On the Borel summability of WKB solutions of certain Schr\"odinger-type differential equations

A class of Schr\"odinger-type second-order linear differential equations with a large parameter $u$ is considered. Analytic solutions of this type of equations can be described via (divergent) formal series in descending powers of $u$. These formal series solutions are called the WKB solutions. We show that under mild conditions on the potential function of the equation, the WKB solutions are Borel summable with respect to the parameter $u$ in large, unbounded domains of the independent variable. It is established that the formal series expansions are the asymptotic expansions, uniform with respect to the independent variable, of the Borel re-summed solutions and we supply computable bounds on their error terms. In addition, it is proved that the WKB solutions can be expressed using factorial series in the parameter, and that these expansions converge in half-planes, uniformly with respect to the independent variable. We illustrate our theory by application to a radial Schr\"odinger equation associated with the problem of a rotating harmonic oscillator and to the Bessel equation.


Introduction and main results
In this paper, we study Schrödinger-type differential equations of the form where u is a real or complex parameter, z lies in some domain D of a Riemann surface, and the potential function f (u, z) is an analytic function of z having the form f (u, z) = f 0 (z) + f 1 (z) u + f 2 (z) u 2 . It is assumed that the coefficient functions f n (z) are analytic functions of z in the domain D, bounded or otherwise, and are independent of u. Furthermore, we suppose that f 0 (z) does not vanish in D. We are interested in asymptotic expansions of analytic solutions of (1.1) when the parameter u becomes large. It is convenient to work in terms of the transformed variables ξ and W (u, ξ) given by 1 The path of integration, except perhaps its endpoint z 0 , must lie entirely within D. In the case that z 0 is a boundary point of D, it is assumed that the integral converges as t → z 0 along the contour of integration. The transformation (1.2) maps D on a domain G, say. In terms of the new variables, equation (1.1) becomes where the coefficients A ± n (ξ), which are analytic functions of ξ in G, can be determined recursively by (1.5) A ± n+1 (ξ) = − with the convention A ± 0 (ξ) = 1. The constants of integration in (1.4) and (1.5) are arbitrary. An alternative method for the evaluation of the coefficients A ± n (ξ), which avoids nested integrations, can be found in Appendix A.
The formal solutions (1.4) are usually referred to as WKB solutions, in honour of physicists Wentzel [37], Kramers [24] and Brillouin [7], who independently discovered these solutions in a quantum mechanical context in 1926. To be strictly accurate, the WKB solutions were discussed earlier by Carlini [8] in 1817, by Liouville [25] and Green [18] in 1837, by Strutt (Lord Rayleigh) [31] in 1912, and again by Jeffreys [21] in 1924. The reader is referred to the book of Dingle [12, Ch. XIII] for a historical discussion and critical survey.
In general, the series in (1.4) diverge, and the most that can be established is that in certain subregions of G they provide uniform asymptotic expansions of solutions of the differential equation (1.3). Olver [30,Ch. 10, §9] demonstrated the asymptotic nature of the expansions (1.4) via the construction of explicit error bounds. In this paper we take a different approach and study the WKB solutions (1.4) from the point of view of Borel summability. Our main aim is to show that the expansions (1.4) are Borel summable in well-defined subdomains Γ ± of G, provided that φ(ξ) and ψ(ξ) satisfy certain mild requirements. We shall construct exact, analytic solutions W ± (u, ξ) of the differential equation (1.3) in terms of Laplace transforms (with respect to the parameter u) of some associated functions, called the Borel transforms of the WKB solutions. The formal expansions in (1.4) will then arise as asymptotic series of these exact solutions. By carefully analysing the Borel transforms, we will derive new, computable bounds for the remainder terms of the asymptotic expansions. In addition, it will be shown that the WKB solutions can be expressed using factorial series in the parameter u, and that these expansions converge for u > 0, uniformly in the independent variable ξ.
The present work is a first step towards the global analysis of the WKB solutions (1.4). Our main result demonstrates that the WKB solutions are Borel summable provided that one stays away from Stokes curves emerging from transition points of f 0 (z) (see Remark (i) following Theorem 1.1). In particular, the problem of global connection formulae for WKB solutions across Stokes curves is not studied in this paper. A possible approach to tackle such problems is discussed briefly in Section 7.
Over the past several decades, there has been increasing interest in the study of Borel summability of asymptotic series, and this field is closely linked with exponential asymptotics (see, for example, [28]). Indeed, there have been extensive developments in summability of singular variable asymptotic expansions, with fewer general results currently existing for the more complicated singular parameter case. The relevance of Borel summability to the WKB analysis was first clearly observed by Bender and Wu [4]. In the seminal work [35], Voros showed how to analyse Borel re-summed WKB solutions and demonstrated the relationship between the singular points of the Borel transforms and the global connection formulae. Dunster et al. [13] studied differential equations of the type (1.3) in the case that φ(ξ) ≡ 0 (or, equivalently, f 1 (z) ≡ 0). Under suitable conditions on ψ(ξ), they established the Borel summability of WKB solutions away from Stokes curves and derived their convergent factorial series expansions. Their analysis was based on a method of Nørlund. Kamimoto and Koike [22] considered the equation (1.1) in the special case that f (u, z) = f (z) is meromorphic and independent of u, and proved the Borel summability of the WKB solutions using WKB theoretic transformation series introduced originally by Aoki et al. [1]. Their results hold in appropriate neighbourhoods of Stokes curves issuing from either a simple turning point or a simple pole of f (z) (see also [32]). For further contributions, see, for example, [2,3,5,9,11,15,16,19,20,33,34] and the references therein.
Before stating our results, we introduce the necessary assumptions, notation and definitions. Hereinafter, we will suppose that the functions φ(ξ) and ψ(ξ) fulfil at least one of the following two conditions. Condition 1.1. The functions φ(ξ) and ψ(ξ) are analytic in a domain ∆ ⊂ G, which has the property that there exist positive constants c and ρ independent of u and ξ, such that Condition 1.2. The functions φ(ξ) and ψ(ξ) are analytic in a domain ∆ ⊂ G, which has the property that there exist positive constants c and ρ independent of u and ξ, such that Often these conditions can be brought about by a suitable normalisation of the differential equation (1.3). (See the second example in Section 6, below.) We denote by Γ ± (d) any (non-empty) subdomains of ∆ that satisfy the following two requirements: (i) The distance between each point of Γ ± (d) and each boundary point of ∆ has a positive lower bound d (which is to be chosen independently of u). 2 (ii) If ξ ∈ Γ ± (d), then ξ ∓ x ∈ Γ ± (d) for all x > 0. Note that condition (ii) requires ∆ to contain at least one infinite strip that is parallel to the real axis. In particular, ∆ (and hence G) has to be unbounded. We shall often integrate along half-lines that are parallel to the real axis. For any given w ∈ Γ + (d), P(−∞, w) denotes the half-line that runs from −∞ + i w to w. Observe that by condition (ii), P(−∞, w) lies entirely in Γ + (d). Similarly, for any w ∈ Γ − (d), P(w, +∞) will denote the half-line that connects w with +∞ + i w. If the orientations of these paths are reversed, we will adopt the notation P(w, −∞) and P(+∞, w), respectively.
2 It will be assumed throughout the paper that d is always chosen so that none of the domains Γ ± (d) is empty.
In (1.6), the contours of integration have to lie entirely within G and can be infinite provided the integrals converge. It is readily seen that if such solutions exist, they must be unique. We would like our solutions to posses asymptotic expansions of the form as |u| → +∞, uniformly with respect to ξ ∈ Γ ± (d). The coefficients A ± n (ξ) must satisfy recurrence relations of the type (1.5) with a suitable choice of integration constants. Note that (1.7) implies lim ξ→∓∞ A ± n (ξ) = 0. We will show in Section 2 that if A ± 0 (ξ) = 1 and for n ≥ 0 and ξ ∈ Γ ± (d), then the requirements lim ξ→∓∞ A ± n (ξ) = 0 are fulfilled. The A ± n (ξ)'s defined in this way will be the coefficients in the asymptotic expansions (1.8) of our solutions (1.6), and throughout the rest of the paper, unless stated otherwise, A ± n (ξ) will always refer to the coefficients specified by the above recurrence relation.
Finally, we define, for any r > 0, Thus, U (r) consists of all points whose distance from the positive real axis is strictly less than r. We are now in a position to formulate our main result.
(i) In order to gain a better understanding of the main result, consider the following special case. Assume that φ(ξ) and ψ(ξ) are (possibly multivalued) holomorphic functions in the ξ-plane, save for a countable number of singularities (which may include branch points), located at ξ = ξ k , k ∈ Z ≥0 . Let us focus on the Borel summability of the WKB solution W + (u, ξ). We introduce brunch cuts from the singularities ξ k to +∞ + i ξ k . Suppose that φ(ξ) and ψ(ξ) are analytic in the resulting domain and satisfy the estimates posed in Conditions 1.1 or 1.2. Then we can set The pre-image in the z-plane of each of the brunch cuts is a Stokes curve emanating from a transition point of f 0 (z) (see, e.g., [23] are also solutions of the differential equation (1.3) for all sufficiently large values of |u|. The right-hand side can be re-expressed in the form where, for each 0 < r < d, the functions f ± (t, ξ, β ± ) are analytic in (t, ξ) ∈ U (2r) × Γ ± (d), and the Laplace transforms converge for u > K ± + σ + σ, where σ is any positive number and K ± = K ± (σ, r) are the corresponding constants given in Theorem 1.
To prove Theorem 1.1, we shall also need an inequality analogous to (1.13) but this inequality will be derived as a direct consequence of Conditions 1.1 or 1.2.
In Theorem 1.2 below, we give explicit bounds for the error terms of the asymptotic expansions (1.12). To state the theorem, we require some notation. For any σ > 0 and ξ ∈ Γ ± (d), we define (1.14) and for any 0 < r < d, we set where f ± (t, ξ) are the Borel transforms appearing in Theorem 1.1. For an upper bound on the quantities C ± , see Section 4. With these notation, we have the following theorem.
To obtain sharp bounds for R ± N (u, ξ), we may choose the parameters σ and r for each u and ξ separately. For example, one can take σ = 1 2 u and r to be any number that is less than the distance of ξ to the boundary of ∆.
In the following theorem, we provide convergent alternatives to the asymptotic expansions (1.12). The coefficients in these expansions depend on an additional (positive) parameter ω and can be generated by the recurrence relations Note that the B ± n (ω, ξ)'s are polynomials in ω of degree n − 1 and are analytic functions of ξ in Γ ± (d). A different expression for these coefficients involving the Stirling numbers of the first kind is given by (5.3).
which are absolutely convergent for u > σ > 0 (σ arbitrary), uniformly for ξ ∈ Γ ± (d). The errors committed by truncating the series (1.18) after N ≥ 0 terms do not exceed in absolute value The quantities C ± are defined by (1.15) with r = π 4ω . The remaining part of the paper is structured as follows. The proof of Theorem 1.1 consists of two steps. In Section 2, we prove that the formal Borel transforms of the WKB expansions converge in a neighbourhood of the origin. This is shown via some Gevrey-type estimates for the coefficients A ± n (ξ). The second step involves analytically continuing the Borel transforms in a strip containing the positive real axis. We construct the continuation as a solution of an integral equation in Section 3, following ideas of Dunster et al. [13]. In Section 4, we provide the proof of the error bounds given in Theorem 1.2. The convergent factorial expansions and the corresponding error bounds stated in Theorem 1.3 are proved in Section 5. In Section 6, we illustrate our results by application to a radial Schrödinger equation associated with the problem of a rotating harmonic oscillator and to the Bessel equation. The paper concludes with a discussion in Section 7.

Pre-Borel summability of the WKB solutions
In this section, we prove the pre-Borel summability of the WKB solutions, i.e., we show that the formal Borel transforms of the WKB expansions (1.8) are analytic near the origin.
In order to prove Theorem 2.1, we shall establish a series of lemmata.
Lemma 2.1. Let d > 0. Assume Conditions 1.1 or 1.2. Then there exists a positive constant c 1 , independent of u and ξ, such that for any non-negative integer j, the following estimates hold: Proof. We shall prove that Lemma 2.1 holds with and let j be a non-negative integer. To prove the first inequality in (2.2), we use Conditions 1.1 or 1.2 and Cauchy's formula, to deduce Here µ = 1 or µ = 1/2 according to whether we assume Conditions 1.1 or 1.2. Now, when |ξ| ≤ 1 + d, We also have which, together with (2.3), imply the first inequality in (2.2). The other bounds in (2.2) can be proved in a similar manner by applying the inequalities which hold for all t ∈ ∆. Here, again, we take µ = 1 or µ = 1/2 according to whether we assume Conditions 1.1 or 1.2.
Then there exists a positive constant c 2 , independent of u and ξ, such that the following estimates hold: Proof. We shall show that Lemma 2.1 holds with c 2 = 4 max(c, Therefore, we have the following series of estimates: If, instead, Condition 1.2 holds, then Now, when ∓ ξ > 1, These two estimates, together with (2.5) or (2.6) and the choice of c 2 , imply the first inequality in (2.4). The third estimate in (2.4) can be obtained in a similar way by employing the inequality To prove the second inequality in (2.4), we can proceed as follows. Using Condition 1.1 and Cauchy's formula, we derive and when |ξ| ≥ 1 + d, Hence, from (2.7), we can infer that With our choice of the constant c 2 , this bound implies the second inequality in (2.4). If, instead, Condition 1.2 is assumed, then By the choice of c 2 , this bound implies the second inequality in (2.4). Proof. The identity can be proved by re-writing the left-hand side as a telescoping sum (with the convention that 1/(−1)! = 0): Proof of Theorem 2.1. We begin by proving that for ξ ∈ Γ ± (d), n ≥ 1 and m ≥ 0, it holds that for n ≥ 1. We proceed via induction on n. By definition, Employing Lemmas 2.1 and 2.2, together with the definitions of c 3 and C 1 (d), we deduce Now let m be an arbitrary positive integer. Differentiating (2.12) m times yields By an application of Lemma 2.1, and the definitions of c 3 and C 1 (d), we obtain Assume that (2.9) holds for all d m A ± k (ξ)/dξ m with 1 ≤ k ≤ n and m ≥ 0. By definition, (2.13) Using the induction hypothesis, Lemmas 2.1 and 2.2, and the definition of C n+1 (d), we deduce Now let m be an arbitrary positive integer. Differentiating (2.13) m times gives Using the induction hypothesis, Lemmas 2.1 and 2.3, and the definition of C n+1 (d), we obtain This completes the proof of the bound (2.9). It is straightforward to show that C n (d) grows at most polynomially in n (see Section 4). Hence, we obtain from (2.9) that lim sup for any ξ ∈ Γ ± (d). Therefore, by the Cauchy-Hadamard theorem, the power series (2.1) are convergent when t ∈ B(2d), and define analytic functions g ± (t, ξ) on B(2d) (for each fixed ξ ∈ Γ ± (d)). From (2.9) we can also infer that the series (2.1) converge uniformly on Γ ± (d) and hence the functions g ± (t, ξ) are analytic with respect to ξ for each fixed t. Since the functions g ± (t, ξ) are continuous in (t, ξ) and are analytic in each of their variables, they are analytic functions in B(2d) × Γ ± (d).

Borel summability of the WKB solutions
In this section, we shall show that the functions η ± (u, ξ) can be represented as Laplace transforms of some associated functions f ± (t, ξ) that are analytic in U (2d) × Γ ± (d). These associated functions will coincide with g ± (t, ξ) of Theorem 2.1 in their common domains of definition. The proof follows closely that in [13]: we make a Laplace "ansatz" to derive a pair of partial differential equations for f ± (t, ξ) with certain boundary conditions. Thus, we seek f ± (t, ξ) such that where u > 0 and ξ ∈ Γ ± (d). By partial integration one easily finds that if f ± (t, ξ) satisfy and the conditions and hence (1.6) satisfy the equation (1.3). We seek solutions of (3.1) in U (2d) × Γ ± (d) such that for all t ∈ U (2d). This is consistent with the requirement (1.7). Using (1.9) and Theorem 2.1, one can show that g ± (t, ξ) are solutions of (3.1) on B(2d) × Γ ± (d). In order to show that g ± (t, ξ) can be continued analytically to the whole of U (2d) × Γ ± (d) and are of exponential type in t as t → +∞ in U (2d), we will transform (3.1) into an integral equation. In what follows, we discuss the construction of f − , an analogous argument can be used to construct f + . We define temporarily Now, we integrate the equation (3.4) in τ from ζ to t and in w from ξ + 1 2 ζ to infinity. Applying the limit condition (3.5) and integrating once by parts, we obtain Here ζ ∈ B(2d), t ∈ U (2d), and ξ, ξ + 1 2 (ζ − t) ∈ Γ − (d). It is not easy to use this integral equation and the initial condition in (3.5) directly to prove the existence of a solution of (3.1) (satisfying (3.3)). Since we would like our solution to coincide with the solution g − (t, ξ) defined in Theorem 2.1 when t ∈ B(2d), and the only restriction on ξ is ξ ∈ Γ − (d), we express t as x + ζ, where x ≥ 0, replace ξ by ξ + 1 2 x, and f − ζ, ξ + 1 2 x by g − ζ, ξ + 1 2 x to obtain the following linear integral equation Note that if t ∈ B(2d), then g − (t, ξ) is a solution of (3.6). Also observe from the definition of for all x ≥ 0. Equation (3.6) will eventually be used to define f − (t, ξ).
Let σ be an arbitrary (fixed) positive number. Denote by B σ the complex vector space of continuous functions h(x, ξ) on R ≥0 × Γ − (d) such that for each h ∈ B σ there exist a constant K (depending only on σ) such that where G − (σ, ξ) is defined in (1.14). Let us define the norm h of h(x, ξ) to be the infimum of all such K for which the inequality (3.7) holds. Equipped with this norm, B σ becomes a Banach space. In the following, we shall use the facts that for fixed ξ ∈ Γ − (d), s → G − (σ, ξ + s) is a monotonically decreasing function, lim s→+∞ G − (σ, ξ + s) = 0 and (We will assume that φ(t) ≡ 0. The case of φ(t) ≡ 0 can be handled in a similar manner.) Consider the operator acting on the space B σ . By the Cauchy-Schwarz inequality, we can assert that Consequently, We also have Thus, L is a B σ → B σ linear operator whose induced operator norm L ≤ 1 √ 6 + 1 12 < 1 2 . Now, for each ζ ∈ B(2d), define the operator T ζ , acting on the space B σ , via Since max(1, sgn( ξ) | ξ| ρ )g − ζ, ξ + 1 2 x ≤ max 1, sgn ξ + 1 2 x ξ + 1 2 x ρ g − ζ, ξ + 1 2 x and the right-hand side is bounded (cf. Theorem 2.1), T ζ h(x, ξ) belongs to the space B σ . We also have for any h 1 (x, ξ), h 2 (x, ξ) ∈ B σ . Consequently, T ζ is a B σ → B σ contraction. Therefore, for each ζ ∈ B(2d) there is a unique function f (x, ξ; ζ) in B σ defined on R ≥0 × Γ − (d) which satisfies To complete the construction of f − (t, ξ), it remains to show that for ζ ∈ B(2d) these functions of x can be combined to yield an analytic solution of (3.6) on U (2d) × Γ − (d). We first claim that when ξ ∈ Γ − (d), ζ ∈ B(2d), x ≥ 0 and x + ζ ∈ B(2d). This follows by uniqueness since each side is a solutions of (3.8), and adding the restriction x + ζ ∈ B(2d) to the definition of B σ and L does not increase the norm. Then, for all real numbers δ > 0 and ζ + δ ∈ B(2d), we find from (3.8) (with x replaced by x + δ) and (3.9) that (3.10) Now, regarding (3.6) as an equation satisfied by g − (x + ζ, ξ), and in that equation replacing x by δ and ξ by ξ + 1 2 x, we see that the first three terms on the right-hand side of (3.10) can be replaced by g − ζ + δ, ξ + 1 2 x . Then, if we replace τ by τ + δ in the third and fourth integrals and w by w + 1 2 δ in the fourth integral on the right-hand side of (3.10), we deduce Hence, by uniqueness, it follows from (3.8) and (3.11) that for all real x ≥ 0, δ > 0 and ζ, ζ + δ ∈ B(2d) since each is a solutions of the same integral equation. The equality (3.12) shows that is well defined for ξ ∈ Γ − (d), ζ ∈ B(2d) and x ≥ 0. Let t = x + ζ be a point in U (2d). We regard x ≥ 0 as fixed and let ζ varying in a neighbourhood of the origin. Since L is independent of ζ, and differentiation with respect to ζ commutes with L, it follows that f (x, ξ; ζ) is an analytic function of ζ. Hence f − (t, ξ) is analytic with respect to t in U (2d) (for each fixed ξ ∈ Γ − (d)), with ∂f − /∂t = ∂f /∂ζ. Since f − (t, ξ) is also continuous in (t, ξ) and analytic in ξ for each fixed t, it follows that Now, by Theorem 2.1, for each 0 < r < d, there exists a number C − > 0, independent of ζ, ξ and x, such that when ζ ∈ B(2r), ξ ∈ Γ − (d) and x ≥ 0. Thus, for each ζ ∈ B(2r), g − ζ, ξ + 1 2 x ∈ B σ and g − ζ, ξ + 1 2 x ≤ 1 2 C − for all σ > 0. Therefore, by (3.8), for ζ ∈ B(2r), ξ ∈ Γ − (d) and x ≥ 0. Upon expressing t ∈ U (2r) in the form t = t + ζ with ζ ∈ B(2r), we obtain (3.15) for all ξ ∈ Γ − (d). The finiteness of the supremum on the right-hand side is guaranteed by Lemma 2.2. This proves (1.11) and thus the convergence of (1.10). The validity of the asymptotic expansions (1.12) follows from Theorem 1.2 which we prove in the forthcoming section.

Error bounds
In this section, we prove the error bounds given in Theorem 1.2. Let N be an arbitrary positive integer. Integrating by parts N times in (1.10), we obtain (1.16) with Let 0 < r < d and σ > 0. From (3.13)-(3.15) and the corresponding results for f + (t, ξ), we can infer that if for all (t, ξ) ∈ U (2r) × Γ ± (d). Thus, by Cauchy's formula, for all t ≥ 0 and ξ ∈ Γ ± (d). Employing this estimate in (4.1) yields It is possible to obtain a simple upper bound for the quantities C ± by referring to the results in Section 2. From the definitions of C ± and g ± (t, ξ), and the estimate (2.9), we can assert that Further simplification is possible by bounding the C n (d)'s. To this end, we note that 1 + x < e x for all x > 0, and for all n ≥ 1 (cf. [30,Ch. 8,§3,Ex. 3.3]). Therefore, using the definitions (2.10) and (2.11), we deduce gives a computable upper bound on C ± .

Convergent factorial series
In this section, we prove the convergent factorial expansions and the corresponding error bounds stated in Theorem 1.3. Let d > 0 and ω > π 4d , and define r > 0 by ω = π 4r . Denote by Λ(2r) the region in the t-plane defined by According to Theorem 1.1, the functions f ± (t, ξ) are analytic in Λ(2r) × Γ ± (d) and satisfy the estimate (1.11) for any fixed σ > 0. Consequently, from the fundamental theorem on factorial series [36,Ch. 11,Theorem 46.2], the functions η ± (u, ξ) posses expansions of the form (1.18) which are absolutely convergent for u > σ > 0 (σ arbitrary), for each ξ ∈ Γ ± (d). It therefore remains to show that the convergence is uniform in ξ. To this end, we show that for ξ ∈ Γ ± (d), 0 < σ < ω and n ≥ 0, it holds that where the implied constants are independent of ξ and n. The coefficients B ± n (ω, ξ) are the following series expansion coefficients with t ∈ Λ(2r) and ξ ∈ Γ ± (d) (cf. [36, pp. 325-326]). Hence, by Cauchy's integral formula, where the path of integration is a small loop that encircles the origin in the positive sense. Next, on nothing the bound (1.11) (and assuming 0 < σ < ω), we deform the contour of integration by expanding it to the boundary of the domain Λ(2r). Then we split the resulting integral into two parts, the first from t = +∞ + π 2ω i to t = − 1 ω log 2, and the second from t = − 1 ω log 2 to t = +∞ − π 2ω i. On making the change of integration variables s = i log(1 − e −ωt ), we obtain 2 |s| π −σ/ω provided 0 < |s| < π, with the constants K ± being independent of s and ξ. Employing these estimates in (5.2) and appealing to Lemma 12.3 and Ex. 12.2 of Olver [30, pp. 99-100], we deduce the desired result (5.1). Therefore, if 0 < σ < ω, the estimates (5.1) and the known asymptotics for the ratio of two gamma functions [29, Eq. 5.11.12] give where the implied constants are independent of ξ and n. An appeal to the Weierstrass M -test establishes that the series (1.18) indeed converge uniformly for ξ ∈ Γ ± (d) provided u > σ.

Applications
In this section, we give two illustrative examples to demonstrate the applicability of our theory.
6.1. A radial Schrödinger equation. As a first application of the theory, we consider the radial Schrödinger equation which is associated with the rotating harmonic oscillator. In physical applications, the parameters u, λ and are real and non-negative, with an integer, and u large (see, e.g., [17]). We introduce a branch cut along the real axis from 1 to −∞ so that z is restricted to D = {z : |arg(z − 1)| < π}. Application of the transformation The domain D is mapped into the sectorial region G = {ξ : |arg ξ| < 2π} on the universal covering of C \ {0}.
We remark that Dunster [14] considered the problem of finding rigorous asymptotic expansions of the eigenvalues of the rotating harmonic oscillator. His analysis relies on WKB theoretic methods applied to the equation (6.1), although his definition of the WKB solutions is different from ours.
6.2. The Bessel equation. As a second application, we consider the Bessel equation where w(ν, z) = z 1/2 H To overcome this obstruction we replace z by νz, and consider instead the equation which has particular solutions w(ν, z) = z 1/2 H (1) ν+κ (νz) and w(ν, z) = z 1/2 H ν+κ (νz). This equation can now be treated by means Theorem 1.1. Similarly to the first example, we make a branch cut along the real axis from 1 to −∞ and restrict z to the domain D = {z : |arg(z − 1)| < π}. The transformation The image of D under the mapping (6.5) can be determined by the following considerations: (i) When z > 1, ξ is purely imaginary; z = 1, +∞ corresponding to ξ = 0, i∞, respectively.

Discussion
We studied the Borel summability of WKB solutions of certain Schrödinger-type differential equations with a large parameter. It was shown that under mild requirements on the potential function of the equation, the WKB solutions are Borel summable with respect to the large parameter in vast, unbounded domains of the independent variable. We demonstrated that the formal WKB expansions are the asymptotic expansions, uniform with respect to the independent variable, of the Borel re-summed solutions and supplied computable bounds on their remainder terms. In addition, it was proved that the WKB solutions can be expressed using factorial series in the parameter, and that these expansions converge in half-planes, uniformly in the independent variable.
The theory presented here is a first step towards the global analysis of the WKB solutions of the differential equation (1.1). The main result of the paper demonstrates that the WKB solutions are Borel summable provided that the independent variable is bounded away from Stokes curves emerging from transition points of f 0 (z). In particular, it does not give any information regarding connection formulae joining the WKB solutions at either side of the Stokes curves.
In what follows, we discuss briefly a possible extension of the theory to turning point problems. Turning points are the simplest types of transition points. Consider the differential equation (1.1), and suppose that the functions f n (z) are analytic in a domain D and f 0 (z) vanishes at exactly one point z 0 , say, of D. For the sake of simplicity, we assume that z 0 is a simple zero of f 0 (z), i.e., z 0 is a simple turning point of (1.1). Following Olver [30,Ch. 11, §11], we make the transformations .
The functions Φ(ζ) and Ψ(ζ) are holomorphic in the corresponding ζ domain H, say. The transformation (7.1) maps the simple turning point z 0 into the origin in H, and the three Stokes curves issuing from z 0 are mapped into the rays arg ζ = 0, ± 2π 3 i. The works of Boyd [6] and Olver [30,Ch. 11, §11] on turning point problems suggest that, under suitable assumptions on Φ(ζ) and Ψ(ζ), the set of all solutions of the differential equation (7.2) is of the form where Ai denotes any solution of Airy's equation. We expect that, for sufficiently large values of |u|, the coefficient functions A (u, ζ) and B(u, ζ) are holomorphic functions of ζ in an appropriate subdomain of H including ζ = 0 and the Stokes rays arg ζ = 0, ± 2π 3 i, and as functions of u are described asymptotically by descending power series in u. Consequently, the continuation formulae for solutions of the form (7.3) follow directly from those for the Airy functions. Then, with the aid of the transformations ξ = 2 3 ζ 3/2 , W (u, ξ) = ζ 1/4 W(u, ζ), connection formulae can be established for formal solutions of the form (1.4). It is also possible to use the foregoing analysis to extend the Borel summability results of the WKB solutions (1.6) to the vicinity of Stokes rays emanating from ξ = 0. First, we identify the solutions of the form (7.3) which correspond to W ± (u, ξ). Next, by employing Theorem 1.1 and techniques similar to those in [15], it is shown that the asymptotic expansions of the coefficient functions A (u, ζ) and B(u, ζ) are Borel summable with respect to the large parameter u, and that the corresponding Borel transforms are analytic functions of ζ in an unbounded domain containing the Stokes rays. The process is completed by an appeal to a theorem on the composition of Borel summable series [26,Ch. 5,Theorem 5.55] and the well-known Borel summability properties of WKB solutions of the Airy equation.
In Section 3, we showed that the Borel transforms f ± (t, ξ) can be continued analytically, in the complex variable t, to a strip containing the positive real axis. The method, however, does not provide us with any information regarding the nature and location of the singularities of f ± (t, ξ) in the t-plane. Explicit knowledge of singularities is an important prerequisite to any investigation in resurgent analysis and exponential asymptotics. By looking at the functional equation (3.6), for example, it is reasonable to expect that the singular points of f − (t, ξ) are located at t = −2ξ + 2ξ k , where the ξ k 's, k ∈ Z ≥0 , are the singularities of the functions φ(ξ) and ψ(ξ). Hence, there seems to be a direct link between the singular points of f − (t, ξ) in the t-plane and those of φ(ξ) and ψ(ξ) in the ξ-plane. This conjecture may be verified for some specific examples where the Borel transforms are explicitly known (e.g., the Airy or the Legendre equation), but the general case requires further investigation.
Finally, it would be of great interest to extend the results of the paper to differential equations of the type (1.1) in which the potential function f (u, z) admits a (Borel summable) uniform asymptotic expansion of the form f (u, z) ∼ The coefficients E ± n (ξ) are found by substitution of (A.1) into (1.3) and equating like powers of u. In this way, we find that for n ≥ 1, with the understanding that empty sums are zero. The constants of integration in (A.1) and (A.2) are arbitrary. Comparing (A.1) with (1.4), it is seen that the following pair of formal relations hold between the coefficients A ± n (ξ) and E ± n (ξ): Application of Ex. 8.3 of Olver [30, p. 22] then leads to the recurrence relations for n ≥ 1. Once the constants of integration in (A.2) are fixed, the A ± n (ξ)'s are uniquely determined by (A.3). For example, to obtain the coefficients generated by the recursive formulae (1.9), the lower limits of integration in (A.2) are taken as ∓∞ + i ξ.