Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

Abstract

We study the effect of the unfolding of a reducible double confluent Heun equation from the point of view of the Stokes phenomenon. We introduce a small complex parameter ε that splits together the non-resonant singular points x = 0 and $x=\infty$ into four different Fuchsian singularities $x_{L}=-\sqrt {\varepsilon }, x_{R}=\sqrt {\varepsilon }$, and $x_{LL}=-1/\sqrt {\varepsilon }, x_{RR}=1/\sqrt {\varepsilon }$, respectively. The perturbed equation is a symmetric general Heun equation and its general solution depends analytically on $\sqrt {\varepsilon }$. Then we prove that when the perturbed equation has exactly two resonant singularities of different type, all the Stokes matrices of the initial double confluent Heun equation are realized as a limit of the upper-triangular parts of the monodromy matrices of the perturbed equation when $\sqrt {\varepsilon } \rightarrow 0$. To establish this result we combine a direct computation with a theoretical approach.

Appendix.: Irregular Singularities, 1-summability and Regular Singularities

To facilitate the reader in this section we review some definitions, facts and notation from the theory of the ordinary differential equations with irregular and regular singularities, as well as from the applications of Borel-Laplace summation.

We write ${\mathbb N}_0={\mathbb N} \cup \{0\}$.

We firstly focus on the Borel-Laplace summation at the origin following the works of Loday-Richaud [22] and Ramis [26]. The Borel-Laplace summation at $z=\infty$ is reduced to summation at x = 0 after the change z = 1/x. The reader can find details for the summation at $x=\infty$ in the work of Sauzin [29].

We denote by ${\mathbb C}[[x]]$ the field of formal power series at the origin

$$ {\mathbb C}[[x]]=\left\{ \sum\limits_{n=0}^{\infty} f_{n} x^{n} | f_{n}\in{\mathbb C} \right\}. $$

Equipping ${\mathbb C}[[x]]$ with the natural derivation $\partial =\frac {d}{d x}$ such that

$$ \partial(\psi \varphi)=\partial (\psi) \varphi + \psi \partial(\varphi),\qquad \psi, \varphi\in{\mathbb C}[[x]] $$

we make ${\mathbb C}[[x]]$ a differential algebra. All singular directions and sectors are defined on the Riemann surface of the natural logarithm.

Definition 6.7

An open sector S with vertex 0 is a set of the form

$$ S=S(\theta, \alpha, \rho)=\left\{ x=r e^{i \delta} | 0 < r < \rho, \theta-\alpha/2 < \delta < \theta + \alpha/2\right\} . $$

Here 𝜃 is an arbitrary real number (the bisector of the sector), α is a positive real (the opening of the sector) and ρ is either a positive real number or $+\infty$ (the radius of the sector).

2. A closed sector $\bar {S}$ with vertex 0 is a set of the form

$$ \bar{S}=\bar{S}(\theta, \alpha, \rho)=\left\{ x=r e^{i \delta} | 0 < r \leq \rho, \theta-\alpha/2 \leq \delta \leq \theta + \alpha/2\right\} . $$

Here 𝜃 and α are as before, but ρ is a positive real number (never equal to $+\infty$).

Among formal power series ${\mathbb C}[[x]]$ we distinguish the formal power series of Gevrey order 1.

Definition 6.8

A formal power series $\hat {f}(x)={\sum }_{n=0}^{\infty } f_n x^n$ is said to be of Gevrey order 1 if there exist two positive constants C,A > 0 such that

$$ |f_{n}| < C A^{n} n! \quad \textrm{for every}\quad n\in{\mathbb N} . $$

We denote by ${\mathbb C}[[x]]_1$ the sets of all power series at x = 0 of Gevrey order 1. This set is a sub-algebra of ${\mathbb C}[[x]]$ as a commutative differential algebra over ${\mathbb C}$, [22, 30].

Definition 6.9

Let the function f(x) be holomorphic on a sector S(𝜃,α,ρ). The (formal) power series $\hat {f}(x)={\sum }_{n=0}^{\infty } f_n x^n$ is said to be 1-Gevrey asymptotic expansion of f(x) as $x \rightarrow 0$ in S, if for every closed sector $\bar {W} \subset S$ and all $N\in \mathbb {N}$ there exist constants C_W,A_W such that

$$ \left| f(x) - \sum\limits_{n=0}^{N-1} f_{n} x^{n} \right| \leq C_{W} {A^{N}_{W}} N! |x|^{N} ,\quad x\in W . $$

The constants C_W and A_W may depend on W but do not depend on N. A series that is 1-Gevrey asymptotic expansion of a function is said to be an 1-Gevrey asymptotic series. It can be proved that an 1-Gevrey asymptotic series is a series of Gevrey order 1. In this case one usually write

$$ f(x) \sim_{1} \hat{f}(x) ,\quad x\in S ,\quad x \rightarrow 0 . $$

Definition 6.10

The formal Borel transform $\hat {{\mathcal B}}_1$ of order 1 of a formal power series $\hat {f}(x)={\sum }_{n=0}^{\infty } f_n x^{n+1}$ is the formal series

$$ \hat{{\mathcal B}}_{1} \hat{f} (\xi)= \sum\limits_{n=0}^{\infty} \frac{f_{n}}{n!} \xi^{n} . $$

If $\hat {f}(x)\in {\mathbb C}[[x]]_1$ then its formal Borel transform $\hat {{\mathcal B}}_1$ converges in a neighborhood of the origin ξ = 0 with a sum f(ξ).

Definition 6.11

Let f(ξ) be analytic and of exponential size at most 1 at $\infty$, i.e., $|f(\xi )| \leq A \exp (B |\xi |), \arg (\xi ) \in \theta$ along a direction 𝜃 from 0 to $+\infty e^{i \theta }$. Then, the integral

$$ \left( \mathcal{L}_{\theta} f\right)(x)= {\int}_{0}^{+\infty e^{i \theta}} f(\xi) \exp\left( -\frac{\xi}{x}\right) d \xi $$

is said to be Laplace complex transform ${\mathscr{L}}_{\theta }$ of order 1 in the direction 𝜃 of f. The integral converges for any complex x in the open disc

$$ \mathcal{D}_{\theta}(A):= \left\{x\in{\mathbb C} | Re \left( \frac{e^{i \theta}}{x}\right) > A \right\}. $$

Moreover ${\mathcal L}_{\theta } f$ is a holomorphic function in $\mathcal {D}_{\theta }(A)$.

Let the function f(ξ) be analytic and satisfies the estimate

$$ \begin{array}{@{}rcl@{}} |f(\xi)| \leq A e^{B |\xi|},\quad B\in{\mathbb R} \end{array} $$

(A.1)

along a direction 𝜃 from 0 to $+\infty e^{i \theta }$. Then the Laplace complex transform ${\mathcal L}_{\theta }$ in the direction 𝜃 satisfies the following useful property.

Lemma 6.12

Let f as above, $\tilde {f}:=({\mathcal L}_{\theta } f)$. Then if moreover f is analytically differentiable along the direction 𝜃 with $\frac {d f}{d \xi }$ satisfying an estimate of the form (A.1)

$$ \left( {\mathcal L}_{\theta} (\frac{d f}{d \xi})\right)=\frac{1}{x} \tilde{f}(x) - f(0) . $$

Now we can give the definitions of the Borel-Laplace 1-summable series in a direction.

Definition 6.13

A series $\hat {f}\in {\mathbb C}[[x]]$ is said to be Borel-Laplace summable in a direction 𝜃 if the following two conditions are satisfied:

• 1. The formal Borel transform $\hat {{\mathcal B}}_1 \hat {f}(\xi )$ of $\hat {f}(x)$ is convergent, i.e., the series $\hat {f}(x)$ is of Gevrey order 1.

• 2. The sum h(ξ) of the Borel transform of $\hat {f}(x)$ can be analytically continued to a sector S neighboring the direction 𝜃 with exponential growth of order 1. We still denote by h its analytic continuation.

When these conditions are satisfied, the Borel-Laplace sum of $\hat {f}(x)$ in the direction 𝜃 is given by the Laplace integral $f_{\theta }(x)=({\mathcal L}_{\theta } h)(x)$.

Remark 6.14

If $\hat {f}(x)$ is convergent, then $\hat {f}(x)$ is k-summable in the direction 𝜃 (any k > 0 and any 𝜃) and the classical sum and k-sum coincide in their respective domain of definition, [26, 29].

In this paper we work with formal fundamental matrix solutions $\hat {\Phi }_0(x, 0)$ and $\hat {\Phi }_{\infty }(x, 0)$ of the initial equation in a slight different form of the theorem of Hukuhara-Turrittin [34]

$$ \begin{array}{@{}rcl@{}} \hat{\Phi}_{0}(x, 0) &=& \exp (G x) \hat{H}(x) x^{\Lambda} \exp\left( -\frac{B}{x}\right) ,\\[0.1ex] \hat{\Phi}_{\infty}(x, 0) &=& \exp \left( -\frac{B}{x}\right) \left( \frac{1}{x}\right)^{-{\Lambda}} \hat{P}(x) \exp (G x) . \end{array} $$

(A.2)

This special form of the formal fundamental matrix solutions, as well as, of the actual fundamental matrix solutions allows us to show in an explicit way haw these solutions are changed under the perturbation. Here the matrix function $x^{\Lambda }, \exp (G x)$ and $\exp (-B/x)$ must be regarded as formal series. The entries of the matrices $\hat {H}(x)$ and $\hat {P}(x)$ are formal power series in x and x^− 1, respectively. The matrices G,Λ and B are diagonal constant matrices as

$$ G={\text{diag}} (0, \gamma),\quad {\Lambda}={\text{diag}} (0, \lambda),\quad B={\text{diag}} (0, \beta) . $$

Suppose that the initial equation is not a reducible one but the points x = 0 and $x=\infty$ are still non-resonant irregular singular points of Poincarè rank 1. Then such an equation admits fundamental sets of solutions of the form

$$ w_{1}(x, 0)=x^{\lambda_{1}} e^{-\frac{\beta_{1}}{x}} \hat{w}_{1}(x),\quad w_{2}(x, 0)=x^{\lambda_{2}} e^{-\frac{\beta_{2}}{x}} \hat{w}_{2}(x) $$

at x = 0 and

$$ w_{1}(x, 0)=x^{\alpha_{1}} e^{\gamma_{1} x} \hat{u}_{1}(x),\quad w_{2}(x, 0)=x^{\alpha_{2}} e^{\gamma_{2} x} \hat{u}_{2}(x) $$

at $x=\infty$. The series $\hat {w}_j(x)$ (resp. u_j(x)) are either convergent or divergent power series in x (resp. x^− 1). Then

Definition 6.15

Under the above assumptions:

• 1. We define, relative to each divergent power series $\hat {w}_k(x)$, the singular direction 𝜃 at the origin as bisector of the angular sector where $Re \left (\frac {\beta _k-\beta _j}{x}\right ) < 0$. In particular $\theta =\arg (\beta _j-\beta _k)$.

• 2. We define, relative to each divergent power series $\hat {u}_k(x)$, the singular direction 𝜃 at $x=\infty$ as a bisector of the angular sector where $Re (e^{-i \arg (\gamma _k-\gamma _j)} x) > 0$. In particular $\theta =\arg (\gamma _k-\gamma _j)$.

Utilizing the summability theory we lift the formal fundamental matrix solutions $\hat {\Phi }_0(x, 0)$ and $\hat {\Phi }_{\infty }(x, 0)$ to actual fundamental matrix solutions. More precisely,

Theorem 6.16

(Hukuhara-Turrittin-Martinet-Ramis) The entries of the matrix $\hat {H}(x)$ (resp. $\hat {P}(x)$) in Eq. (A.2) are 1-summable in every non-singular direction 𝜃. If we denote by H_𝜃(x) (resp. P_𝜃(x)) the 1-sum of $\hat {H}(x)$ (resp. $\hat {P}(x)$) along 𝜃 obtained from $\hat {H}(x)$ (resp. $\hat {P}(x)$) by a Borel - Laplace transform, then ${\Phi }^{\theta }_0(x, 0)=e^{G x} H_{\theta }(x) x^{\Lambda } e^{-B/x}$ (resp. ${\Phi }^{\theta }_{\infty }(x, 0)=e^{-B/x} x^{\Lambda } P_{\theta }(x) e^{G x}$) is an actual fundamental matrix solution at the origin (resp. at $x=\infty$) of the initial equation.

Let 𝜃 be a singular direction of the initial equation at the origin. Let 𝜃⁺ = 𝜃 + χ and 𝜃⁻ = 𝜃 − χ, where χ > 0 is a small number, be two non-singular neighboring directions of the singular direction 𝜃. Denote by ${\Phi }^{\theta +}_k(x, 0)$ and ${\Phi }^{\theta -}_k(x, 0), k=0, \infty$ the actual fundamental matrix solutions of the initial equation corresponding to the direction 𝜃⁺ and 𝜃⁻ in the sense of Theorem 6.16. Then

Definition 6.17

With respect to the given formal fundamental matrix solution $\hat {\Phi }_k(x, 0)$ the Stokes matrix $St^{\theta }_k\in GL_2({\mathbb C})$ corresponding to the singular direction 𝜃 is defined as

$$ St^{\theta}_{k}=({\Phi}^{\theta+}_{k}(x, 0))^{-1} {\Phi}^{\theta-}_{k}(x, 0),\quad k=0, \infty . $$

The considered in this paper perturbed equation is a Fuchsian equation of order 2 with 4 finite singular points taken at $x_R=\sqrt {\varepsilon }, x_L=-\sqrt {\varepsilon }, x_{RR}=1/\sqrt {\varepsilon }$ and $x_{LL}=-1/\sqrt {\varepsilon }$. Now we recall the necessary local theory of Fuchsian singularities following the book of Golubev [12]. Recall that for a scalar differential equation, a singularity is a regular one if and only if it is a Fuchsian one. With every Fuchsian (regular) singularity of a given n-order scalar linear differential equation we associate an n-order algebraic equation, the so-called characteristic (or indicial) equation. More precisely, consider a n-order linear differential equation

$$ y^{(n)}(x) + b_{n-1}(x) y^{(n-)}(x)+ {\cdots} + b_{0}(x) y(x)=0,\quad b_{j}(x)\in{\mathbb C}(x) . $$

Let $x=x_0\in {\mathbb C}$ be a regular singularity for this equation. Recall that due to the theorem of Fuchs this means that all the functions b_n−k(x)(x − x₀)^k are holomorphic functions at x = x₀. Then

Definition 6.18

• 1. The n-order algebraic equation

  $$ \rho (\rho-1) (\rho-2) {\ldots} (\rho-(n-1)) + c_{n-1} \rho (\rho-1){\ldots} (\rho-(n-2)) + {\cdots} + c_{1} \rho + c_{0}=0 , $$

  where

  $$ c_{k}=\lim_{x \rightarrow x_{0}} b_{n-k}(x) (x-x_{0})^{k},\quad 0 \leq k \leq n-1 $$

  is called the characteristic (or the indicial) equation at the regular singularity $x_0\in {\mathbb C}$. Its roots ρ_k, 1 ≤ k ≤ n are called the characteristic exponents at the singularity x₀.

• 2. The characteristic equation at the point t = 0 of the equation obtained after the transformation x = 1/t, is called the characteristic equation at $x=\infty$. Its roots are called the characteristic exponents at the regular point $x=\infty$.

Denote by $\rho ^j_i$ and $\rho ^{jj}_i, i=1, 2, j=R, L$ the characteristic exponents at the singular points x_j and x_jj,j = R,L, respectively. In [32], Proposition 4.6 we have proved that if we know the coefficients of the differential operators L_j,ε then we can directly determine the characteristic coefficients at every singular points. The restriction of Proposition 4.6 to the perturbed equation leads to

Proposition 6.19

The coefficients of the operators L_j,ε in Eq. (1.7) are uniquely determined only by the characteristic exponents

$$ \begin{array}{@{}rcl@{}} L_{1, \varepsilon} &=& \partial - \left( \frac{{\rho^{R}_{1}}}{x-x_{R}} + \frac{{\rho^{L}_{1}}}{x-x_{L}} + \frac{\rho^{RR}_{1}}{x-x_{RR}} + \frac{\rho^{LL}_{1}}{x-x_{LL}}\right)\\[0.2ex] L_{2, \varepsilon} &=& \partial - \left( \frac{{\rho^{R}_{2}}-1}{x-x_{R}} + \frac{{\rho^{L}_{2}}-1}{x-x_{L}} + \frac{\rho^{RR}_{2}-1}{x-x_{RR}} + \frac{\rho^{LL}_{2}-1}{x-x_{LL}}\right) . \end{array} $$

Thanks to Proposition 6.19 the characteristic exponents $\rho ^j_i$ and $\rho ^{jj}_i, i=1, 2, j=R, L$ are

$$ \begin{array}{@{}rcl@{}} {\rho^{R}_{1}}=0, {\rho^{R}_{2}}=\frac{\beta}{2 \sqrt{\varepsilon}}; & & {\rho^{L}_{1}}=0, {\rho^{L}_{2}}=-\frac{\beta}{2 \sqrt{\varepsilon}};\\[0.4ex] \rho^{RR}_{1}=0, \rho^{RR}_{2}=1-\frac{\gamma}{2 \sqrt{\varepsilon}}; & & \rho^{LL}_{1}=0, \rho^{LL}_{2}=1+\frac{\gamma}{2 \sqrt{\varepsilon}} . \end{array} $$

The exponents differences ${\Delta }^j_{12}=\rho ^j_1-\rho ^j_2$ and ${\Delta }^{jj}_{12}=\rho ^{jj}_1-\rho ^{jj}_2$ corresponding to the above characteristic exponents are defined as follows,

$$ \begin{array}{@{}rcl@{}} {\Delta}^{R}_{12}=-\frac{\beta}{2 \sqrt{\varepsilon}},\quad {\Delta}^{L}_{12}=\frac{\beta}{2 \sqrt{\varepsilon}},\quad {\Delta}^{RR}_{12}=-1+\frac{\gamma}{2 \sqrt{\varepsilon}},\quad {\Delta}^{LL}_{12}=-1 - \frac{\gamma}{2 \sqrt{\varepsilon}} . \end{array} $$

The local theory of the Fuchsian singularities ensures a necessary condition for the existence of the logarithmic term near the singular point x_j or x_jj in the solution w₂(x,ε) of the perturbed equation. If the exponent difference ${\Delta }^j_{12}\in {\mathbb N}_0$ or ${\Delta }^{jj}_{12}\in {\mathbb N}_0$ then the solution w₂(x,ε) can contain logarithmic term near the point x_j or x_jj, respectively. Classically, such a Fuchsian singularity x_j (resp. x_jj) for which ${\Delta }^j_{12}\in \mathbb {N}_0$ (resp. ${\Delta }^{jj}_{12}\in {\mathbb N}_0$) is called a resonant Fuchsian singularity.