\widehat{\mathbf{U}}(z)$ satisfies a RH problem with jumps along
$\gamma_{k}$, $k=1,3,4,5,7,8$ as shown in Figure 1. Subsequently, we deform
the jump contours to the anti-Stokes curves of $\widehat{g}(z)$ illustrated in
Figure 7. We rewrite the RH problem as the following problem formulated on the
anti-Stokes curves of $\widehat{g}(z)$.

Figure 7: The jump contour $\Sigma_{\widehat{\mathbf{U}}}$

### RH problem for $\widehat{\mathbf{U}}(z)$

(1) $\widehat{\mathbf{U}}(z)$ is analytic for
$z\in\mathbb{C}\setminus\Sigma_{\widehat{\mathbf{U}}}$, with
$\Sigma_{\widehat{\mathbf{U}}}$ is shown in Figure 7. Note that we have
deformed $\gamma_{k}$ to the anti-Stokes curves of $\widehat{g}(z)$, namely
$\widehat{\gamma}_{k}$, $k=1,3,5,7$.

(2) On the contour $\Sigma_{\widehat{\mathbf{U}}}$, we have
$\widehat{\mathbf{U}}_{+}(z)=\widehat{\mathbf{U}}_{-}(z)J_{\widehat{\mathbf{U}}}(z)$,
where

$J_{\widehat{\mathbf{U}}}(z)=\left\\{\begin{aligned}
&\begin{pmatrix}1&s_{k}|x|^{2\alpha}e^{2x^{2}\widehat{g}(z)}\\\
0&1\end{pmatrix},\quad&z&\in\widehat{\gamma}_{k},\ k=1,3,5,7,\\\
&\begin{pmatrix}-1&0\\\ s_{0}e^{2\pi
i\alpha}|x|^{-2\alpha}e^{-2x^{2}\widehat{g}(z)}&-1\end{pmatrix},\quad&z&\in\widehat{\gamma}_{2},\\\
&\begin{pmatrix}-e^{-2\pi i\alpha}&0\\\ s_{0}e^{2\pi
i\alpha}|x|^{-2\alpha}e^{-2x^{2}\widehat{g}(z)}&-e^{2\pi
i\alpha}\end{pmatrix},\quad&z&\in\widehat{\gamma}_{6},\\\
&\begin{pmatrix}1&0\\\ -s_{0}e^{2\pi
i\alpha}|x|^{-2\alpha}e^{-2x^{2}\widehat{g}(z)}&1\end{pmatrix},\quad&z&\in\gamma_{4},\\\
&\begin{pmatrix}e^{-2\pi i\alpha}&0\\\ s_{0}e^{2\pi
i\alpha}|x|^{-2\alpha}e^{-2x^{2}\widehat{g}(z)}&e^{2\pi
i\alpha}\end{pmatrix},\quad&z&\in\gamma_{8}.\end{aligned}\right.$

(3)
$\widehat{\mathbf{U}}(z)=\left(\mathbf{I}+O(z^{-1})\right)z^{\alpha\sigma_{3}}$
as $z\rightarrow\infty$, where $\arg z\in(0,2\pi)$.

(4) $\widehat{\mathbf{U}}(z)$ possesses the following asymptotic behavior near
the origin

$\widehat{\mathbf{U}}(z)=\widehat{\mathbf{U}}_{0}(z)z^{\alpha\sigma_{3}}E_{0}S_{1}|x|^{-\alpha\sigma_{3}}e^{-x^{2}\widehat{g}(z)\sigma_{3}},\qquad\arg
z\in(\pi/4,\pi/2),$ (5.4)

where $\widehat{\mathbf{U}}_{0}(z)$ is analytic in the neighborhood of
$z_{0}=0$. The asymptotic behaviors of $\widehat{\mathbf{U}}(z)$ in other
regions are determined by (5.4) and the jump relations satisfied by
$\widehat{\mathbf{U}}(z)$.

From the properties of $g$-function $\widehat{g}(z)$, it is readily seen that
the jump matrices on $\gamma_{4}$ and the anti-Stokes curves
$\widehat{\gamma}_{k}$, $k=1,3,5,7$ approach to identity matrix exponentially
fast as $x\rightarrow-\infty$. Thus, the task is to construct a global
parametrix satisfying the remaining jump along $(z_{-},+\infty)$ and three
local parametrices in the neighborhoods of the saddle points $z_{0}$,
$z_{\pm}$.

###  5.1 Global parametrix

We need to solve the following RH problem for a $2\times 2$ matrix-valued
function $\widehat{\mathbf{P}}^{(\infty)}(z)$.

### RH problem for $\widehat{\mathbf{P}}^{(\infty)}(z)$

(1) $\widehat{\mathbf{P}}^{(\infty)}(z)$ is analytic for
$z\in\mathbb{C}\setminus(z_{-},+\infty)$.

(2) $\widehat{\mathbf{P}}^{(\infty)}(z)$ satisfies the following jump
relations

$\widehat{\mathbf{P}}^{(\infty)}_{+}(z)=\widehat{\mathbf{P}}^{(\infty)}_{-}(z)\left\\{\begin{aligned}
&\begin{pmatrix}-e^{-2\pi i\alpha}&0\\\ 0&-e^{2\pi
i\alpha}\end{pmatrix},\quad&z&\in(z_{-},0),\\\
&-\mathbf{I},\quad&z&\in(0,z_{+}),\\\ &\begin{pmatrix}e^{-2\pi i\alpha}&0\\\
0&e^{2\pi i\alpha}\end{pmatrix},\quad&z&\in(z_{+},+\infty)\\\
\end{aligned}\right.$ (5.5)

(3) $\widehat{\mathbf{P}}^{(\infty)}(z)$ has at most singularities of order
less than $|\alpha|+\frac{3}{2}$ at $z_{\pm}=\pm\sqrt{2}$.

(4) As $z\rightarrow\infty$, we have
$\widehat{\mathbf{P}}^{(\infty)}(z)=\left(\mathbf{I}+O(z^{-1})\right)z^{\alpha\sigma_{3}}$.

A solution to the above RH problem is given by

$\widehat{\mathbf{P}}^{(\infty)}(z)=\widehat{\mathbf{H}}(z)z^{-\alpha\sigma_{3}}\left(z+\sqrt{2}\right)^{(\alpha+\frac{1}{2})\sigma_{3}}\left(z-\sqrt{2}\right)^{(\alpha-\frac{1}{2})\sigma_{3}},$
(5.6)

where the branches of the powers are chosen such that $\arg z\in(0,2\pi)$ and
$\arg(z\pm\sqrt{2})\in(0,2\pi)$. The function $\widehat{\mathbf{H}}(z)$,
similar to (4.3), takes the rational form

$\widehat{\mathbf{H}}(z)=\mathbf{I}+\frac{\widehat{A}}{z+\sqrt{2}}+\frac{\widehat{B}}{z-\sqrt{2}},\quad\det\widehat{\mathbf{H}}(z)=1,$
(5.7)

where constant matrices $\widehat{A}$ and $\widehat{B}$ are to be determined.
We point out that the factor $\widehat{\mathbf{H}}(z)$ in (5.6) is introduced
to meet the matching conditions (5.8) and (5.15) below.

###  5.2 Local parametrices at $z_{\pm}=\pm\sqrt{2}$

We seek two functions $\widehat{\mathbf{P}}^{(r)}(z)$ and
$\widehat{\mathbf{P}}^{(l)}(z)$, satisfying the same jumps as
$\widehat{\mathbf{\mathbf{U}}}(z)$ respectively in the neighborhoods
$U(z_{\pm},\delta)=\\{z\in\mathbb{C}\mid|z-z_{\pm}|<\delta\\}$ of the saddle
points $z_{\pm}$, and matching with $\widehat{\mathbf{P}}^{(\infty)}(z)$ on
the boundaries $\partial
U(z_{\pm},\delta)=\\{z\in\mathbb{C}\mid|z-z_{\pm}|=\delta\\}$.

### RH problem for $\widehat{\mathbf{P}}^{(r)}(z)$

(1) $\widehat{\mathbf{P}}^{(r)}(z)$ is analytic for $z\in
U(z_{+},\delta)\setminus\Sigma_{\widehat{\mathbf{U}}}$.

(2) $\widehat{\mathbf{P}}^{(r)}(z)$ shares the same jumps as
$\widehat{\mathbf{U}}(z)$ on $\Sigma_{\widehat{\mathbf{U}}}\cap
U(z_{+},\delta)$.

(3) On the boundary $\partial U(z_{+},\delta)$, we have

$\widehat{\mathbf{P}}^{(r)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\widehat{\mathbf{P}}^{(\infty)}(z)\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (5.8)

Firstly, we introduce a conformal mapping

$\varphi_{5}(z)=2\widehat{g}(z)^{\frac{1}{2}}=2(z-z_{\pm})(1+o(1)),\quad
z\rightarrow z_{\pm}.$ (5.9)

We will make use of the parabolic cylinder function to construct the solution.
Let $\Phi^{(\mathrm{PC})}$ be the parabolic cylinder parametrix with parameter
$\beta=\frac{1}{2}-\alpha$ as given in Appendix A.2. Then, the parametrix
$\widehat{\mathbf{P}}^{(r)}(z)$ is defined as

$\widehat{\mathbf{P}}^{(r)}(z)=\mathbf{E}^{(r)}(z)\Phi^{(\mathrm{PC})}(|x|\varphi_{5}(z))\left(\frac{h_{1}^{(r)}}{s_{1}}\right)^{\frac{\sigma_{3}}{2}}\mathbf{D}^{(r)}(z)\sigma_{3}|x|^{-\alpha\sigma_{3}}e^{-x^{2}\widehat{g}(z)\sigma_{3}},$
(5.10)

where $h_{1}^{(r)}=\frac{\sqrt{2\pi}e^{-(\alpha-\frac{1}{2})\pi
i}}{\Gamma(\alpha-\frac{1}{2})}$ is the Stokes multiplier defined in (A.4),

$\mathbf{D}^{(r)}(z)=\left\\{\begin{aligned}
&e^{2\pi(\alpha+\frac{1}{4})\sigma_{3}},&\arg z&\in(-\frac{\pi}{4},0),\\\
&e^{\frac{1}{2}\pi i\sigma_{3}},&\arg z&\in(0,\pi),\\\ &e^{-\frac{1}{2}\pi
i\sigma_{3}},&\arg z&\in(\pi,\frac{7\pi}{4}).\end{aligned}\right.$ (5.11)

and $\mathbf{E}^{(r)}(z)$ is given by

$\displaystyle\mathbf{E}^{(r)}(z)=\widehat{\mathbf{P}}^{(\infty)}(z)\sigma_{3}$
$\displaystyle\left(\frac{s_{1}}{h_{1}^{(r)}}\right)^{\frac{\sigma_{3}}{2}}|x|^{\frac{\sigma_{3}}{2}}e^{\mp\frac{1}{2}\pi
i\sigma_{3}}$
$\displaystyle\times\varphi_{5}(z)^{(\frac{1}{2}-\alpha)\sigma_{3}}\begin{pmatrix}1&\frac{\alpha-\frac{1}{2}}{|x|\varphi_{5}(z)}\\\
0&1\end{pmatrix}2^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}|x|\varphi_{5}(z)&1\\\
1&0\end{pmatrix},~{}~{}\pm\mathop{\rm Im}\nolimits z>0$ (5.12)

with $\arg\varphi_{5}(z)\in(0,2\pi)$.

It is readily seen from (5.5), (5) and (A.5) that $\mathbf{E}^{(r)}(z)$ is
analytic in the deleted neighborhood $U(z_{+},\delta)\setminus\\{z_{+}\\}$ and
the matching condition (5.8) is satisfied. To guarantee that
$\mathbf{E}^{(r)}(z)$ is also analytic at the isolated point $z_{+}=\sqrt{2}$,
we find, by computing the Laurent expansion at $z_{+}=\sqrt{2}$ using (5.6),
(5.7), (5.9), (5.11) and (5), that the constant matrices $\widehat{A}$ and
$\widehat{B}$ in (5.7) should satisfy the following algebraic equation

$\left(I+\frac{\widehat{A}}{2\sqrt{2}}\right)\begin{pmatrix}0&c_{r}\\\
0&0\end{pmatrix}=-\widehat{B},$ (5.13)

where the constant $c_{r}$ is given by

$c_{r}=\frac{2s_{1}\Gamma(\alpha+\frac{1}{2})e^{i\pi(\alpha-\frac{1}{2})}}{\sqrt{\pi}}.$
(5.14)

### RH problem for $\widehat{\mathbf{P}}^{(l)}(z)$

(1) $\widehat{\mathbf{P}}^{(l)}(z)$ is analytic for $z\in
U(z_{-},\delta)\setminus\Sigma_{\widehat{\mathbf{U}}}$.

(2) $\widehat{\mathbf{P}}^{(l)}(z)$ satisfies the same jumps as
$\widehat{\mathbf{U}}(z)$ on $\Sigma_{\widehat{\mathbf{U}}}\cap
U(z_{-},\delta)$.

(3) On the circle $\partial U(z_{-},\delta)$, it holds that

$\widehat{\mathbf{P}}^{(l)}(z)=\left(\mathbf{I}+O(|x|^{-2})\right)\widehat{\mathbf{P}}^{(\infty)}(z)\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (5.15)

Similarly, the solution $\widehat{\mathbf{P}}^{(l)}(z)$ can also be built in
terms of the parabolic cylinder function. We choose the parameter
$\beta=-\frac{1}{2}-\alpha$ in Appendix A.2. More precisely, we define

$\widehat{\mathbf{P}}^{(l)}(z)=\mathbf{E}^{(l)}(z)\Phi^{(\mathrm{PC})}(|x|\varphi_{5}(z))\left(\frac{h_{1}^{(l)}}{s_{1}}\right)^{\frac{\sigma_{3}}{2}}\mathbf{D}^{(l)}(z)\sigma_{3}|x|^{-\alpha\sigma_{3}}e^{-x^{2}\widehat{g}(z)\sigma_{3}},$
(5.16)

where $h_{1}^{(l)}=\sqrt{2\pi}e^{-(\alpha+\frac{1}{2})\pi
i}/\Gamma(\alpha+\frac{1}{2})$ is the Stokes multiplier given in (A.4),

$\mathbf{D}^{(l)}(z)=\left\\{\begin{aligned}
&e^{2\pi(\alpha+\frac{1}{2})\sigma_{3}},&\arg z&\in(-\frac{\pi}{4},0),\\\
&\mathbf{I},&\arg z&\in(0,\frac{7\pi}{4}).\end{aligned}\right.$ (5.17)

and $\mathbf{E}^{(l)}(z)$ is given by

$\displaystyle\mathbf{E}^{(l)}(z)=\widehat{\mathbf{P}}^{(\infty)}(z)\sigma_{3}$
$\displaystyle\left(\frac{s_{1}}{h_{1}^{(l)}}\right)^{\frac{\sigma_{3}}{2}}|x|^{-\frac{\sigma_{3}}{2}}\varphi_{5}(z)^{-(\alpha+\frac{1}{2})\sigma_{3}}\begin{pmatrix}1&0\\\
-\frac{1}{|x|\varphi_{5}(z)}&1\end{pmatrix}2^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}|x|\varphi_{5}(z)&1\\\
1&0\end{pmatrix}$ (5.18)

with $\arg\varphi_{5}(z)\in(0,2\pi)$.

Using the jumps (5.5), it is straightforward to check that
$\mathbf{E}^{(l)}(z)$ is analytic in the deleted neighborhood
$U(z_{-},\delta)\setminus\\{z_{-}\\}$ and the matching condition (5.15) is
also satisfied. To ensure that $\mathbf{E}^{(l)}(z)$ is also analytic at the
isolated point $z_{-}=-\sqrt{2}$, by calculating the Laurent expansion at
$z_{-}$ using (5.6), (5.7), (5.9), (5.18) and (5.17), we obtain another
algebraic equation

$\left(I-\frac{\widehat{B}}{2\sqrt{2}}\right)\begin{pmatrix}0&0\\\
c_{l}&0\end{pmatrix}=-\widehat{A},$ (5.19)

where the constant $c_{l}$ is given by

$c_{l}=-\frac{4\sqrt{\pi}}{s_{1}\Gamma(\alpha+\frac{1}{2})e^{i\pi(\alpha+\frac{1}{2})}}.$
(5.20)

Combining (5.13), (5.19) with the fact that $c_{r}c_{l}=8$, we can now derive
explicit expressions of $\widehat{A}$ and $\widehat{B}$ as follows:

$\widehat{A}=\begin{pmatrix}-\sqrt{2}&0\\\
-\frac{1}{2}c_{l}&0\end{pmatrix},\quad\widehat{B}=\begin{pmatrix}0&-\frac{1}{2}c_{r}\\\
0&\sqrt{2}\end{pmatrix},$ (5.21)

where $c_{r}$ and $c_{l}$ are given by (5.14) and (5.20), respectively.

After determining $\widehat{A}$ and $\widehat{B}$ as given in (5.21), it is
straightforward to verify that the determinant condition
$\det\widehat{\mathbf{H}}(z)=1$ is also satisfied.

###  5.3 Local parametrix near the origin

Now, we look for a function $\widehat{\mathbf{P}}^{(0)}(z)$ satisfying the
same jumps as $\widehat{\mathbf{U}}(z)$ in the neighborhood
$U(z_{0},\delta)=\\{z\in\mathbb{C}\mid|z-z_{0}|<\delta\\}$ of $z_{0}=0$ with
some constant $0<\delta<\sqrt{2}$, and matching with
$\widehat{\mathbf{P}}^{(\infty)}(z)$ on the boundary $\partial
U(z_{0},\delta)=\\{z\in\mathbb{C}\mid|z-z_{0}|=\delta\\}$.

### RH problem for $\widehat{\mathbf{P}}^{(0)}(z)$

(1) $\widehat{\mathbf{P}}^{(0)}(z)$ is analytic for $z\in
U(z_{0},\delta)\setminus[z_{-},z_{+}]$.

(2) $\widehat{\mathbf{P}}^{(0)}(z)$ satisfies the following jump relations

$\widehat{\mathbf{P}}^{(0)}_{+}(z)=\widehat{\mathbf{P}}^{(0)}_{-}(z)\left\\{\begin{aligned}
&\begin{pmatrix}-e^{-2\pi i\alpha}&0\\\ s_{0}e^{2\pi
i\alpha}|x|^{-2\alpha}e^{-2x^{2}\widehat{g}(z)}&-e^{2\pi
i\alpha}\end{pmatrix},\quad&z&\in(-\delta,0),\\\ &\begin{pmatrix}-1&0\\\
s_{0}e^{2\pi
i\alpha}|x|^{-2\alpha}e^{-2x^{2}\widehat{g}(z)}&-1\end{pmatrix},\quad&z&\in(0,\delta).\end{aligned}\right.$

(3) On the boundary $\partial U(z_{0},\delta)$, we have

$\widehat{\mathbf{P}}^{(0)}(z)=(\mathbf{I}+O(|x|^{-2\alpha}e^{-x^{2}}))\widehat{\mathbf{P}}^{(\infty)}(z)\quad\mathrm{as}\quad
x\to-\infty.$ (5.22)

(4) $\widehat{\mathbf{P}}^{(0)}(z)$ has the same behavior as
$\widehat{\mathbf{U}}(z)$ near the origin; see (5.4).

A solution to the above RH problem can be constructed explicitly as follows:

$\widehat{\mathbf{P}}^{(0)}(z)=\widehat{\mathbf{H}}(z)\begin{pmatrix}1&0\\\
r(z)&1\end{pmatrix}z^{-\alpha\sigma_{3}}\left(z+\sqrt{2}\right)^{(\alpha+\frac{1}{2})\sigma_{3}}\left(z-\sqrt{2}\right)^{(\alpha-\frac{1}{2})\sigma_{3}},$
(5.23)

where $\widehat{\mathbf{H}}(z)$ is given by (5.7) and (5.21). The function
$r(z)$ is defined by

$r(z)=\left\\{\begin{aligned}
&-\frac{s_{0}|x|^{-2\alpha}e^{-2x^{2}\widehat{g}(z)}z^{2\alpha}}{(1+e^{-2\pi
i\alpha})(z+\sqrt{2})^{2\alpha+1}(z-\sqrt{2})^{2\alpha-1}},&&|z|<\delta,~{}\mathop{\rm
Im}\nolimits z>0,\\\
&\frac{s_{0}|x|^{-2\alpha}e^{-2x^{2}\widehat{g}(z)}e^{2\pi
i\alpha}z^{2\alpha}}{(1+e^{-2\pi
i\alpha})(z+\sqrt{2})^{2\alpha+1}(z-\sqrt{2})^{2\alpha-1}},&&|z|<\delta,~{}\mathop{\rm
Im}\nolimits z<0,\end{aligned}\right.$ (5.24)

where the branches are chosen such that $\arg z\in(0,2\pi)$ and
$\arg(z\pm\sqrt{2})\in(0,2\pi)$.

###  5.4 Final transformation

The final transformation is now defined as

$\widehat{\mathbf{R}}(z)=\left\\{\begin{aligned}
&\widehat{\mathbf{U}}(z)\left[\widehat{\mathbf{P}}^{(l)}(z)\right]^{-1},\quad&z&\in
U(z_{-},\delta)\setminus\Sigma_{\widehat{\mathbf{U}}},\\\
&\widehat{\mathbf{U}}(z)\left[\widehat{\mathbf{P}}^{(r)}(z)\right]^{-1},\quad&z&\in
U(z_{+},\delta)\setminus\Sigma_{\widehat{\mathbf{U}}},\\\
&\widehat{\mathbf{U}}(z)\left[\widehat{\mathbf{P}}^{(0)}(z)\right]^{-1},\quad&z&\in
U(z_{0},\delta)\setminus\Sigma_{\widehat{\mathbf{U}}},\\\
&\widehat{\mathbf{U}}(z)\left[\widehat{\mathbf{P}}^{(\infty)}(z)\right]^{-1},\quad&\mathrm{e}&\mathrm{lsewhere}.\\\
\end{aligned}\right.$ (5.25)

Then, $\widehat{\mathbf{R}}(z)$ satisfies the following RH problem.

### RH problem for $\widehat{\mathbf{R}}(z)$

(1) $\widehat{\mathbf{R}}(z)$ is analytic for
$z\in\mathbb{C}\setminus\Sigma_{\widehat{\mathbf{R}}}$, where
$\Sigma_{\widehat{\mathbf{R}}}$ is depicted in Figure 8.

(2) On the contour $\Sigma_{\widehat{\mathbf{R}}}$, we have
$\widehat{\mathbf{R}}_{+}(z)=\widehat{\mathbf{R}}_{-}(z)J_{\widehat{\mathbf{R}}}(z)$,
where

$J_{\widehat{\mathbf{R}}}(z)=\left\\{\begin{aligned}
&\widehat{\mathbf{P}}^{(l)}(z)\widehat{\mathbf{P}}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(z_{-},\delta),\\\
&\widehat{\mathbf{P}}^{(r)}(z)\widehat{\mathbf{P}}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(z_{+},\delta),\\\
&\widehat{\mathbf{P}}^{(0)}(z)\widehat{\mathbf{P}}^{(\infty)}(z)^{-1},\quad&z&\in\partial
U(z_{0},\delta)\\\
&\widehat{\mathbf{P}}^{(\infty)}_{-}(z)J_{\widehat{\mathbf{U}}}(z)\widehat{\mathbf{P}}^{(\infty)}_{+}(z)^{-1},\quad&\mathrm{e}&\mathrm{lsewhere}.\end{aligned}\right.$
(5.26)

(3) As $z\rightarrow\infty$, $\widehat{\mathbf{R}}(z)$ admits the expansion

$\widehat{\mathbf{R}}(z)=\mathbf{I}+\frac{\widehat{\mathbf{R}}_{1}}{z}+\frac{\widehat{\mathbf{R}}_{2}}{z^{2}}+O(z^{-3}).$
(5.27)

Figure 8: The jump contour $\Sigma_{\widehat{\mathbf{R}}}$

By virtue of the matching conditions (5.8), (5.15) and (5.22), the jump matrix
(5.26) has the following estimations as $x\rightarrow-\infty$

$J_{\widehat{\mathbf{R}}}(z)=\left\\{\begin{aligned}
&\mathbf{I}+O(|x|^{-2}),&z&\in\partial U(z_{\pm},\delta),\\\
&\mathbf{I}+O(|x|^{-2\alpha}e^{-x^{2}}),&z&\in\partial
U(z_{0},\delta)\cup\widehat{\gamma}_{2}\cup\widehat{\gamma}_{6},\\\
&\mathbf{I}+O(e^{-c_{3}x^{2}}),&\mathrm{e}&\mathrm{lsewhere},\end{aligned}\right.$
(5.28)

where $c_{3}$ is some positive constant. As a result, we get that for any
$z\in\mathbb{C}\setminus\Sigma_{\widehat{\mathbf{R}}}$,

$\widehat{\mathbf{R}}(z)=\mathbf{I}+O(|x|^{-2}),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (5.29)

##  6 Proof of Theorems 1.1 \- 1.4

To derive the asymptotics for the PIV solution $q(x;\alpha,\kappa)$ and the
associated Hamiltonian $\mathcal{H}(x;\alpha,\kappa)$ as
$x\rightarrow-\infty$, we use the identities (3.3) and (3.4) and the
asymptotic analysis of the RH problem for $\Psi(\xi,x)$ performed in Sections
3-5.

###  6.1 Derivations of (1.18) and (1.25)

Let us focus on the case $\kappa(\kappa-\kappa^{*})<0$. Tracing back the
series of invertible transformations

$\Phi\mapsto\mathbf{U}\mapsto\mathbf{T}\mapsto\mathbf{R}$

as defined in (3.7), (3.14) and (3.63)), respectively, we obtain that for
large $z$

$e^{\frac{x^{2}}{3}\sigma_{3}}\Phi(z)e^{-x^{2}g(z)\sigma_{3}}=\mathbf{R}(z)\mathbf{P}^{(\infty)}(z).$

It follows from the asymptotic expansions (3.2) and (3.6) that

$e^{\frac{x^{2}}{3}\sigma_{3}}\Phi(z)e^{-x^{2}g(z)\sigma_{3}}z^{-\alpha\sigma_{3}}=\mathbf{I}+\frac{e^{\frac{x^{2}}{3}\sigma_{3}}\Phi_{1}e^{-\frac{x^{2}}{3}\sigma_{3}}}{z}+\frac{e^{\frac{x^{2}}{3}\sigma_{3}}\Phi_{2}e^{-\frac{x^{2}}{3}\sigma_{3}}-\frac{4x^{2}}{27}\sigma_{3}}{z^{2}}+O(z^{-3}).$
(6.1)

To compute the coefficients $\Phi_{1}$ and $\Phi_{2}$, we need to know the
asymptotics of $\mathbf{R}(z)$ and $\mathbf{P}^{(\infty)}(z)$ as $z\to\infty$.
The large-$z$ asymptotics of $\mathbf{R}(z)$ has already been given in (3.65).
As for $\mathbf{P}^{(\infty)}(z)$, we obtain from (3.18), (3.24) and (3.25)
that

$\mathbf{P}^{(\infty)}(z)z^{-\alpha\sigma_{3}}=\mathbf{I}+\frac{\mathbf{P}_{1}^{(\infty)}}{z}+\frac{\mathbf{P}_{2}^{(\infty)}}{z^{2}}+O\left(\frac{1}{z^{3}}\right),~{}~{}~{}\mbox{as}~{}~{}z\rightarrow\infty,$
(6.2)

where

$\mathbf{P}_{1}^{(\infty)}=\sqrt{\frac{2}{3}}s_{0}^{-\frac{\sigma_{3}}{2}}f_{\infty}^{-\sigma_{3}}\sigma_{2}f_{\infty}^{\sigma_{3}}s_{0}^{\frac{\sigma_{3}}{2}},\quad\mathbf{P}_{2}^{(\infty)}=\frac{\mathbf{I}}{3}-\frac{2}{3}\left(\alpha+\sqrt{3}i{\beta}\right)\sigma_{3}.$
(6.3)

A combination of (3.65), (6.1) and (6.2) gives

$\Phi_{1}=e^{-\frac{x^{2}}{3}\sigma_{3}}\left[\mathbf{R}_{1}+\mathbf{P}_{1}^{(\infty)}\right]e^{\frac{x^{2}}{3}\sigma_{3}}$
(6.4)

and

$\Phi_{2}=e^{-\frac{x^{2}}{3}\sigma_{3}}\left[\mathbf{R}_{1}\mathbf{P}_{1}^{(\infty)}+\mathbf{R}_{2}+\mathbf{P}_{2}^{(\infty)}\right]e^{\frac{x^{2}}{3}\sigma_{3}}+\frac{4x^{2}}{27}\sigma_{3}.$
(6.5)

Using formulas (3.3), (3.4) and combining them with (3.65), (6.1), (6.3),
(6.4) and (6.5), we get the following expressions for $q(x;\alpha,\kappa)$ and
$\mathcal{H}(x;\alpha,\kappa)$

$q(x;\alpha,\kappa)=-\frac{2}{3}x-i\sqrt{\frac{2}{3}}x\Big{[}s_{0}f_{\infty}^{2}(\mathbf{R}_{1})_{12}-s_{0}^{-1}f_{\infty}^{-2}(\mathbf{R}_{1})_{21}\Big{]}-x(\mathbf{R}_{1})_{12}(\mathbf{R}_{1})_{21}$
(6.6)

and

$\displaystyle\mathcal{H}(x;\alpha,\kappa)=$
$\displaystyle-\frac{8}{27}x^{3}+\frac{4}{3}(\alpha+\sqrt{3}i{\beta})x-x\sqrt{\frac{2}{3}}i\Big{[}s_{0}f_{\infty}^{2}(\mathbf{R}_{1})_{12}+s_{0}^{-1}f_{\infty}^{-2}(\mathbf{R}_{1})_{21}\Big{]}$
$\displaystyle-x\big{[}(\mathbf{R}_{2})_{11}-(\mathbf{R}_{2})_{22}\big{]}.$
(6.7)

Thus, the remaining task is to compute the asymptotics of $\mathbf{R}_{1}$ and
$\mathbf{R}_{2}$. It is observed from (3.66) and (3.67) that, neglecting a
uniform error term $O(|x|^{-2})$, the contribution to
$\mathbf{R}(z)-\mathbf{I}$ comes from the jumps on $\partial
U(z_{1,\pm},\delta)$. Therefore, it follows from (3), (3) and (3.64) that as
$x\to-\infty$,

$\mathbf{R}(z)=\mathbf{I}+\frac{\mathbf{R}^{(1)}(z)}{|x|}+O\left(\frac{1}{|x|^{2}}\right),\quad\mathrm{for}\quad
z\in\partial U(z_{1,\pm},\delta),$ (6.8)

where the coefficient $\mathbf{R}^{(1)}(z)=O(z^{-1})$ as $z\rightarrow\infty$,
and satisfies the jump relation

$\mathbf{R}^{(1)}_{+}(z)-\mathbf{R}^{(1)}_{-}(z)=\mathbf{G}(z),\quad
z\in\partial U(z_{1,\pm},\delta),$ (6.9)

in which $\mathbf{G}(z)=\mathbf{G}^{(\pm)}(z)$ are given by (3.44) and (3.54),
respectively. Using the Sokhotskii-Plemelj formula and keeping in mind the
clockwise orientations of the boundaries $\partial U(z_{1,\pm},\delta)$, the
solution to the above RH problem is explicitly given by

$\mathbf{R}^{(1)}(z)=\left\\{\begin{aligned}
&\frac{L_{+}}{z-\sqrt{\frac{2}{3}}}+\frac{L_{-}}{z+\sqrt{\frac{2}{3}}},&z&\notin\overline{U(z_{1,+},\delta)}\cup\overline{U(z_{1,-},\delta)},\\\
&\frac{L_{+}}{z-\sqrt{\frac{2}{3}}}+\frac{L_{-}}{z+\sqrt{\frac{2}{3}}}-\mathbf{G}(z),&z&\in
U(z_{1,+},\delta)\cup U(z_{1,-},\delta),\end{aligned}\right.$ (6.10)

where
$L_{\pm}=\mathrm{Res}\,\left(\mathbf{G}^{(\pm)}(z),z=\pm\sqrt{\frac{2}{3}}\;\right)$
are given by

$\displaystyle L_{+}$
$\displaystyle=\mathbf{W}^{(+)}(\textstyle\sqrt{\frac{2}{3}})\begin{pmatrix}0&-\frac{{\beta}h_{0}e^{i\frac{\sqrt{3}}{3}x^{2}}|x|^{2{\beta}}}{2\cdot
3^{-\frac{1}{4}}e^{\frac{3\pi i}{4}}s_{*}}\\\
-\frac{s_{*}|x|^{-2{\beta}}}{2\cdot 3^{-\frac{1}{4}}e^{\frac{3\pi
i}{4}}h_{0}e^{i\frac{\sqrt{3}}{3}x^{2}}}&0\end{pmatrix}\mathbf{W}^{(+)}(\sqrt{\frac{2}{3}})^{-1},$
(6.11) $\displaystyle L_{-}$
$\displaystyle=\mathbf{W}^{(-)}(-\textstyle\sqrt{\frac{2}{3}})\begin{pmatrix}0&\frac{{\beta}h_{0}e^{i\frac{\sqrt{3}}{3}x^{2}}|x|^{2{\beta}}}{2\cdot
3^{-\frac{1}{4}}e^{\frac{3\pi i}{4}}s_{*}}\\\
\frac{s_{*}|x|^{-2{\beta}}}{2\cdot 3^{-\frac{1}{4}}e^{\frac{3\pi
i}{4}}h_{0}e^{i\frac{\sqrt{3}}{3}x^{2}}}&0\end{pmatrix}\mathbf{W}^{(-)}(-\textstyle\sqrt{\frac{2}{3}})^{-1},$
(6.12)

with $\mathbf{W}^{(\pm)}(\pm\sqrt{\frac{2}{3}})$ given by (3.42)) and (3.52),
respectively.

Using (6.10) to expand $\mathbf{R}^{(1)}(z)$ in (6.8) into the Taylor series
at infinity, we obtain the asymptotics for the coefficients $\mathbf{R}_{1}$
and $\mathbf{R}_{2}$ in the expansion (3.65):

$\mathbf{R}_{1}=\frac{\mathbf{R}_{1}^{(1)}}{|x|}+O(|x|^{-2}),\quad\mathbf{R}_{2}=\frac{\mathbf{R}_{2}^{(1)}}{|x|}+O(|x|^{-2}),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (6.13)

The coefficients $\mathbf{R}_{1}^{(1)}=L_{+}+L_{-}$ and
$\mathbf{R}_{2}^{(1)}=\sqrt{\frac{2}{3}}\left(L_{+}-L_{-}\right)$, after a
direct computation from (6.11) and (6.12), are explicitly given by

$\mathbf{R}_{1}^{(1)}=\begin{pmatrix}0&\left(C_{1}-C_{2}+\frac{i}{\sqrt{3}}(C_{1}+C_{2})\right)s_{0}^{-1}f_{\infty}^{-2}\\\
\left(C_{2}-C_{1}+\frac{i}{\sqrt{3}}(C_{1}+C_{2})\right)s_{0}f_{\infty}^{2}&0\end{pmatrix},$
(6.14)

and

$\mathbf{R}_{2}^{(1)}=\begin{pmatrix}\frac{2\sqrt{2}}{3}(C_{1}+C_{2})&0\\\
0&-\frac{2\sqrt{2}}{3}(C_{1}+C_{2})\end{pmatrix},$ (6.15)

with

$C_{1}=\frac{{\beta}h_{0}e^{\frac{i\sqrt{3}}{3}x^{2}}2^{{\beta}-1}3^{\frac{2{\beta}+1}{4}}}{s_{*}e^{\frac{3\pi
i}{4}+\frac{2\pi i\alpha}{3}+\frac{\pi i{\beta}}{2}}}|x|^{2{\beta}},\qquad
C_{2}=\frac{s_{*}e^{-\frac{3\pi i}{4}+\frac{2\pi i\alpha}{3}+\frac{\pi
i{\beta}}{2}}}{h_{0}e^{\frac{i\sqrt{3}}{3}x^{2}}2^{{\beta}+1}3^{\frac{2{\beta}-1}{4}}}|x|^{-2{\beta}}.$
(6.16)

Now, we are ready to derive the asymptotics for $q(x;\alpha,\kappa)$ and
$\mathcal{H}(x;\alpha,\kappa)$. Substituting (6.13), (6.14), (6.15) and (6.16)
into (6.6) and (6.1) yields

$\displaystyle q(x;\alpha,\kappa)$
$\displaystyle=-\frac{2}{3}x+2^{\frac{1}{2}}3^{-\frac{1}{4}}e^{-\frac{\pi
i}{4}}\Big{[}{\beta}h_{0}s_{*}^{-1}e^{\frac{\sqrt{3}x^{2}i}{3}-\frac{2\pi
i\alpha}{3}-\frac{\pi i{\beta}}{2}+{\beta}\ln\left(2\sqrt{3}x^{2}\right)}$
$\displaystyle\qquad\qquad\
-h_{0}^{-1}s_{*}e^{-\frac{\sqrt{3}x^{2}i}{3}+\frac{2\pi i\alpha}{3}+\frac{\pi
i{\beta}}{2}-{\beta}\ln\left(2\sqrt{3}x^{2}\right)}\Big{]}+O(|x|^{-1}),\quad\mathrm{as}\quad
x\rightarrow-\infty,$ (6.17)

and

$\displaystyle\mathcal{H}(x;\alpha,\kappa)$
$\displaystyle=-\frac{8}{27}x^{3}+\frac{4}{3}\left(\alpha+\sqrt{3}i{\beta}\right)x+2^{\frac{1}{2}}3^{-\frac{3}{4}}e^{-\frac{3\pi
i}{4}}\Big{[}{\beta}h_{0}s_{*}^{-1}e^{\frac{\sqrt{3}x^{2}i}{3}-\frac{2\pi
i\alpha}{3}-\frac{\pi i{\beta}}{2}+{\beta}\ln\left(2\sqrt{3}x^{2}\right)}$
$\displaystyle\qquad\qquad\
+h_{0}^{-1}s_{*}e^{-\frac{\sqrt{3}x^{2}i}{3}+\frac{2\pi i\alpha}{3}+\frac{\pi
i{\beta}}{2}-{\beta}\ln\left(2\sqrt{3}x^{2}\right)}\Big{]}+O(|x|^{-1}),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (6.18)

Recalling the definition (A.4) of the Stokes multiplier $h_{0}$ and using the
reflection formula

$\Gamma(1-{\beta})\Gamma({\beta})=\frac{\pi}{\sin(\pi{\beta})},$ (6.19)

we can rewrite (6.1) and (6.1) in the following symmetric form

$\displaystyle q(x;\alpha,\kappa)=$
$\displaystyle-\frac{2}{3}x+3^{-\frac{1}{4}}\pi^{-\frac{1}{2}}e^{\frac{\pi
i{\beta}}{2}}{\beta}|s_{*}|\Big{[}\Gamma(-{\beta})e^{\frac{\sqrt{3}x^{2}i}{3}-\frac{2\pi
i\alpha}{3}-\frac{\pi i}{4}-i\arg
s_{*}+{\beta}\ln\left(2\sqrt{3}x^{2}\right)}$
$\displaystyle-\Gamma({\beta})e^{-\frac{\sqrt{3}x^{2}i}{3}+\frac{2\pi
i\alpha}{3}+\frac{\pi i}{4}+i\arg
s_{*}-{\beta}\ln\left(2\sqrt{3}x^{2}\right)}\Big{]}+O(|x|^{-1}),\quad\mathrm{as}\quad
x\rightarrow-\infty,$ (6.20)

and

$\displaystyle\mathcal{H}(x;\alpha,\kappa)=$
$\displaystyle-\frac{8}{27}x^{3}+\frac{4}{3}\left(\alpha+\sqrt{3}i{\beta}\right)x$
$\displaystyle-3^{-\frac{3}{4}}\pi^{-\frac{1}{2}}e^{\frac{\pi
i{\beta}}{2}}{\beta}|s_{*}|\Big{[}\Gamma(-{\beta})e^{\frac{\sqrt{3}x^{2}i}{3}-\frac{2\pi
i\alpha}{3}+\frac{\pi i}{4}-i\arg
s_{*}+{\beta}\ln\left(2\sqrt{3}x^{2}\right)}$
$\displaystyle-\Gamma({\beta})e^{-\frac{\sqrt{3}x^{2}i}{3}+\frac{2\pi
i\alpha}{3}-\frac{\pi i}{4}+i\arg
s_{*}-{\beta}\ln\left(2\sqrt{3}x^{2}\right)}\Big{]}+O(|x|^{-1}),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (6.21)

Remember that ${\beta}$ in the case under consideration is purely imaginary
(see (3.23)), we thus have the following complex conjugate relation

$\overline{\Gamma({\beta})}=\Gamma(-{\beta}).$

Denote by

$a^{2}=2i{\beta}=-\frac{1}{\pi}\ln(1-|s_{*}|^{2}),\quad a>0.$ (6.22)

It is seen from (3.22), (6.19) and (6.22) that

$|\Gamma({\beta})|^{2}=\Gamma({\beta})\Gamma(-{\beta})=-\frac{\pi}{{\beta}\sin\pi{\beta}}=-\frac{2\pi
i}{{\beta}e^{\pi i\beta}(1-e^{-2\pi
i{\beta}})}=4\pi|s_{*}|^{-2}a^{-2}e^{-\frac{1}{2}\pi a^{2}}.$ (6.23)

By substituting (6.22), (6.23) into (6.1) and (6.1), we arrive at the final
formulas (1.18) and (1.25) for the case $0<|s_{*}|<1$.

Similarly, by using the asymptotic analysis of the RH problem for
$\Psi(\xi,x)$ performed in Section 3.7, we obtain in the special case
$|s_{*}|=0$, the following asymptotic formulas

$\displaystyle q(x;\alpha,\kappa)$ $\displaystyle=-\frac{2}{3}x+O(x^{-1}),$
(6.24) $\displaystyle\mathcal{H}(x;\alpha,\kappa)$
$\displaystyle=-\frac{8}{27}x^{3}+\frac{4\alpha}{3}x+O(x^{-1}),$ (6.25)

as $x\rightarrow-\infty$. Notice that it follows from (3.22) that ${\beta}=0$
if $|s_{*}|=0$. Therefore, (6.24) and (6.25) can be respectively regarded as
the limits of (1.18) and (1.25) as $b_{1}\rightarrow 0$.

###  6.2 Derivations of (1.20) and (1.27)

Tracing back the above modified RH analysis for the case
$\kappa(\kappa-\kappa^{*})<0$, it follows from (3.7), (3.14) and (4.18) that

$e^{\frac{x^{2}}{3}\sigma_{3}}\Phi(z)e^{-x^{2}g(z)\sigma_{3}}=\widetilde{\mathbf{R}}(z)\widetilde{\mathbf{P}}^{(\infty)}(z),$
(6.26)

for large $z$. Accordingly, to compute the coefficients $\Phi_{1}$ and
$\Phi_{2}$ in expansion (3.2), we need the asymptotic approximation of
$\widetilde{\mathbf{P}}^{(\infty)}(z)$ as $z\rightarrow\infty$. By (4.3) and
(6.2), we have

$\widetilde{\mathbf{P}}^{(\infty)}(z)z^{-\alpha\sigma_{3}}=\mathbf{I}+\frac{\widetilde{\mathbf{P}}_{1}^{(\infty)}}{z}+\frac{\widetilde{\mathbf{P}}_{2}^{(\infty)}}{z^{2}}+O\left(\frac{1}{z^{3}}\right),\quad\mathrm{as}\quad
z\rightarrow\infty,$ (6.27)

where

$\widetilde{\mathbf{P}}_{1}^{(\infty)}=\mathbf{P}_{1}^{(\infty)}+\widetilde{A}+\widetilde{B},\qquad\widetilde{\mathbf{P}}_{2}^{(\infty)}=(\widetilde{A}+\widetilde{B})\mathbf{P}_{1}^{(\infty)}+\sqrt{\frac{2}{3}}(\widetilde{A}-\widetilde{B})+\mathbf{P}_{2}^{(\infty)},$
(6.28)

with $\widetilde{A}$, $\widetilde{B}$ and $\mathbf{P}_{1}^{(\infty)}$,
$\mathbf{P}_{2}^{(\infty)}$ given by (4.4) and (6.3), respectively. Using
(4.20), (6.26) and (6.27), we now obtain

$e^{\frac{x^{2}}{3}\sigma_{3}}\Phi_{1}e^{-\frac{x^{2}}{3}\sigma_{3}}=\mathbf{P}_{1}^{(\infty)}+\widetilde{\mathbf{R}}_{1}+\widetilde{A}+\widetilde{B},$
(6.29)

and

$e^{\frac{x^{2}}{3}\sigma_{3}}\Phi_{2}e^{-\frac{x^{2}}{3}\sigma_{3}}=\frac{4x^{2}}{27}\sigma_{3}+\widetilde{\mathbf{R}}_{1}(\widetilde{A}+\widetilde{B}+\mathbf{P}_{1}^{(\infty)})+\widetilde{\mathbf{R}}_{2}+(\widetilde{A}+\widetilde{B})\mathbf{P}_{1}^{(\infty)}+\sqrt{\frac{2}{3}}(\widetilde{A}-\widetilde{B})+\mathbf{P}_{2}^{(\infty)}.$
(6.30)

Inserting (6.29), (6.30) into the formulas (3.3) and (3.4), we obtain the
following expressions for $q(x;\alpha,\kappa)$ and
$\mathcal{H}(x;\alpha,\kappa)$

$\displaystyle q(x;\alpha,\kappa)$
$\displaystyle=-x\,\Big{[}(\mathbf{P}^{(\infty)}_{1})_{12}+\widetilde{A}_{12}+\widetilde{B}_{12}+(\widetilde{\mathbf{R}}_{1})_{12}\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad\times\Big{[}(\mathbf{P}^{(\infty)}_{1})_{21}+\widetilde{A}_{21}+\widetilde{B}_{21}+(\widetilde{\mathbf{R}}_{1})_{21}\Big{]},$
(6.31) $\displaystyle\mathcal{H}(x;\alpha,\kappa)$
$\displaystyle=-x\,\bigg{\\{}\frac{8}{27}x^{2}+\left[\widetilde{\mathbf{R}}_{1}(\widetilde{A}+\widetilde{B}+\mathbf{P}_{1}^{(\infty)})\right]_{11}-\left[\widetilde{\mathbf{R}}_{1}(\widetilde{A}+\widetilde{B}+\mathbf{P}_{1}^{(\infty)})\right]_{22}$
$\displaystyle\qquad\qquad+(\widetilde{\mathbf{R}}_{2})_{11}-(\widetilde{\mathbf{R}}_{2})_{22}+\left[(\widetilde{A}+\widetilde{B})\mathbf{P}_{1}^{(\infty)}\right]_{11}-\left[(\widetilde{A}+\widetilde{B})\mathbf{P}_{1}^{(\infty)}\right]_{22}$
$\displaystyle\qquad\qquad\qquad+\sqrt{\frac{2}{3}}(\widetilde{A}_{11}-\widetilde{A}_{22}+\widetilde{B}_{22}-\widetilde{B}_{11})+(\mathbf{P}_{2}^{(\infty)})_{11}-(\mathbf{P}_{2}^{(\infty)})_{22}\bigg{\\}}.$
(6.32)

Next, we compute the asymptotics of $\widetilde{\mathbf{R}}_{1}$ and
$\widetilde{\mathbf{R}}_{2}$ as $x\rightarrow-\infty$. In view of the error
estimate (4.22) and following similar analysis as we have done in Section 6.1,
we easily get that

$\widetilde{\mathbf{R}}_{1}=O(|x|^{-2}),\qquad\widetilde{\mathbf{R}}_{2}=O(|x|^{-2}),\quad\mathrm{as}\quad
x\rightarrow-\infty.$ (6.33)

Now, substituting (4.4), (6.3) and (6.33) into (6.2) and (6.32) gives

$\displaystyle q(x;\alpha,\kappa)$
$\displaystyle=-2x-4\sqrt{\frac{2}{3}}i\cdot\frac{l-\sqrt{\frac{2}{3}}i}{l^{2}-2}x+O(x^{-1}),$
(6.34) $\displaystyle\mathcal{H}(x;\alpha,\kappa)$
$\displaystyle=-\frac{8}{27}x^{3}+\frac{4}{3}\left(\alpha+\sqrt{3}i{\beta}\right)x+i\frac{4l}{\sqrt{3}}x\cdot\frac{l-\sqrt{\frac{2}{3}}i}{l^{2}-2}+O(x^{-1}),$
(6.35)

where the error terms are uniform for $x$ bounded away from the zeros of
$l^{2}-2$. From the definition of the function $l$ in (4.5), we may rewrite

$\frac{l-\sqrt{\frac{2}{3}}i}{l^{2}-2}=\frac{i}{4}\sqrt{\frac{2}{3}}\frac{(2+e^{i\phi})(1+2e^{i\phi})}{1+e^{i\phi}+e^{2i\phi}}=\frac{i}{4}\sqrt{\frac{2}{3}}\left(2+\frac{3}{2\cos\phi+1}\right).$

By inserting the above equation into the expressions (6.34) and (6.35), we
arrive at the formulas (1.20) and (1.27).

###  6.3 Derivations of (1.19) and (1.26)

We now consider the case $\kappa=\kappa^{*}$. By tracing back the
transformations (5.3) and (5.25), it follows that for large $z$

$e^{\frac{x^{2}}{2}\sigma_{3}}|x|^{\alpha\sigma_{3}}\Phi(z)|x|^{-\alpha\sigma_{3}}e^{-x^{2}\widehat{g}(z)\sigma_{3}}=\widehat{\mathbf{R}}(z)\widehat{\mathbf{P}}^{(\infty)}(z).$
(6.36)

To compute the coefficients $\Phi_{1}$ and $\Phi_{2}$ in expansion (3.2), we
need to write down the asymptotic approximation of
$\widehat{\mathbf{P}}^{(\infty)}(z)$ as $z\rightarrow\infty$. By (5.6), (5.7)
and (5.21), we get that as $z\rightarrow\infty$

$\widehat{\mathbf{P}}^{(\infty)}(z)z^{-\alpha\sigma_{3}}=\mathbf{I}+\frac{\widehat{\mathbf{P}}_{1}^{(\infty)}}{z}+\frac{\widehat{\mathbf{P}}_{2}^{(\infty)}}{z^{2}}+O\left(\frac{1}{z^{3}}\right),$
(6.37)

where

$\widehat{\mathbf{P}}_{1}^{(\infty)}=\begin{pmatrix}0&-\frac{1}{2}c_{r}\\\
-\frac{1}{2}c_{l}&0\end{pmatrix},\qquad\widehat{\mathbf{P}}_{2}^{(\infty)}=\begin{pmatrix}1-2\alpha&0\\\
0&1+2\alpha\end{pmatrix}$ (6.38)

with $c_{r}$, $c_{l}$ given by (5.14) and (5.20), respectively.

From (6.36) and the large-$z$ expansions (5.27) and (6.37), we have

$\Phi_{1}=e^{-\frac{x^{2}}{2}\sigma_{3}}|x|^{-\alpha\sigma_{3}}\left(\widehat{\mathbf{P}}_{1}^{(\infty)}+\widehat{\mathbf{R}}_{1}\right)|x|^{\alpha\sigma_{3}}e^{\frac{x^{2}}{2}\sigma_{3}}$
(6.39)

and

$\Phi_{2}=e^{-\frac{x^{2}}{2}\sigma_{3}}|x|^{-\alpha\sigma_{3}}\left(\widehat{\mathbf{P}}_{2}^{(\infty)}+\widehat{\mathbf{P}}_{1}^{(\infty)}\widehat{\mathbf{R}}_{1}+\widehat{\mathbf{R}}_{2}\right)|x|^{\alpha\sigma_{3}}e^{\frac{x^{2}}{2}\sigma_{3}}.$
(6.40)

Now, recalling (3.3) and (3.4), we obtain from (6.38), (6.39) and (6.40) that

$q(x;\alpha,\kappa)=-x\left[-\frac{c_{r}}{2}+(\widehat{\mathbf{R}}_{1})_{12}\right]\left[-\frac{c_{l}}{2}+(\widehat{\mathbf{R}}_{1})_{21}\right]$
(6.41)

and

$\mathcal{H}(x;\alpha,\kappa)=-x\bigg{\\{}-4\alpha-\frac{1}{2}\left[c_{r}(\widehat{\mathbf{R}}_{1})_{21}-c_{l}(\widehat{\mathbf{R}}_{1})_{12}\right]+(\widehat{\mathbf{R}}_{2})_{11}-(\widehat{\mathbf{R}}_{2})_{22}\bigg{\\}}.$
(6.42)

In view of the estimation (5.29), we easily obtain the following asymptotics
for $\widehat{\mathbf{R}}_{1}$ and $\widehat{\mathbf{R}}_{2}$ as
$x\rightarrow-\infty$

$\widehat{\mathbf{R}}_{1}=O(|x|^{-2}),\quad\widehat{\mathbf{R}}_{2}=O(|x|^{-2}).$
(6.43)

Substituting the asymptotics (6.43) into the expressions (6.41) and (6.42) and
using the fact that $c_{r}c_{l}=8$, we obtain (1.19) and (1.26).

###  6.4 Derivation of (1.24)

It is seen from (1.17) and (2.17) that

$\frac{d}{dx}\mathcal{H}(x;\alpha,\kappa)=\kappa\,2^{\alpha+\frac{1}{2}}x^{2\alpha}e^{-x^{2}}\left(1+O\left(x^{-2}\right)\right),\quad\mathrm{as}~{}~{}x\to+\infty.$
(6.44)

Moreover, it follows from (1.17) and (2.14) that

$\mathcal{H}(x;\alpha,\kappa)\to 0,\quad\mathrm{as}~{}~{}x\to+\infty.$ (6.45)

Thus, integrating both sides of (6.44) and using the boundary condition
(6.45), we obtain (1.24).

###  6.5 Derivations of (1.4) and (1.4)

Recalling from (2.12) and using the fact that $F(x)\to\mathbf{I}$ as
$x\to+\infty$, as can be seen from the asymptotic analysis of $\Psi(\xi,x)$ as
$x\to+\infty$ performed in [18], we have

$F(x):=\Psi^{(0)}(0,x)=\exp\left(\int_{+\infty}^{x}q(t)dt\sigma_{3}\right),$
(6.46)

with $\Psi^{(0)}(\xi,x)$ given in (2.6). Hence, the evaluation of the total
integral of $q(x)$ is boiled down to computing the asymptotic of $F(x)$ as
$x\to-\infty$. In the remaining part of this section, we derive the
asymptotics of $F(x)$ as $x\to-\infty$ and prove Theorem 1.4 based on the
asymptotic analysis of the RH problem for $\Psi(\xi,x)$ performed in Sections
3 and 5.

####  6.5.1 Derivation of (1.4)

In the case $\kappa(\kappa-\kappa^{*})<0$, by inverting the transformations
(3.1), (3.7), (3.14) and (3.63), we obtain that for $z$ small and $\arg
z\in(\frac{\pi}{4},\frac{\pi}{2})$,

$\displaystyle\Psi(|x|^{\frac{1}{2}}z,x)$
$\displaystyle=e^{-\frac{x^{2}}{3}\sigma_{3}}|x|^{\frac{\alpha}{2}\sigma_{3}}\mathbf{R}(z)\mathbf{P}^{(0)}(z)\begin{pmatrix}1&0\\\
\frac{\overline{s}_{*}e^{-2\pi i\alpha}e^{-2x^{2}g(z)}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix}e^{x^{2}g(z)\sigma_{3}}$
$\displaystyle=e^{-\frac{x^{2}}{3}\sigma_{3}}|x|^{\frac{\alpha}{2}\sigma_{3}}\mathbf{R}(z)\mathbf{E}^{(0)}(z)\Phi^{(\mathrm{Bes})}(x^{2}\varphi_{4}(z))$
$\displaystyle\qquad\qquad\qquad\times\begin{pmatrix}1&0\\\ -e^{-2\pi
i\alpha}&1\end{pmatrix}\left[s_{1}(e^{-2\pi
i\alpha}+s_{*})\right]^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}1&0\\\
\frac{\bar{s}_{*}e^{-2\pi i\alpha}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix}.$ (6.47)

It is readily seen from (2.6) and (6.5.1) that

$\displaystyle F(x)$
$\displaystyle=e^{-\frac{x^{2}}{3}\sigma_{3}}|x|^{\frac{\alpha}{2}\sigma_{3}}\mathbf{R}(0)\mathbf{E}^{(0)}(0)\lim\limits_{z\to
0}\Phi^{(\mathrm{Bes})}(x^{2}\varphi_{4}(z))\begin{pmatrix}1&0\\\ -e^{-2\pi
i\alpha}&1\end{pmatrix}$
$\displaystyle\qquad\qquad\qquad\times\left[s_{1}(e^{-2\pi
i\alpha}+s_{*})\right]^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}1&0\\\
\frac{\bar{s}_{*}e^{-2\pi i\alpha}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix}S_{1}^{-1}E_{0}^{-1}z^{-\alpha\sigma_{3}}|x|^{-\frac{\alpha}{2}\sigma_{3}}.$
(6.48)

By (2.8), (3.57) and (A.9), we have

$\displaystyle\lim\limits_{z\to
0}\Phi^{(\mathrm{Bes})}(x^{2}\varphi_{4}(z))\begin{pmatrix}1&0\\\ -e^{-2\pi
i\alpha}&1\end{pmatrix}[s_{1}(e^{-2\pi
i\alpha}+s_{*})]^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}1&0\\\
\frac{\bar{s}_{*}e^{-2\pi i\alpha}}{s_{1}(e^{-2\pi
i\alpha}+s_{*})}&1\end{pmatrix}S_{1}^{-1}E_{0}^{-1}z^{-\alpha\sigma_{3}}$
$\displaystyle\qquad=C_{\alpha}^{\sigma_{3}}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}2^{\frac{3}{2}\alpha\sigma_{3}}3^{-\frac{3}{2}\alpha\sigma_{3}}s_{1}^{-\frac{\sigma_{3}}{2}}(e^{-2\pi
i\alpha}+s_{*})^{\frac{\sigma_{3}}{2}}(1+e^{-2\pi
i\alpha})^{-\sigma_{3}}|x|^{2\alpha\sigma_{3}},$ (6.49)

where the constant $C_{\alpha}$ is given by (A.10). Recalling the definition
(3.60) of $\mathbf{E}^{(0)}(z)$, it follows from (3.18), (3.21) and (3.22)
that

$\displaystyle\mathbf{E}^{(0)}(0)$
$\displaystyle=\frac{1}{2}s_{0}^{-\frac{\sigma_{3}}{2}}2^{\frac{\alpha}{2}\sigma_{3}}3^{-\frac{\alpha}{2}\sigma_{3}}e^{-\frac{\pi
i{\beta}}{3}\sigma_{3}}\begin{pmatrix}1&-1\\\ 1&1\end{pmatrix}e^{-\frac{3\pi
i}{4}\sigma_{3}}\begin{pmatrix}1&i\\\ i&1\end{pmatrix}.$ (6.50)

Substituting (6.5.1) and (6.50) into (6.5.1) yields

$\displaystyle
e^{\frac{x^{2}}{3}\sigma_{3}}|x|^{-\frac{\alpha}{2}\sigma_{3}}F(x)|x|^{-\frac{3\alpha}{2}\sigma_{3}}=\mathbf{R}(0)2^{(\alpha-\frac{1}{2})\sigma_{3}}3^{-2\alpha\sigma_{3}}\pi^{-\frac{\sigma_{3}}{2}}\Gamma(\textstyle\frac{1}{2}-\alpha)^{\sigma_{3}}[(1-s_{*})e^{\pi
i\alpha}]^{-\sigma_{3}}e^{-\frac{4}{3}\pi i{\beta}\sigma_{3}}.$ (6.51)

We choose some constants $c<0$ and $d>0$ such that all real poles of
$q(x;\alpha,\kappa)$ lie in the $x$-interval $(c,d)$. Using the estimation
(3.67), the expression (6.46) and letting $x\to-\infty$ in (6.51), we obtain

$\displaystyle\exp\Bigg{\\{}\int^{d}_{+\infty}q(t)dt+\int_{\Upsilon}q(t)dt+\int^{-\infty}_{c}\left(q(t)+\frac{2t}{3}-\frac{2\alpha}{t}\right)dt+\frac{c^{2}}{3}-2\alpha\ln|c|\Bigg{\\}}$
$\displaystyle\qquad\qquad=2^{\alpha-\frac{1}{2}}3^{-2\alpha}\pi^{-\frac{1}{2}}\Gamma\left(\textstyle\frac{1}{2}-\alpha\right)[(1-s_{*})e^{\pi
i\alpha}]^{-1}e^{-\frac{4}{3}\pi i{\beta}}.$ (6.52)

where $\Upsilon$ is any contour in the complex plane from $d$ to $c$ that
avoids the real poles of $q(x;\alpha,\kappa)$. Indeed, we may take for
$\Upsilon$ a path along the real axis with infinitesimal semicircular
indentations centered at each real pole of $q(x;\alpha,\kappa)$ in the lower
half complex plane. Using the fact that all poles of $q(x;\alpha,\kappa)$ are
simple with residues $\pm 1$, we have

$\int_{\Upsilon}q(t)dt=\mathrm{P.V.}\int^{c}_{d}q(t)dt+\pi i(N_{+}-N_{-}),$
(6.53)

where $\mathrm{P.V.}$ denotes the Cauchy principal value and $N_{\pm}$ are the
numbers of real poles of $q(x;\alpha,\kappa)$ of residue $\pm 1$,
respectively.

By inserting (6.53) into (6.5.1) and using the definitions of $\kappa$,
${\beta}$ as given by (2.21), (3.22), respectively, we arrive at the total
integral (1.4).

####  6.5.2 Derivation of (1.4)

Now, we concentrate on the case $\kappa=\kappa^{*}$. Tracing back the
transformations (3.1), (5.3) and (5.25) gives

$\displaystyle\Psi(|x|^{\frac{1}{2}}z,x)$
$\displaystyle=e^{-\frac{x^{2}}{2}\sigma_{3}}|x|^{-\frac{\alpha}{2}\sigma_{3}}\widehat{\mathbf{R}}(z)\widehat{\mathbf{P}}^{(0)}(z)|x|^{\alpha\sigma_{3}}e^{x^{2}\widehat{g}(z)\sigma_{3}}.$
(6.54)

It follows from (2.6), (2.12), (5.23) and (6.54) that

$\displaystyle
F(x)=e^{-\frac{x^{2}}{2}\sigma_{3}}|x|^{-\frac{\alpha}{2}\sigma_{3}}$
$\displaystyle\widehat{\mathbf{R}}(0)\widehat{\mathbf{H}}(0)\lim\limits_{z\to
0}\begin{pmatrix}1&0\\\
r(z)&1\end{pmatrix}z^{-\alpha\sigma_{3}}(z+\sqrt{2})^{(\alpha+\frac{1}{2})\sigma_{3}}$
$\displaystyle\qquad\qquad\times(z-\sqrt{2})^{(\alpha-\frac{1}{2})\sigma_{3}}|x|^{\alpha\sigma_{3}}e^{x^{2}\widehat{g}(z)\sigma_{3}}S_{1}^{-1}E_{0}^{-1}z^{-\alpha\sigma_{3}}|x|^{-\frac{\alpha}{2}\sigma_{3}},$
(6.55)

for $z\to 0$ with $\arg{z}\in(\frac{\pi}{4},\frac{\pi}{2})$.

Notice that by (5.7), (5.21) and (5.24),

$\widehat{\mathbf{H}}(0)=(2\pi)^{-\frac{\sigma_{3}}{2}}e^{\pi
i(\alpha-\frac{1}{2})\sigma_{3}}\Gamma(\alpha+\textstyle\frac{1}{2})^{\sigma_{3}}\begin{pmatrix}0&s_{1}\\\
-s_{1}^{-1}&0\end{pmatrix}$ (6.56)

and

$\displaystyle
e^{-x^{2}\widehat{g}(z)\sigma_{3}}|x|^{-\alpha\sigma_{3}}(z+\sqrt{2})^{-(\alpha+\frac{1}{2})\sigma_{3}}(z-\sqrt{2})^{-(\alpha-\frac{1}{2})\sigma_{3}}z^{\alpha\sigma_{3}}\begin{pmatrix}1&0\\\
r(z)&1\end{pmatrix}$ $\displaystyle\qquad\qquad\qquad\times
z^{-\alpha\sigma_{3}}(z+\sqrt{2})^{(\alpha+\frac{1}{2})\sigma_{3}}(z-\sqrt{2})^{(\alpha-\frac{1}{2})\sigma_{3}}|x|^{\alpha\sigma_{3}}e^{x^{2}\widehat{g}(z)\sigma_{3}}=\begin{pmatrix}1&0\\\
-\frac{s_{0}e^{2\pi i\alpha}}{1+e^{2\pi i\alpha}}&1\end{pmatrix}.$ (6.57)

Substituting (6.5.2) and (6.56) into (6.5.2) and combining with the facts

$\begin{pmatrix}0&s_{1}\\\ -s_{1}^{-1}&0\end{pmatrix}=\begin{pmatrix}1&0\\\
-s_{1}^{-1}&1\end{pmatrix}\begin{pmatrix}1&s_{1}\\\
0&1\end{pmatrix}\begin{pmatrix}1&0\\\ -s_{1}^{-1}&1\end{pmatrix}$

and

$E_{0}=\begin{pmatrix}1&0\\\ \frac{s_{0}e^{2\pi i\alpha}}{1+e^{2\pi
i\alpha}}&1\end{pmatrix}=\begin{pmatrix}1&0\\\ -s_{1}^{-1}&1\end{pmatrix},$

which follows from (2.8) and (5.1), we have

$\displaystyle e^{\frac{x^{2}}{2}\sigma_{3}}$
$\displaystyle|x|^{\frac{\alpha}{2}\sigma_{3}}F(x)|x|^{\frac{3\alpha}{2}\sigma_{3}}e^{\frac{x^{2}}{2}\sigma_{3}}=\widehat{\mathbf{R}}(0)(2\pi)^{-\frac{\sigma_{3}}{2}}e^{\pi
i(\alpha-\frac{1}{2})\sigma_{3}}\Gamma(\alpha+\textstyle\frac{1}{2})^{\sigma_{3}}2^{-\alpha\sigma_{3}}e^{\pi
i(\frac{1}{2}-\alpha)\sigma_{3}}.$ (6.58)

Similarly, take $c<0$ and $d>0$ such that all real poles of
$q(x;\alpha,\kappa)$ lie in the interval $(c,d)$. Using the estimation (5.29)
and letting $x\to-\infty$ in (6.58), we obtain (1.4).

##  7 Proof of Theorem 1.6

Denote

$\displaystyle\mathbf{f}(\lambda)$
$\displaystyle=\begin{pmatrix}f_{1}(\lambda)\\\
f_{2}(\lambda)\end{pmatrix}=\begin{pmatrix}D_{\nu}\left(\sqrt{2}(\lambda+x)\right)\\\
-D_{\nu-1}\left(\sqrt{2}(\lambda+x)\right)\end{pmatrix},$ (7.1)
$\displaystyle\mathbf{h}(\lambda)$
$\displaystyle=\begin{pmatrix}h_{1}(\lambda)\\\
h_{2}(\lambda)\end{pmatrix}=\gamma\begin{pmatrix}D_{\nu-1}\left(\sqrt{2}(\lambda+x)\right)\\\
D_{\nu}\left(\sqrt{2}(\lambda+x)\right)\end{pmatrix},$ (7.2)

where $D_{\nu}$ is the parabolic cylinder function. Then, the parabolic
cylinder kernel (1.30) can be written as

$\gamma
K_{\nu,x}(\lambda,\mu)=\frac{\mathbf{f}(\lambda)^{T}\mathbf{h}(\mu)}{\lambda-\mu}.$
(7.3)

For general parameter $\nu\in\mathbb{R}$, the logarithmic derivative of
$\gamma K_{\nu,x}(\lambda,\mu)$ can be expressed in terms of the solution of
the following RH problem for $Y(\lambda):=Y(\lambda,x)$.

### RH problem for $Y(\lambda)$

  * (1)

$Y(\lambda)$ is analytic in $\mathbb{C}\setminus[0,\infty)$.

  * (2)

$Y(\lambda)$ satisfies the jump condition

$Y_{+}(\lambda)=Y_{-}(\lambda)\left(\mathbf{I}-2\pi
i\mathbf{f}(\lambda)\mathbf{h}(\lambda)^{T}\right),$ (7.4)

where $\mathbf{f}$ and $\mathbf{h}$ are defined by (7.1) and (7.2),
respectively.

  * (3)

The behavior of $Y(\lambda)$ at infinity is

$Y(\lambda)=\mathbf{I}+\frac{Y_{1}}{\lambda}+O\left(\frac{1}{\lambda^{2}}\right).$
(7.5)

  * (4)

The behavior of $Y(\lambda)$ at the origin is

$Y(\lambda)=O\left(\ln\lambda\right).$ (7.6)

Actually, the solution to the RH problem for $Y(\lambda)$ can be expressed as
follows [17]

$Y(\lambda)=\mathbf{I}-\int_{0}^{+\infty}\frac{\mathbf{V}(\mu)\mathbf{h}(\mu)^{T}}{\mu-\lambda}d\mu,$
(7.7)

where

$\mathbf{V}(\mu)=(\mathbf{I}-\gamma K_{\nu,x})^{-1}\mathbf{f}(\mu).$ (7.8)

The equation (7.3) implies

$\frac{d}{dx}\ln\det(\mathbf{I}-\gamma
K_{\nu,x})=-\mathrm{tr}\left(\left(\mathbf{I}-\gamma
K_{\nu,x}\right)^{-1}\gamma\frac{d}{dx}K_{\nu,x}\right).$ (7.9)

Using the recurrence relations satisfied by the parabolic cylinder function
(cf. [24, Equation (12.8)])

$D_{\nu}^{\prime}(\lambda)=\frac{\lambda}{2}D_{\nu}(\lambda)-D_{\nu+1}(\lambda),$
(7.10)

and

$D_{\nu}^{\prime}(\lambda)=-\frac{\lambda}{2}D_{\nu}(\lambda)+\nu
D_{\nu-1}(\lambda),$ (7.11)

we find after some direct calculation that

$\gamma\frac{d}{dx}K_{\nu,x}(\lambda,\mu)=-\mathbf{h}(\mu)^{T}\sigma_{3}\mathbf{f}(\lambda)=-\left(f_{1}(\lambda)h_{1}(\mu)-f_{2}(\lambda)h_{2}(\mu)\right).$
(7.12)

Substituting (7.12) into (7.9) and applying (7.7)-(7.8), we arrive at the
differential identity

$\frac{d}{dx}\ln\det(\mathbf{I}-\gamma
K_{\nu,x})=(Y_{1})_{11}-(Y_{1})_{22}=2(Y_{1})_{11},$ (7.13)

with $Y_{1}$ given in (7.5).

To proceed, we define

$P(z)=\begin{pmatrix}e^{i\frac{\pi}{2}\nu}D_{-\nu}(e^{i\frac{\pi}{2}}z)&D_{\nu-1}(z)\\\
\nu
e^{i\frac{\pi}{2}(\nu+1)}D_{-\nu-1}(e^{i\frac{\pi}{2}}z)&D_{\nu}(z)\end{pmatrix}$
(7.14)

for $\arg z\in(-\frac{1}{4}\pi,0)$. In view of the asymptotic behavior (1.4)
for $D_{\nu}$, we see that $P(z)$ satisfies the following asymptotic behavior
at infinity

$P(z)=\left(\mathbf{I}+\frac{P_{1}}{z}+O\left(\frac{1}{z^{2}}\right)\right)z^{-\nu\sigma_{3}}\exp\left(\frac{1}{4}z^{2}\sigma_{3}\right),$
(7.15)

where the diagonal entries of $P_{1}$ are zero. Define

$\widehat{\Psi}(\lambda)=e^{-\frac{1}{2}x^{2}\sigma_{3}}2^{\frac{\nu}{2}\sigma_{3}}Y(\lambda)^{-T}P\left(\sqrt{2}(\lambda+x)\right)\prod_{i=0}^{k-1}J_{i},$
(7.16)