The Baker-Akhiezer function and factorization of the Chebotarev-Khrapkov matrix

# The Baker-Akhiezer function and factorization of the Chebotarev-Khrapkov matrix

Yuri A. Antipov
Department of Mathematics, Louisiana State University
Baton Rouge, LA 70803
###### Abstract

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient , , , is a zero-trace polynomial matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires the finding of the zeros of the Baker-Akhiezer function ( is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree- polynomial and solution of a certain linear algebraic system of equations.

AMS subject classifications. 30E25, 30F99, 45E

Key words. Riemann-Hilbert problem, Baker-Akhiezer function, Riemann surfaces

## 1 Introduction

Many problems of elasticity [19], [24], [5], [1], [2], electromagnetic diffraction [12], [16], [9], [21], [7], [8], [3], [4], and acoustic diffraction [18], [25], [6] require the solution of the vector Riemann-Hilbert problem (RHP) of the theory of analytic functions [27] , , where is either the whole real axis, or a finite segment, when the matrix has the Chebotarev-Khrapkov (also known as Daniele-Khrapkov) structure [10], [19], [12],

 G(t)=α1(t)I+α2(t)Q(t). (1.1)

Here, and are Hölder functions on , , and is a zero-trace polynomial matrix. In the case (, and has simple zeros only) the problem was solved in [19]. For a particular case of the matrix (1.1) and when , the exact solution was derived in [12]. For any finite , the vector problem is reduced [23] to a scalar RHP on a hyperelliptic surface of genus . A theory of the RHP on compact Riemann surfaces and a constructive procedure for the solution of the associated Jacobi inversion problem was proposed in [28] (see also [29]). This technique was further developed and adjusted to specific needs of the RHPs on hyperelliptic surfaces arising in elasticity [24], [5], diffraction theory in [6], [7], [8] and for symmetric vector RHPs in [3], [4]. The method for the vector RHP with the coefficient (1.1) in the elliptic and hyperelliptic cases first factorizes the coefficient of the associated scalar RHP using the Weierstrass analogue of the Cauchy kernel. In general, that solution has an essential singularity at the infinite points of the surface due to unavoidable poles of the Weierstrass kernel. The next step of the procedure, the removal of the essential singularities, leads to the classical problem of the inversion of abelian integrals and, eventually, to the finding of the zeros of a certain degree- polynomial.

The main goal of this paper was to develop a new factorization procedure for matrices of the form (1.1) based on the use of the Baker-Akhiezer function. The Baker-Akhiezer function plays an important role in the study of analytic properties of eigenfunctions of ordinary differential operators with periodic coefficients [13], [17], [15], [20], [14]. The representation of the Baker-Akhiezer function on a genus- hyperelliptic surface

 F(P)=eΩ(P)θ(u1(P)−σ1+V∘1,…,uρ(P)−σρ+V∘ρ)θ(u1(P)−σ1,…,uρ(P)−σρ) (1.2)

that we employ for the solution of the Wiener-Hopf matrix factorization problem was first written by A. R. Its in context of the finite gap solutions of the KdV equation [22]. Here, , is an abelian integral of the second kind with zero -periods and a certain prescribed polynomial growth at the infinite point of the surface , is the theta Riemann function, form the canonical basis of abelian integrals of the first kind, , are simple poles of the Baker-Akhiezer function, are the Riemann constants associated with the homology basis , , and , .

In section 2 we state the vector RHP in the real axis with the matrix coefficient (1.1) and reduce it to a scalar RHP on a hyperelliptic surface of the algebraic function . We derive a particular solution, , to the scalar RHP in section 3. This solution satisfies the boundary condition but has inadmissible essential singularities at the two infinite points and of the surface. In section 4 we construct the Baker-Akhiezer function (1.2) of the surface . This function is associated with an abelian integral of the second type with zero--periods used to remove the essential singularities and two Riemann -functions which serve to make the solution continuous through the -cross-sections. We find the Wiener-Hopf factors in terms of the functions and and the general solution to the vector RHP in section 5.

## 2 Scalar RHP on a Riemann surface associated with the Chebotarev-Khrapkov matrix

Motivated by numerous applications in acoustics, electromagnetic theory, fluid mechanics and elasticity we assume that the Riemann-Hilbert contour, , is the whole real axis which splits the plane of a complex variable into two half-planes, and . Let be a matrix which is nonsingular in and whose structure is

 G(t)=(α1(t)+α2(t)l0(t)α2(t)l1(t)α2(t)l2(t)α1(t)−α2(t)l0(t)), (2.1)

where , , , and are polynomials, and is the class of all Hölder functions in any finite interval in which tend to a definite limit as . For large , they satisfy the condition , , . Without loss of generality assume that . Let be an order-2 -vector-function on such that is the zero-vector. Consider the following RHP.

Given and find two vectors, and , analytic in the domains and , respectively, bounded at infinity, -continuous up to the contour and satisfying the boundary condition

 \boldmathΦ+(t)=G(t)\boldmathΦ−(t)+g(t),t∈L. (2.2)

Denote and . All zeros, , of are simple, while the zeros of the polynomial , , have multiplicity , respectively, and . Some or all zeros of the polynomial may have an odd multiplicity . In this case the -th zero is counted as a simple zero of and an order- zero of the polynomial . Assume that none of the zeros of and falls in the contour (we refer to [1] otherwise). In addition, we assume that is even, (this is true for all known applications of the problem (2.2) with the matrix coefficient (2.1) to elasticity and diffraction theory). This implies . Denote , , and for simplicity, accept that and , ().

Choose a single branch of in the plane cut along simple smooth disjoint arcs , , such that , . The functions

 λ1(t)=α1(t)+α2(t)h(t)√f(t),λ2(t)=α1(t)−α2(t)h(t)√f(t) (2.3)

are the eigenvalues of the matrix , and their product is the determinant of . To pursue the Wiener-Hopf factorization of , we split it as

 G(t)=T(t)Λ(t)[T(t)]−1, (2.4)

where ,

 T(z)=⎛⎝11−l0(z)−h(z)√f(z)l1(z)−l0(z)+h(z)√f(z)l1(z)⎞⎠, (2.5)

and reduce the problem of matrix factorization to a scalar RHP on a Riemann surface [23]. First we introduce two new vectors, and ,

 \boldmathψ(z)=[T(z)]−1\boldmathΦ(z),g∘(t)=[T(t)]−1g(t), (2.6)

where

 [T(z)]−1=⎛⎜ ⎜⎝l0(z)2h(z)√f(z)+12l1(z)2h(z)√f(z)−l0(z)2h(z)√f(z)+12−l1(z)2h(z)√f(z)⎞⎟ ⎟⎠. (2.7)

The components of the vector are expressed through the components of the vector as

 ψ1(z)=12[1+l0(z)h(z)√f(z)]Φ1(z)+l1(z)2h(z)√f(z)Φ2(z),
 ψ2(z)=12[1−l0(z)h(z)√f(z)]Φ1(z)−l1(z)2h(z)√f(z)Φ2(z). (2.8)

Similar formulas can be written for the components of the vectors and . The new functions and may grow at infinity if . Let . Then since the functions and are bounded as , we have , , , .

Due to continuity of the vector through the branch cuts (), we have , . This implies that the components of the vector satisfy the following Riemann-Hilbert boundary conditions:

 ψ+1(t)=ψ−2(t),ψ+2(t)=ψ−1(t),t∈γj,j=1,2,…,ρ+1,
 ψ+j(t)=λj(t)ψ−j(t)+g∘j(t),t∈L,j=1,2, (2.9)

and may have poles of multiplicity , at the zeros of the polynomial .

We wish to reformulate (2.9) as a scalar RHP on a Riemann surface. Let be the two-sheeted Riemann surface of the algebraic function formed by gluing two copies, and , of the extended complex plane along the cuts () such that

 w={√f(z),z∈C1,−√f(z),z∈C2, (2.10)

is a single-valued function on the surface . Here, is the branch chosen before. Let , () be a homology basis of the genus- surface (Figure 1).

Denote the contour on the surface with () being two copies of the contour . With each pair of the functions , and we associate the following functions on the surface :

 Ψ(z,w)=ψj(z),(z,w)∈Cj,
 λ(t,ξ)=λj(t),g∗(t,ξ)=g∘j(t),(t,ξ)∈Lj,j=1,2,ξ=w(t). (2.11)

The function may have simple poles at the branch points of the surface , (recall [26] that a branch point of the Riemann surface is called an order- pole of the function if , , const, and is a local uniformizing parameter of the point ). We also assert that the function is continuous through the contours (), and therefore the vector RHP (2.2) on the plane is equivalent to the following scalar RHP on the surface .

Find a piece-wise analytic function with the discontinuity contour , -continuous up to the contour , satisfying the boundary condition

 Ψ+(t,ξ)=λ(t,ξ)Ψ−(t,ξ)+g∗(t,ξ),(t,ξ)∈L, (2.12)

and having poles of multiplicity , in both sheets of the surface and simple poles at the branch points . In neighborhoods of the two infinite points of the surface the function satisfies the inequality , , .

## 3 Solution with an essential singularity at the infinite points

We begin with factorization of the function . For an analogue of the Cauchy kernel we choose the Weierstrass kernel

 dW=w+ξ2ξdtt−z (3.1)

and analyze the integral

 12πi∫Llogλ(t,ξ)dW=14πi∫L[logλ1(t)+logλ2(t)]dtt−z
 +w4πi∫L[logλ1(t)−logλ2(t)]dt√f(t)(t−z). (3.2)

Pick a point on , , and treat it as the starting point, , of the contour (it is convenient to take ). Let

 κj=indλj(t)=12π[argλj(t)]|L, (3.3)

where is the index of the function , and is the increment of as traverses the contour in the positive direction with being the starting point. Because of the continuity of the functions and in the contour both numbers, and , are integers. Fix branches of the logarithmic functions and by the condition , Then at the terminal point of the contour (to distinguish the terminal and starting points, we denote the former point as ), . Analysis of the singular integrals in the right-hand side (3.2) implies

 14πi∫L[logλ1(t)+logλ2(t)]dtt−z∼κ1+κ22log(z−z0),z→z0,
 w4πi∫L[logλ1(t)−logλ2(t)]dt√f(t)(t−z)∼κ1−κ22(−1)j−1log(z−z0),z→z0,(z,w)∈Cj. (3.4)

Consequently, the integral in the left-hand side (3.2) has a logarithmic singularity at the point in both sheets of the surface

 12πi∫Llogλ(t,ξ)dW∼κjlog(z−z0),z→z0,(z,w)∈Cj,j=1,2. (3.5)

It is an easy matter to move the singularity from the contour to the surface by adding the extra term

 I(z,w)=2∑m=1sgnκm|κm|∑j=1∫qmjqm0dW. (3.6)

Here, are smooth simple contours which do not intersect the contours , , , are arbitrary fixed points, are their affixes, and , . The function is continuous through the contour except for the points and at which the integral has logarithmic singularities. In addition, the integral has logarithmic singularities at the internal points ,

 I(z,w)∼−κmlog(z−z0),z→z0,(z,w)∈Cm,
 I(z,w)∼sgnκmlog(z−zmj),z→zmj,(z,w)∈Cm,j=1,…,|κm|,m=1,2. (3.7)

At the same time, the sum of the integral (3.2) and (3.6) does not have the singularity at the points . Now, to factorize the function , we use the function

 χ0(z,w)=exp⎧⎨⎩12πi∫Llogλ(t,ξ)dW+2∑m=1sgnκm|κm|∑j=1∫qmjqm0dW⎫⎬⎭. (3.8)

The function satisfies the homogeneous boundary condition

 χ+0(t,ξ)=λ(t,ξ)χ−0(t,ξ),(t,ξ)∈L, (3.9)

does not have singularities at the points and , but has inadmissible essential singularities at the points and . Also, it has simple zeros on the sheet if and simple poles on if (, ).

## 4 Baker-Akhiezer function

Our aim is to quench the essential singularities at the infinite points of the function by employing the Baker-Akhiezer function, , on the genus- surface associated with the function . The function has to satisfy the following two conditions:

(i) it is meromorphic everywhere on except at the points and ,

(ii) the function is bounded at the points and .

Setting

 χ0(z,w)=eβ0(z)+wβ1(z), (4.1)

where

 β0(z)=14πi∫L[logλ1(t)+logλ2(t)]dtt−z+122∑m=1sgnκm|κm|∑j=1∫zmjz0dtt−z,
 β1(z)=14πi∫L[logλ1(t)−logλ2(t)]dt√f(t)(t−z)+122∑m=1sgnκm|κm|∑j=1∫qmjqm0dtξ(t−z), (4.2)

we study the behavior of the function at the infinite points. For the branch chosen we have

 √f(z)= ⎷2ρ+2∏j=1(z−rj)=zρ+1∞∑m=0cmz−m, (4.3)

Here,

 c0=1,c1=(−1/2)11!2ρ+2∑j=1rj,
 c2=(−1/2)22!2ρ+2∑j=1r2j+[(−1/2)1]2(1!)22ρ+2∑j=1rj2ρ+2∑m=1,m≠jrm,
 c3=(−1/2)33!2ρ+2∑j=1r3j+(−1/2)1(−1/2)21!2!2ρ+2∑j=1r2j2ρ+2∑m=1,m≠jrm,…, (4.4)

where is the factorial symbol. By virtue of (4.2)

 β1(z)=∞∑j=0~cjzj+1, (4.5)

where

 ~cj=−14πi∫L[logλ1(t)−logλ2(t)]tjdt√f(t)−122∑m=1sgnκm|κm|∑j=1∫qmjqm0tjdtξ, (4.6)

and therefore, as ,

 √f(z)β1(z)=zρ∞∑m=0dmzm,dm=m∑k=0ck~cm−k. (4.7)

This brings us to the expansion of the function at the infinite points

 χ0(z,w)=exp{(−1)j−1M(z)+O(1)},(z,w)→∞j,(z,w)∈Cj,j=1,2, (4.8)

where

 M(z)=d0zρ+d1zρ−1+…+dρ−1z. (4.9)

Our next step is to construct a special abelian integral of the second kind,

 Ω(P)=∫PP0dΩ,P0=(r2ρ+2,0),P=(z,w). (4.10)

Determine by the following properties:

(a) , , ,

(b) .

We seek the abelian differential in the form

 dΩ=e0z2ρ+e1z2ρ−1+…+e2ρwdz, (4.11)

where the coefficients are to be determined. We wish to exploit this formula in order to study the behavior of the integral at the infinite points. Because of (4.3) we have

 dΩ=(−1)j−1(~e0zρ−1+~e1zρ−2+…+~eρ−1+~eρz+…)dz, (4.12)

where are defined recursively by

 ~em=−m∑k=1~em−kck+em,m=0,1,…,ρ. (4.13)

By integrating (4.12) we determine the asymptotic expansion of the abelian integral

 Ω(P)=(−1)j−1(~e0zρρ+~e1zρ−1ρ−1+…+~eρ−1z+~eρlogz−~eρ+1z−…)+Kj,
 P→∞j∈Cj,j=1,2, (4.14)

where and are constants. On satisfying the property (a) of the integral we find the coefficients , …,

 ~e0=−ρd0,~e1=−(ρ−1)d1,~e2=−(ρ−2)d2,…,~eρ−1=−dρ−1,~eρ=0. (4.15)

Due to (4.13) we can express the coefficients () through

 em=~em+m∑k=1~em−kck,m=0,1,…,ρ. (4.16)

The remaining coefficients in the representation (4.11) of the abelian differential are fixed by solving the system of linear algebraic equations

 2ρ∑m=ρ+1Ujmem=^dj,j=1,2,…,ρ, (4.17)

which follows from the property (b) of the integral . Here,

 ^dj=−ρ∑m=0Ujmem,Ujm=∫ajz2ρ−mw(z)dz. (4.18)

This completes the construction of the abelian integral .

It becomes evident that the product is bounded as , . This function is continuous through the cross-sections of the surface because of the zero -periods and discontinuous through the cross-sections () due to the non-zero -periods of the integral . Our efforts will now be directed towards annihilating the jumps ,

 Vm=∫bmdΩ,m=1,2,…,ρ, (4.19)

of the function through the cross-sections , .

Let () be the canonical basis of Abelian differentials of the first kind

 dωj=c(1)jzρ−1+c(2)jzρ−2+…+c(ρ)jwdz, (4.20)

where the constants () are chosen such that

 ∫akdωj=δjk. (4.21)

Denote the periods of the basis by

 Bjk=∫bkdωj. (4.22)

The matrix () is symmetric and is positive definite. The principal tool we shall use to suppress the discontinuities of is the Riemann -function

 θ(s(P))=θ(s1(P),s2(P),…,sρ(P)) (4.23)

defined by

 θ(s(P))=∞∑n1,…,nρ=−∞exp{ρ∑j=1ρ∑k=1Bjknjnk+2πiρ∑j=1njsj(P)}. (4.24)

Because of the positive definiteness of the matrix the series converges for all . The -function has periods , are integers, and quasiperiods , ,

 θ(s1+n1,…,sρ+nρ)=θ(s1,…,sρ),
 θ(s1+Bj1,…,sρ+Bjρ)=exp{−πiBjj−2πisj}θ(s1,…,sρ). (4.25)

Introduce next the function

 F0(P)=θ(u1(P)−σ1+V∘1,…,uρ(P)−σρ+V∘ρ)θ(u1(P)−σ1,…,uρ(P)−σρ). (4.26)

Here, and are the integrals

 uj(P)=∫PP0dωj,j=1,2,…,ρ, (4.27)

which form the canonical basis of abelian integrals of the first kind. It is convenient to choose as the branch point . The numbers are chosen to be

 σj=ρ∑m=1uj(Pm)+kj,j=1,…,ρ, (4.28)

where () are some arbitrary distinct fixed points on say, on ,