Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])$$\end{document}HL:L2([bL,0])→L2([0,bR]) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])$$\end{document}HR:L2([0,bR])→L2([bL,0]). These operators arise when one studies the interior problem of tomography. The diagonalization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_R,\mathcal {H}_L$$\end{document}HR,HL has been previously obtained, but only asymptotically when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_L\ne -b_R$$\end{document}bL≠-bR. We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_R,\mathcal {H}_L$$\end{document}HR,HL explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.


Introduction
Let I 1 , I 2 ⊂ R be multiintervals, i.e. the unions of finitely many non-overlapping intervals (but the sets I 1 , I 2 can overlap). The intervals can be bounded or unbounded. Consider the Finite Hilbert transform (FHT) and its adjoint: The general problem we consider is to study the spectral properties of H (and the associated self-adjoint operators H * H, HH * ) depending on the geometry of the sets I 1 , I 2 . The properties we are interested in include finding the spectrum, establishing the nature of the spectrum (e.g., discrete vs. continuous), and finding the associated resolution of the identity. In the case when I 1 = I 2 , these problems were studied starting in the 50s and 60s, see e.g. [20][21][22][27][28][29]33].
More recently, the problem of the diagonalization of H * H and HH * occurred when solving the problem of image reconstruction from incomplete tomographic data, e.g. when solving the interior problem of tomography [4,8,23,[34][35][36]. In these applications, a significant diversity of different arrangements of I 1 , I 2 was encountered: the two sets can be disjoint (i.e., dist(I 1 , I 2 ) > 0), touch each other (i.e., have a common endpoint), or overlap over an interval. Each of these arrangements leads to different spectral properties of the associated FHT. Generally, if dist(I 1 , I 2 ) > 0, the operators H * H and HH * are Hilbert-Schmidt with discrete spectrum [6,17]. In one particular case of overlap, when I 1 = [a 1 , a 3 ], I 2 = [a 2 , a 4 ], a 1 < a 2 < a 3 < a 4 , the spectrum is discrete, but has two accumulation points: λ = 0 and λ = 1 [3,4], where λ denotes the spectral parameter. In two cases when I 1 , I 2 touch each other, the spectrum is purely absolutely continuous. The case I 1 = [a 1 , a 2 ], I 2 = [a 2 , a 3 ], a 1 < a 2 < a 3 , was considered in [19]. The case when I 1 , I 2 are unions of more than one sub-interval each and I 1 ∪ I 2 = R was considered in [7].
To obtain the above mentioned results, three methods have been employed. In very few exceptional cases, e.g. when I 1 ∪ I 2 = R, it is possible to diagonalize the FHT explicitly via some ingenious transformations. When I 1 , I 2 each consist of a single interval (the intervals can be separated, touch, or overlap), the method of a commuting differential operator is used [3,4,[17][18][19]. Here the associated singular functions (and kernels of the unitary transformations) are obtained as solutions of the special Sturm-Liouville problems for the differential operator that commutes with the FHT. As is seen, both of these approaches are fairly limited.
In [6], a new powerful approach based on the method of the Riemann-Hilbert Problem (RHP) and the nonlinear steepest descent method of Deift-Zhou was proposed. This approach allows one to treat fairly general cases of I 1 , I 2 . However, the limitation of this approach so far was the assumption dist(I 1 , I 2 ) > 0, which ensured that the associated operators H * H and HH * are Hilbert-Schmidt. In the case when I 1 , I 2 touch each other, the RHP approach encountered several challenges that have not been studied before. Here are the three main ones: 1. In the case of a purely discrete spectrum, residues are used to compute singular functions in [6]. However, in the limit dist(I 1 , I 2 ) → 0 the spectra of the operators change from discrete to continuous. Thus, a different technique is needed to extract the spectral properties of the FHT from the solution of the corresponding RHP; 2. Construction of small λ approximations of the RHP solution near the common endpoints (parametrices) was not known, and it had to be developed; 3. The proper boundary conditions near the common endpoints in the formulation of the RHP was not well understood.
As stated earlier, in the particular case when I 1 = [a 1 , a 2 ], I 2 = [a 2 , a 3 ], a 1 < a 2 < a 3 , the spectral analysis of the FHT was performed in [19]. Since the method in [19] is based on a commuting differential operator, this method cannot be extended to more general situations of touching intervals, for example, to the case when I 2 consists of two disjoint intervals. The goal of this paper is to extend the RHP and Deift-Zhou approach from [6] to the case when I 1 , I 2 touch each other. This problem in its most general setting is very complicated. For example, if I 1 , I 2 touch each other at several points, the continuous spectrum may have multiplicity greater than 1 (see [7]). If there are also pieces of I 1 that are at a positive distance from I 2 (or, vice versa), then there will be discrete spectrum accumulating at λ = 0 embedded in the continuous spectrum. In this paper we build the foundation for using the RHP/Deift-Zhou method by studying the case I 1 = [a 1 , a 2 ], I 2 = [a 2 , a 3 ], a 1 < a 2 < a 3 , as a model example. Our results include: 1. Formulating the corresponding RHP (including the proper boundary conditions) and explicitly calculating its solution Γ (z; λ) in terms of hypergeometric functions; 2. Complete spectral analysis of the operators H * H and HH * as well as diagonalization of the operators H, H * ; 3. Calculating the leading order asymptotics of Γ (z; λ) in the limit λ → 0 in various regions of the complex z plane. These regions include, in particular, a small annulus centered at the common endpoint a 2 . 4. Finally, we also show that the spectral asymptotics of [19] match the explicit spectral results of this paper.
The asymptotics in the annulus around a 2 from Item 3 allows us to use Γ (z; λ) as a parametrix near any common endpoint for more general multiintervals I 1 , I 2 .
This parametrix is the key missing link that is needed to construct the leading order asymptotics of Γ (z; λ) for the general I 1 , I 2 . This will be the subject of future research.
The paper is organized as follows. In Sect. 2 we introduce the integral operator which, when restricted to L 2 ([b L , 0]) and L 2 ([0, b R ]), coincides with 1 2i H L and 1 2i H R , respectively (see equation (3)). Here H L is the FHT from L 2 ([b L , 0]) → L 2 ([0, b R ]), and H R is the FHT from L 2 ([0, b R ]) → L 2 ([b L , 0]). In the spirit of the above notation, we have a 1 = b L , a 2 = 0, and a 3 = b R . It can easily be shown that the knowledge of the spectrum ofK 2 allows one to find the spectra of H * L H L and H R H * R . Similarly to the case of disjoint intervals studied in [5], the key observation is that the kernel K of K , see (4), is a kernel of integrable type in the sense of [16]. Therefore, the kernel of the resolventR ofK is readily available in terms of the solution Γ (z; λ) of the matrix RHP 1.
Thus, RHP 1 plays a fundamental role in our paper. Because the jump matrix of this RHP is constant in z, its solution Γ (z; λ) satisfies a Fuchsian system of linear differential equations with three singular points and, therefore, can be expressed in terms of hypergeometric functions. An explicit expression for Γ (z; λ) is obtained in Theorem 1. Using this expression, we show that the matrix Γ (z; λ) is analytic for λ ∈ C\[−1/2, 1/2] and has analytic continuations across (−1/2, 0) and (0, 1/2) from above and from below (Proposition 2).
In Sect. 3 we explicitly find the unitary operators U L : , which diagonalize the operators H * L H L and H * R H R , respectively, see Theorem 8. Here σ χ L , σ χ R denote the corresponding spectral measures. We prove in Theorems 6 and 7 that the spectrum of the operators H * L H L and H * R H R is simple, purely absolutely continuous, and consists of the interval [0, 1]. Our approach is based on the resolution of the identity Theorem 3 (see, for example, [2]) and explicit calculation of the jump of the kernel of the resol-ventR over the spectral set in terms of the hypergeometric functions (Theorem 5). In the process, we find that Γ (z; λ) satisfies the jump condition in the spectral variable λ over the segment [− 1 2 , 1 2 ], see Theorem 4, which, in some sense, is dual to the jump condition in the z variable, see RHP 1. This observation is a RHP analogue of certain bispectral problems [15].
The spectrum and diagonalization of the operators H L , H R was studied in [19], where the authors used a second order differential operator L, see (90), which commutes with H L and H R . The Titchmarsh-Weyl theory was utilized in [19] to obtain the small-λ asymptotics of the unitary operators U 1 , U 2 that diagonalize H L , H R , i.e. U 2 H L U * 1 and U 1 H R U * 2 are multiplication operators. In Sect. 4, we construct the operators U 1 , U 2 , see (115), (116), explicitly in terms of the hypergeometric functions everywhere on the continuous spectrum. We also show, see Theorem 13, that the diagonalization of H * L H L and H * R H R , obtained in Sect. 3, is equivalent to the diagonalization obtained through the operators U 1 , U 2 .
In Sect. 5 we obtain the leading order behavior of Γ (z; λ) as λ → 0 in different regions of the complex z-plane, see Theorem 17. The main tool we use here is the Deift-Zhou nonlinear steepest descent method combined with the g-function mechanism, which reduces (asymptotically) the original RHP 1 for Γ (z; λ) to the so called model RHP 4 for Ψ (z). The latter represents the leading order approximation of Γ (z; λ) on compact subsets of C\[− 1 2 , 1 2 ]. Since the jump matrices in RHP 4 commute with each other, the model RHP has a simple algebraic solution (Theorem 16). However, due to a singularity at the common endpoint z = 0, this solution is not unique. In order to select the appropriate Ψ (z), we need to match it with the leading order behavior of Γ (z; λ) as λ → 0 in a small annulus centered at z = 0, which is derived in Theorem 15. This, in turn, requires calculating the leading order approximation of the hypergeometric function F(a, b; c; η), where the parameters a, b, c go to infinity in a certain way as λ → 0. Moreover, this approximation must be uniform in a certain large radius annulus Ω in the complex η-plane. Such asymptotics was recently obtained in [26] based on the saddle point method for complex integrals of the type of (133), but error estimates and uniformity needed for our purposes were not addressed there. Thus, we state and prove Theorem 14 for such complex integrals.
Details of the construction of Γ (z; λ), the proof of Theorem 14, and other auxiliary material can be found in the Appendix.

Integral operatorK and RHP
Let us begin by defining the finite Hilbert transforms H L : Notice that the adjoint of H L is −H R .

Definition and properties ofK
We define the integral operatorK : Explicitly, Proof The boundedness ofK follows from the boundedness of the Hilbert transform on L 2 (R) and we can see thatK is self-adjoint because K (z, x) = K (x, z).

Resolvent ofK and the Riemann-Hilbert problem
The operatorK falls within the class of "integrable kernels" (see [16]) and it is known that its spectral properties are intimately related to a suitable Riemann-Hilbert problem. In particular, the kernel of the resolvent integral operatorR =R(λ) : can be expressed through the solution Γ (z; λ) of the following RHP.

Riemann-Hilbert
The principle branch of the log is taken and √ 4λ 2 − 1 = 2λ + O (1) as λ → ∞, so it can be shown that a(λ) is analytic for λ ∈ C\[−1/2, 1/2]. This function a(λ) will occur frequently throughout this paper so we have listed its relevant properties in Appendix B. We will often write a in place of a(λ) for convenience. Recall that the standard Pauli matrices are In Appendix A, we describe how to construct the (unique!) solution to RHP 1.
We now show the relation betweenK and Γ (z; λ).

Theorem 2
With the resolvent operatorR defined by (5), let the kernel ofR be denoted by R. Then, The matrix Γ (z; λ) is defined in (17) and functions χ L , χ R are indicator functions on The proof is the same as in [6] (Lemma 3.16) so it will be omitted here. An important ingredient of the proof is the observation that the jump of Γ (z; λ) can be compactly written as where K (z, x) is the kernel ofK , see (4).

Spectral properties and diagonalization H * R H R and H * L H L
The goal of this section is to construct unitary operators U R : where λ 2 is a multiplication operator (the space is clear by context), J := {λ 2 : 0 ≤ λ 2 ≤ 1}, and the spectral measures σ L , σ R are to be determined. This is to be understood in the sense of operator equality on L 2 (J , σ R ), L 2 (J , σ L ), respectively. We will begin this section with a brief summary of the spectral theory for a self-adjoint operator with simple spectrum.

Basic facts about diagonalizing a self-adjoint operator with simple spectrum
For an in-depth review of the spectral theorem for self-adjoint operators, see [2,11,32]. We present a short summary of this topic which is directly related to the needs of this paper. Let K be a Hilbert space and let A be a self-adjoint operator with simple spectrum acting on K. Recall from [2], that a self-adjoint operator has simple spectrum if there is a vector g ∈ K so that the span ofÊ Δ [g], where Δ runs through the set of all subintervals of the real line, is dense in K. Here the family of operatorsÊ t denotes the so-called resolution of the identity for the operator A, which we define in (25). Definê R, the resolvent of A, via the formulâ for t / ∈ R. Then, according to [11] p.921, the resolution of the identity is computed by the formulâ where α < β. Once we obtainÊ t , we can construct the unitary operators which will diagonalize A, as described in the following Theorem from [2] p.279.

Theorem 3
If A is a self-adjoint operator with simple spectrum, if g is any generating element, and if σ (t) = Ê t [g], g , then the formula associates with each functionf ∈ L 2 (R, σ ) a vector f ∈ K, and this correspondence is an isometric mapping of L 2 (R, σ ) onto K. It maps the domain D(Q) of the multiplication operator Q in L 2 (R, σ ) into the domain D(A) of the operator A, and if the element f ∈ D(A) corresponds to the functionf ∈ L 2 (R, σ ), then the element A f corresponds to the function tf (t).

Remark 2
In short, Theorem 3 says that σ (t) := Ê t [g], g defines the spectral measure (g is any generating element) and the operator U * : is unitary. Moreover, U AU * = t (28) in the sense of operator equality on L 2 (R, σ ).
Thus our immediate goal moving forward is to construct the resolution of the identity for H * R H R and H * L H L .

Resolution of the identity for H *
where π R :  (29), (30), slightly differ from the standard definition given by (24). It is easy to see that and the same is true for the resolvent of H * L H L .

Proof
In the direct sum decomposition ,K has the block structurê For λ sufficiently large, we can write (recall thatK is bounded, see Proposition 1) where all the even powers in the right hand side of (33) are block diagonal and all the odd powers in (33) are block off-diagonal. The result is extended to all λ / ∈ R via analytic continuation. Similarly, we can write and comparing with the series in (33) gives our result for the resolvent of H * R H R . The proof for the resolvent of H * L H L is nearly identical.
To construct the resolution of the identity (see (25)), we need to compute the jump of the resolvent of H * R H R and H * L H L in the λ-plane. The kernel of the resolvent is expressed in terms of Γ (z; λ) (see Theorem 2), so we need to compute the jump of Γ (z; λ) in the λ-plane.

Remark 4
In the remaining sections of this paper we will frequently encounter the following Möbius transformations: Define functions where h ∞ , s ∞ is defined (12), (13), a := a(λ) is defined in (15), and coefficients α(λ), β(λ) are The functions d L , d R will play a major role in this section so we have compiled all of their relevant properties in Appendix C. Importantly, it is shown that both d L (z; λ) and d R (z; λ) are single valued when λ ∈ (−1/2, 0). For λ ∈ (−1/2, 0) ∪ (0, 1/2), define vectors , We are now ready to compute the jump of Γ (z; λ) in the λ−plane.

Recall from Proposition 3 and Theorem 2 that the resolvents of H
In light of the previous Theorem, we can now compute the jump of the resolvents of H * L H L , H * R H R in the λ plane, which is required to construct the resolution of the identity, see (25).

Remark 5 From Theorem 5 we can immediately see that when
and when x, z ∈ (b L , 0), where d R , d L are defined in (37).

Proposition 4 The operators H
Proof We will prove this statement for H * R H R only, as the proof for H * L H L is similar. We show that the resolvent of H * R H R has no poles in the λ plane. Recall from (29) that and using (2), we can see that the kernel of R R is Using Proposition 2, we know that Γ (z; λ/2) can (potentially) have a pole only We can see that λ 2 = 0 is not an eigenvalue of H * R H R because the null space of H R contains only the zero vector. Likewise, it is easy to see that λ 2 = 1 is not an eigenvalue as well. Otherwise proves the desired assertion. Here we used the convention that whenever the norm · R is computed, the left-most Hilbert transform inside the norm is evaluated over the entire line.
where φ L , φ R and σ L , σ R are defined in (63) where R R is the resolvent of H * R H R , see (29). Thus the spectral set is {λ 2 ∈ [0, 1]}. It was shown in Proposition 4 that H * R H R has no eigenvalues. Next, we constructÊ R,λ 2 . It was shown in Theorem 5 thatR(λ) is single-valued for λ ∈ C\[−1/2, 1/2], thus from (25) and Proposition 3 we havê We have shown in Proposition 4 that H * R H R has no eigenvalues. According to [2] section 82, the lack of eigenvalues guarantees thatÊ R,λ 2 has no points of discontinuity. Now returning to (68), we can take δ = 0 and we can move the limit inside the integral as the kernel of R R has analytic continuation above and below the interval (0, 1). So from (25), (29), and Remark 3 we obtain and now plugging in (57) gives the result. Note that Δ λ 2 R R (λ 2 ) = sgn(λ)Δ λR (λ/2), because when λ is on the upper shore of (−1, 0), λ 2 is on the lower shore of (0, 1).

Nature of the spectrum of H * R H R and H * L H L
In this subsection, we show that the spectrum of H * R H R and H * L H L is simple and purely absolutely continuous. We will prove statements in this section for H * R H R only because the statements and ideas for proofs are nearly identical for H * L H L . Notice that the resolution of the identity of H * R H R [see (65)] can be compactly written aŝ where φ R , σ R are defined in (63), (64), respectively and the operator U R : For any interval Δ ⊂ [0, 1], which is at a positive distance from 0 and 1 (to avoid the singularities of φ R (z, μ 2 ) as μ 2 → 0 or 1), we have where we have used thatÊ R,Δ is a self-adjoint projection operator, see [2] p.214.
Using (70) and (71), it is easy to show that (73) By (72) and (73) and by continuity, we can extend U R to all of L 2 ([0, b R ]) and Taking the limit Δ → [0, 1] in (72), (73) and using thatÊ R,λ 2 is the resolution of the identity, the spectrum is confined to [0, 1], and there are no eigenvalues, we prove the following Lemma.

Lemma 1 The operator U R extends to an isometry from L
We are now ready to conclude this section.

Theorem 7
The spectrum of H * R H R , H * L H L is simple and purely absolutely continuous.
Proof To prove that the spectrum is simple we will show that is a generating vector. So for any where α jn are some constants, and I jn := [( j − 1)/n, j/n). Thus, the intervals I jn , 1 ≤ j ≤ n, partition the spectral interval [0, 1]. Using the properties ofÊ R,λ 2 , we calculate whereφ n is the simple functioñ Using the properties ofÊ R,λ 2 , we can write the left hand side of (77) as Now using (74), (78), (79), and the fact thatÊ R,Δ jÊ R,Δ k = 0 whenever Δ j ∩ Δ k = ∅ (see [2], p. 214), we see that It is clear that any U R [ f ] can be approximated by a sequence of simple functionφ n , so we have as desired. Thus, the spectrum of H * R H R is simple and g = χ R is a generating vector. Lastly, to show that the spectrum of H * R H R is purely absolutely continuous, we need to show that the function [2], Vol. 2, Section 95. Similarly to the proof of Lemma 1, we have for 2 and the kernel of U R are real analytic for μ 2 ∈ (0, 1) (see Proposition 8) and , the desired assertion follows immediately.

Diagonalization of H * R H R and H * L H L
We are now ready to use Theorem 3 and build the unitary operators which will diagonalize H * R H R and H * L H L . Recall from Theorem 7 that χ R , χ L are generating vectors for H * R H R , H * L H L , respectively. Following Theorem 3 and Remark 2, we define

Remark 7
Again, using Proposition 8, it can be verified that where φ L , φ R are defined in (63). It is now clear that the adjoint of U R , defined in (71), is U * R , and We conclude this section with the following main result. (71), (88), respectively, are unitary and

Theorem 8 The operators U
, respectively, where λ 2 is to be understood as a multiplication operator.
Proof We state the proof for H * R H R only as the proof for H * L H L is nearly identical. The resolution of the identity of H * R H R was constructed in Theorems 6 and in Theorem 7 it was shown that the spectrum of H * R H R is simple and χ R is a generating vector. Notice that both U * R , defined in (85), and σ χ R = σ R , defined in (64), were constructed in accordance with Theorem 3 and Remark 2. The combination of Remarks 6, 7 and Theorem 3 complete the proof.
We later obtain a different proof of this Theorem, see Corollary 1 and Theorem 13.

Diagonalization of H R , H L via Titchmarsh-Weyl theory
Using recent developments in the Titchmarsh-Weyl theory obtained in [13], it was shown in [19] that the operator has only continuous spectrum and commutes with the FHTs H L , H R , defined in (2). We now state the main result of [19] and refer the reader to this paper for more details.

Theorem 9
The operators U 1 : Moreover, in the sense of operator equality on L 2 (J , ρ 2 ) one has where There is a minor typo in this theorem in [19]; when describing σ (λ), the factor a 3 2 a 1 is incorrect and should be − a 2 a 1 . The operators U 1 , U 2 in Theorem 9 were obtained asymptotically when ω → ∞. Here we obtain these operators explicitly. According to [13], the kernels of U 1 , U 2 and the spectral measures ρ 1 , ρ 2 are defined through particular solutions of L f = ω f . Such solutions will be constructed in the following subsections.

Interval [0, b R ]
Define the function where M 4 (x) and μ are defined in Remark 4 and Theorem 9, respectively. Notice that (93), where a is a constant, we obtain (4.9) of [19]. Now define where Here γ is Euler's constant and Ψ is the Digamma function, see [1] 6.3.1.

Remark 8
Using properties of the Gamma functions, see [1] 6.1.30, it can be shown that provided μ ≥ 0.

Interval [b L , 0]
This subsection will be similar to the last so many proofs will be omitted, as the ideas have been previously presented. Define the function where M 4 (x) and μ(ω) are defined in Remark 4 and Theorem 9, respectively. Now define where k is defined in (95) and (108)

Remark 10
The functions f L , f R , defined in (106), (93), respectively, share the relation where M 2 (x) is defined in (4). This relation combined with Remark 9 shows that where D L , ϕ 2 are defined in (62), (94), respectively, and the relation between ω and λ is described in (97).

Diagonalization of H L , H R
According to the spectral theory developed in [13], we have gathered nearly all the necessary ingredients to diagonalize H L , H R . It remains to construct two functions m 1 (ω) and m 2 (ω) so that (111) whenever ω > 0. It can be verified that where l 1 and l 2 , k are defined in (108), (95), respectively. The spectral measures ρ 1 , ρ 2 are constructed via the formula for j = 1, 2 (see [13] for more details). From (113) we obtain where we have used (112), Remark 8, and (102). Define the operators U 1 : where ϕ 1 , ϕ 2 are defined in (107), (94), respectively. We are now ready to prove the main result of this section.

Theorem 12
The operators U 1 , U 2 , defined in (115), (116), are unitary and in the sense of operator equality on L 2 (J , where ρ 2 is defined in (114).
Proof First, the operators U 1 , U 2 are unitary by [13]. According to Proposition 8 and Remark 10, since (97) implies that |λ| = sech(μπ ). Using (118), we calculate that for anyf ∈ L 2 (J , ρ 1 ), which is equivalent to Since the adjoint of H L is −H R , we have an immediate Corollary.

Corollary 1
In the sense of operator equality on L 2 (J , ρ 1 ) one has and in the sense of operator equality on L 2 (J , ρ 2 ) one has Proof The proof follows quickly from Theorem 12 because and (what follows is the multiplication operator) This corollary can be used to recover Theorem 8. We have now obtained two (seemingly) different diagonalizations of H * R H R and H * L H L in Theorem 8 and Corollary 1. We show that these diagonalizations are equivalent in the sense of change of spectral variable. Proof We will relate the operators U L , U 1 by using the change of variable λ → ω in (97), which implies that Now using this change of variable, we see that where c(ω) = −b L √ π |λ|D R (∞; λ). Similarly, So using (127) and (128) we obtain as desired.

Small asymptotics of 0(z; )
In this section we only consider the symmetric scenario when b R = −b L = 1, but the results can be obtained for general endpoints via Möbius transformations. The main results of this section are Theorems 15 and 17, which describes the small λ asymptotics of Γ (z; λ) first in a small annulus around z = 0 and then in the rest of C respectively. The proof of Theorem 15 is based on the asymptotics of hypergeometric functions that appear in Γ (z; λ), whereas the prove of Theorem 17 is based on Theorem 15 and the Deift-Zhou nonlinear steepest descent method [10].

Modified saddle point method uniform with respect to parameters
According to (17), we are interested in h ∞ (η), where In view of the integral representation ([1], 15.3.1) of h ∞ (η), given by (12), where a(λ) → ∞ as λ → 0, we want to use the saddle point method to find the small λ asymptotics of h ∞ (η). We start with the case λ ≥ 0 which implies [a] → −∞ as λ → 0, see Appendix B for more information about a(λ). For λ ≤ 0 the results are similar, see Remark 18. With that in mind, define function where the branch cuts of S η (t) in t variable are chosen to be (−∞, 0), (1, ∞), and the ray from t = η to t = ∞ with angle arg η. The integral from (131) can be now written as Define closed regions where M is a large, positive, fixed number that is to be determined. Notice that the set of all z such that z+1 2z ∈ Ω is a small annulus about the origin. The large a asymptotics of the integral in (133) that is uniform in η ∈ Ω is technically not covered by standard saddle point theorems (see, for example, [12,25,30]). Therefore, in Appendix D we present a proof of Theorem 14 for such integrals, that will be used later for the small lambda asymptotics of hypergeometric functions h ∞ (η), s ∞ (η) and their derivatives. The obtained results in Theorem 14 leading order term of the hypergeometric function is consistent with the results of Paris [26], where the formal asymptotic expansion in the large parameter a(λ) was derived, but the error estimates and uniformity in η were not addressed.
The following proposition identifying the saddle points of S η (t) is a simple exercise. We need the saddle point t * − (η) to state Theorem 14.
The idea of the proof is as follows: we deform the contour of integration in (139) from [0, 1] to a path we call γ η , which passes through a relevant saddle point t * − (η) of S η (t). We then show that the leading order contribution in (139) comes from a small neighborhood of t * − (η). Remark 12 One can simplify equation (139) in Theorem 14 by substituting (138).

Small asymptotics of 0(z; ) for z ∈Ä
In this subsection we use Theorem 14 to calculate the leading order asymptotics of Γ (z; λ) given by (17) as λ → 0, provided that z ∈Ω, wherẽ Notice thatΩ is a small annulus about the origin. In this section, we will often use the variable κ = − ln λ (141) instead of λ and the function The properties of the g-function can be found in Proposition 6.
Proof According to Theorem 14, where we have used F(t, η, λ) = e i [a(λ)]S η (t) and η = z+1 2z . By Proposition 7 as λ → 0 with λ ≥ 0 and z ∈Ω + . Using [24] 5.5.5, 5.11.13 we have Combining the two previous equations, we see that The perk of this equation is that the right hand side has an integral representation for −1/2 ≤ [a] ≤ 1/2. Thus we can apply Theorem 14 twice and obtain the leading order asymptotics. So we have shown A similar process can be repeated for s ∞ (η) and we obtain We have an immediate Corollary.
It remains to find the small λ leading order asymptotics of the remaining factors of Γ (z; λ). This is a tedious, but straightforward exercise.
(161) Combining that with Proposition 7 and 5.11.13 from [24], we obtain the following Lemma.
We are ready to put the pieces from this section together and obtain the asymptotics of Γ (z; λ) as λ → 0 for z ∈Ω. Define the matrix This matrix is a particular solution of the so-called model RHP 4 corresponding to x = y = i/2 in (188). As we will see in Theorem 15 below, the limit of Γ (z; λ) as λ → 0, λ > 0 distinguishes Φ(z) among all other solutions of the model RHP.
Proof First take λ ≥ 0; the leading order term of D −1Γ −1 1 e κgσ 3 D from Lemmas and Corollaries 3, 2, 3, we have When λ ≤ 0, observe that the leading order term of D −1Γ −1 1 (165) and thus we have the leading order term. Since Φ(z) is uniformly bounded away from 0 when z ∈Ω, we immediately obtain the lower order term.

Define matrix
Note that the matrix σ 1 Φ(z)σ 1 is also a solution to RHP 4 with x = y = −i/2 in (188). Now we are ready to prove one of the main results of this section.

Deift-Zhou steepest descent method
The g-function, defined in (142), will play an important role so we list its relevant properties, all of which follow directly from Proposition 7.

Transformation 0(z; ) → Z(z; κ)
Our first transformation will be where Γ (z; λ) was defined in (17). Since Γ (z; λ) is the solution of RHP 1, it is easy to show that Y (z; κ) solves the following RHP.
The endpoint behavior is listed column-wise.
The jumps for Y (z; κ) on (−1, 0) and (0, 1) can be written as This decomposition can be verified by direct matrix multiplication and by using the jump properties of g(z) in Proposition 6. We define the 'lense' regions L (±) L,R as in Fig. 2.

Remark 13
Assume Ψ (z) is a solution of the RHP 4. Then det Ψ (z) ≡ 1 if and only if y = x in the representation (188). If, additionally, x ∈ iR in this representation then Ψ (z) has the symmetry Both properties can be easily verified.

Approximation of Z(z; κ) and main result
We will construct a piecewise (in z) approximation of Z (z; κ) when κ → ∞. Our approach is very similar to that in [6]. Denote by D j a disc of small radius l centered L,R (see Fig. 2) outside the discs D j , j = 0, ±1, the jumps of Z (z; κ) are uniformly close to the identity matrix thus Ψ 0 (z; κ) (a solution to model RHP, see (166)) is a 'good' approximation of Z (z; κ). Inside D j , j = 0, ±1, we construct local approximations that are commonly called 'parametrices'. The solution of the so-called Bessel RHP is necessary.

Riemann-Hilbert Problem 5
Let ν ∈ (0, π) be any fixed number. Find a matrix B ν (ζ ) that is analytic off the rays R − , e ±iθ R + and satisfies the following conditions.
This RHP has an explicit solution in terms of Bessel functions and can be found in [31]. Define local coordinates at points z = ±1 as Proof The behavior for z ∈ ∂D ±1 , ∂D 0 is a direct consequence of (198), Corollary 4, respectively. The behavior on the lenses is clear via inspection of (204). Proof Given J , the disks D 0,±1 can be taken sufficiently small in order to not intersect J . Corollaries 4, 5 and the so-called small norm theorem, see [9] Theorem 7.171, can now be applied to conclude that E(z; κ) = 1 + O M 2 κ uniformly for z ∈ J . This is equivalent to the stated result.
We are now ready to prove the main result of this section.

For z in compact subsets of C\[−1, 1] we have the uniform approximation
2. For z in compact subsets of (−1, 0) ∪ (0, 1) we have the uniform approximation where ± denotes the upper/lower shore of the real axis in the z-plane.
Proof This Theorem is a direct consequence of Corollary 6. We simply need to revert the transforms that took Γ → Z . Doing so, we find that Since z is in a compact subset of C\[−1, 1] or (−1, 0) ∪ (0, 1), we apply Corollary 6 to obtain the result.
which has exactly three regular singular points at η = 0, 1, ∞. The idea is to choose parameters a, b, c so that the monodromy matrices of the fundamental matrix solution solution of the ODE will match (up to similarity transformation) the jump matrices of RHP 1. We orient the real axis of the η−plane as described in Fig. 4.

A.2 Selection of parameters a, b, c
DefineΓ Notice that for any η ∈ C det Γ (η) = e (a+b)πi (a − b) according to (214). Our solution Γ (z; λ) to RHP 1 has singular points at z = b L , 0, b R . Notice that the Möbius transform maps b L → 0, b R → 1, and 0 → ∞ where the orientation of the z−axis is described in Fig. 5. Thus we are interested in the matrix We need to determine parameters a, b, c such thatΓ (M 1 (z)) is L 2 loc at z = b L , 0, b R , so we are interested in the bi-resonant case, which is When z = b L , to guarantee thatΓ is L 2 loc we must have that (use the connection formula of section A.1 to easily inspect the local behavior) Since c ∈ Z, it must be so that c = 0. Now for z = b R , we must have that where a + b = r ∈ Z. Since r ∈ Z, the only possibility is r = −1. So we have that b = −1 − a and c = 0. Lastly, as z → 0, so we see that it is not possible forΓ (M 1 (z)) to have L 2 behavior at z = 0. On the other hand, observe that Thus the matrix 1 is L 2 loc as z → 0 provided that | (a)| < 1/2. In the next section, we solve for a explicitly in terms of λ and the condition | (a)| < 1/2 will be met provided that λ / ∈ [−1/2, 1/2], see Appendix B.

A.3 Monodromy
The monodromy matrices ofΓ (z) about the singular points z = 0, 1, ∞ are where C ∞0 , C 0∞ , C ∞1 , C 1∞ are defined in (217), (218), (221), (222), respectively. With some effort it can be shown that From the previous section, we take c = 0 and b = −1 − a. It is important to note that the connection matrices C ∞0 , C 0∞ , C ∞1 , C 1∞ are singular when c = 0 and/or b = −1 − a, but we can see that M 0 , M 1 are not. Taking c = 0 and b = −a − 1, we obtain In Appendix B we have listed all the important properties of a(λ).

A.4 Proof of Theorem 1
We will now construct Γ (z; λ) so that it is the solution of RHP 1.
Proof Notice that the matrix Thus, we conclude that the matrix is a solution of RHP 1 provided that λ / ∈ [−1/2, 1/2]. Uniqueness follows immediately from the L 2 behavior of Γ (z; λ) at the endpoints z = b L , 0, b R .

Appendix B: Definition and properties of a( )
Recall, from (15), that where √ 4λ 2 − 1 = 2λ + O (1) as λ → ∞ and the principle branch of the logarithm is taken. The following proposition lists all relevant properties of a(λ), none of which are difficult to prove.
Using the definition of Γ (z; λ), see (17), some simple algebra shows that Since Γ (z; λ) is a solution of RHP 1, we know M(z, λ) is analytic for z ∈ C\[b L , b R ] and and a = a(λ), M 1 (z) are Schwarz symmetric. Thus d R (z; λ) = d R (z; λ) and the symmetry of d L (z; λ) follows from the relation d R (M 2 (x); λ) = d L (x; λ). 4. This was proven above. 5. Let γ R be the circle with center and radius of b R /2 with negative orientation. Then, where we have deformed γ R through z = ∞ and into the circle of center and radius b L /2 with positive orientation and then apply residue theorem. The remaining computation is nearly identical. 6. The idea is similar to that of the last proof; and the remaining identity is proven analogously.
Proof Let us fix some η ∈ Ω + . The function S η (t) is a harmonic function in t ∈ B(1/2, 1/2), so the level curves S η (t) = S η (t * − (η)) can not form closed loops there. Since, according to Lemma 5, there exactly four level curves S η (t) = S η (t * − (η)) entering the disc B(1/2, 1/2), and exactly four legs of this level curve emanating from t = t * − (η), the only possible topology of these level curves is shown on Fig. 6. We can now deform [0, 1] to an 'appropriate' path γ η as stated in the theorem. The new path of integration, γ η , is the black curve in Fig. 7.

D.2 Estimates
Here we will split γ η into 3 pieces (see Fig. 7) and show that the leading order contribution in Theorem 14 comes from the piece containing t * − (η). Define radius so that t * − (η) ∈ B(1/2, r ) for all η ∈ Ω + . The latter estimate follows from Prposition 5. Also define function v η (t) := v (t, η) = S η (t * − (η)) − S η (t) where the square root is defined so that v η (t) > 0 along the path γ η lying between t * − (η) and 1. The function v η (t) will be an essential change of variables in the integral (139) so we mention its important properties.
Obviously c * (η) is a continuous and negative function of η ∈ Ω + . Since Ω + is a compact set, the maximal value c * of c * (η) on Ω + is negative. Then is finite by hypotheses of Theorem 14.
The combination of Lemmas 7 and 8 proves Theorem 14 provided that λ is in the upper half plane.

Remark 18
When λ → 0 in the lower half plane, the key difference is that [a(λ)] → ∞ (see Proposition 7). With that in mind, rewrite the integrand of (139) as We can replace S with −S and carry out the same analysis as before. The only difference will be that the regions in the t plane where [S η (t) − S η (t * − (η))] < 0 and [S η (t) − S η (t * − (η))] > 0 will swap, so we deform [0, 1] to a different contour, see Fig. 8. All previous ideas from this section can now be applied using the new contour γ .