Global Solution of the Inverse Spectral Problem for Differential Operators on
a Finite Interval with Complex Weights
V.A. Yurko

Abstract. Non-self-adjoint second-order ordinary differential operators on a
finite interval with complex weights are studied. Properties of spectral
characteristics are established and the inverse problem of recovering
operators from their spectral characteristics are investigated. For this class
of nonlinear inverse problems an algorithm for constructing the global
solution is obtained. To study this class of inverse problems, we develop
ideas of the method of spectral mappings.

Keywords: differential operators, complex weight, spectral characteristics,
inverse problem, method of spectral mappings

Mathematics Subject Classification: 34A55, 34B24, 47E05.

1\. Introduction

We consider the boundary value problem $L$ for the differential equation

$-y^{\prime\prime}(x)+q(x)y(x)=\lambda r(x)y(x),\quad 0<x<T,$ $None$

subject to the Robin boundary conditions

$U(y):=y^{\prime}(0)-hy(0)=0,\quad V(y):=y^{\prime}(T)+Hy(T)=0,$ $None$

and the jump conditions at an interior point $b\in(0,T)$:

$y(b+0)=d_{1}y(b-0),\quad y^{\prime}(b+0)=y(b-0)/d_{1}+d_{2}y(b-0).$ $None$

Here $\lambda$ is the spectral parameter, $q(x)$ and $r(x)$ are complex-valued
functions, $q(x)\in L(0,T),$ and $r(x)=a_{k}^{2}$ for $x\in(b_{k-1},b_{k}),$
where $0=b_{0}<b_{1}=b<b_{2}=T.$ The numbers $h,H,a_{k}$ and $d_{k}$ are
complex, and $a_{k}\neq 0,d_{1}\neq 0.$ For definiteness, let $\arg
d_{1}\in[0,\pi).$

We study the inverse spectral problem for the boundary value problem
(1.1)(1.3). Inverse spectral problems consist in recovering operators from
their spectral characteristics. Such problems play an important role in
mathematics and have many applications in natural science and technology.
Inverse spectral problems are also used for solving nonlinear integrable
evolution equations of mathematical physics. Inverse problems for the
classical SturmLiouville operators (when $r(x)\equiv 1,$ $d_{1}=1,$ and
$d_{2}=0$) have been studied fairly completely (see [1] and the historical
review therein). Inverse problems for arbitrary order differential operators
and systems with arbitrary characteristic numbers are more difficult. They
have been solved later by the method of spectral mappings (see the monographs
[2][3] and the references therein). Inverse problems on spatial networks are
an important and popular part of the inverse problem theory; in the review
paper [4] one can find the main results on inverse problems on spatial
networks. Boundary value problems with discontinuous weights and jump
conditions at interior points have been considered in many papers, but mostly
for the case with real weights. In the case when $r(x)\equiv 1$ (i.e.
$a_{k}=1$), the boundary value problem $L$ satisfying conditions (3) was
studied in [5][9] and other papers. Inverse problems for a real weight $r(x)$
were studied in [10][14] and other works. Inverse problems for the boundary
value problem $L$ with complex-valued weights were studied in [15]-[16] where
only uniqueness results were obtained. Note that complex-valued weights
appear, in particular, in the study of the interaction of electromagnetic
waves with layered media possessing both dielectric and magnetic properties
[17]. Moreover, a number of problems for SturmLiouville equations on curves in
the complex plane can be reduced to the boundary-value problem $L$ of the form
(1)(3) on a real interval. In the present paper, we establish properties of
the spectral characteristics for $L,$ and study the inverse spectral problem
of recovering parameters of $L$ from the given spectral characteristics. For
this class of nonlinear inverse problems an algorithm for constructing the
global solution is obtained. To study this class of inverse problems, we
develop ideas related to the method of spectral mappings [2].

2\. Spectral data

Let $l_{k}:=b_{k}-b_{k-1}$ and $a_{k}=r_{k}\exp(i\varphi_{k}),$ $r_{k}>0,$
$0\leq\varphi_{2}<\varphi_{1}<\pi.$ We assume that the following regularity
condition holds: $\omega_{\pm}:=d_{1}a_{2}\pm a_{1}/d_{1}\neq 0.$ Denote by
$\Phi(x,\lambda)$ the solution of (1.1) such that (1.3) holds and $U(\Phi)=1,$
$V(\Phi)=0.$ Let $M(\lambda):=\Phi(0,\lambda).$ We will also use the solutions
$\varphi(x,\lambda),\psi(x,\lambda),S(x,\lambda)$ of Eq. (1.1) satisfying
(1.3) and the conditions
$\varphi(0,\lambda)=1,\varphi^{\prime}(0,\lambda)=h,S(0,\lambda)=0,S^{\prime}(0,\lambda)=1,\psi(T,\lambda)=1,\psi^{\prime}(T,\lambda)=-H.$
Denote
$D(x,\lambda,\mu):=(\lambda-\mu)^{-1}\langle\varphi(x,\lambda),\varphi(x,\mu)\rangle,$
where $\langle y(x),z(x)\rangle:=y(x)z^{\prime}(x)-y^{\prime}(x)z(x).$ The
function

$\Delta(\lambda):=\langle\varphi(x,\lambda),\psi(x,\lambda)\rangle=-V(\varphi)=U(\psi)$
$None$

does not depend on $x,$ and it is called the characteristic function for $L.$
The eigenvalues $\Lambda:=\\{\lambda_{k}\\}_{k\geq 0}$ of $L$ coincide with
the zeros of the entire function $\Delta(\lambda).$ Clearly,

$\Phi(x,\lambda)=S(x,\lambda)+M(\lambda)\varphi(x,\lambda)=\psi(x,\lambda)/\Delta(\lambda),\;M(\lambda)=\Delta_{0}(\lambda)/\Delta(\lambda),$
$None$

where $\Delta_{0}(\lambda):=\psi(0,\lambda)=V(S).$ Using (2.1) and (2.2) one
gets

$\langle\varphi(x,\lambda),\Phi(x,\lambda)\rangle\equiv 1.$ $None$

Let $\lambda=\rho^{2},\lambda_{k}=\rho_{k}^{2}.$ Consider the half-planes
$\Pi_{k}^{\pm}:=\\{\rho:\;\pm Im(\rho a_{k})>0\\},$ $k=1,2,$ and denote

$S_{1}=\Pi_{1}^{+}\cup\Pi_{2}^{+},\;S_{2}=\Pi_{1}^{-}\cup\Pi_{2}^{+},\;S_{3}=\Pi_{1}^{-}\cup\Pi_{2}^{-},\;S_{4}=\Pi_{1}^{+}\cup\Pi_{2}^{-}.$

Then $S_{j}=\\{\rho:\;\arg\rho\in(\theta_{j},\theta_{j+1})\\},$ where
$\theta_{1}=\theta_{5}=-\varphi_{2},\theta_{2}=\pi-\varphi_{1},\theta_{3}=\pi-\varphi_{2},\theta_{4}=-\varphi_{1}$.
For sufficiently small $\delta>0$ we construct the sectors
$S_{j,\delta}:=\\{\rho:\;\arg\rho\in(\theta_{j}+\delta,\theta_{j+1}-\delta)\\}.$

Let $\\{e_{k}(x,\rho)\\}_{k=1,2},\;x\in[0,b]$ and
$\\{E_{k}(x,\rho)\\}_{k=1,2},\;x\in[b,T]$ be the Birkhoff-type fundamental
systems of solutions (FSS’s) of Eq. (1.1) with the asymptotics as
$|\rho|\to\infty,$ $\rho\in\overline{S_{j}},$ $\nu=0,1$ (see [1]):

$e^{(\nu)}_{k}(x,\rho)=((-1)^{k-1}i\rho a_{1})^{\nu}\exp((-1)^{k-1}i\rho
a_{1}x)[1],\quad x\in[0,b],$ $E^{(\nu)}_{k}(x,\rho)=((-1)^{k-1}i\rho
a_{2})^{\nu}\exp((-1)^{k-1}i\rho a_{2}(x-b))[1],\quad x\in[b,T].$

where $[1]=1+O(1/\rho).$ The functions $e^{(\nu)}_{k}(x,\rho)$ and
$E^{(\nu)}_{k}(x,\rho)$ are regular for $\rho\in S_{j}$, $|\rho|>\rho^{*}$ and
continuous for $\rho\in\overline{S_{j}}$, $|\rho|\geq\rho^{*}$ for some
$\rho^{*}>0.$ Using these FSS’s and the jump conditions (1.3) we get the
following asymptotical formulas as $|\rho|\to\infty,$ $\nu=0,1$:

$\varphi^{(\nu)}(x,\lambda)=\Big{(}(i\rho a_{1})^{\nu}\exp(i\rho
a_{1}x)[1]+(-i\rho a_{1})^{\nu}\exp(-i\rho a_{1}x)[1]\Big{)}/2,\;x\in[0,b],$
$\varphi^{(\nu)}(x,\lambda)=\Big{(}\Big{(}\omega_{+}\exp(i\rho
a_{1}l_{1})[1]+\omega_{-}\exp(-i\rho a_{1}l_{1})[1]\Big{)}(i\rho
a_{2})^{\nu}\exp(i\rho a_{2}(x-b))[1]+$ $\Big{(}\omega_{-}\exp(i\rho
a_{1}l_{1})[1]+\omega_{+}\exp(-i\rho a_{1}l_{1})[1]\Big{)}(-i\rho
a_{2})^{\nu}\exp(-i\rho a_{2}(x-b))[1]\Big{)}/(4a_{2}),\;x\in[b,T],$
$\psi^{(\nu)}(x,\lambda)=\Big{(}\Big{(}\omega_{+}\exp(i\rho
a_{2}l_{2})[1]-\omega_{-}\exp(-i\rho a_{2}l_{2})[1]\Big{)}(i\rho
a_{1})^{\nu}\exp(i\rho a_{1}(b_{1}-x))[1]+$ $\Big{(}-\omega_{-}\exp(i\rho
a_{2}l_{2})[1]+\omega_{+}\exp(-i\rho a_{2}l_{2})[1]\Big{)}(-i\rho
a_{1})^{\nu}\exp(-i\rho a_{1}(b_{1}-x))[1]\Big{)}/(4a_{1}),\;x\in[0,b],$
$\psi^{(\nu)}(x,\lambda)=\Big{(}(-i\rho a_{2})^{\nu}\exp(i\rho
a_{2}(T-x))[1]+(i\rho a_{2})^{\nu}\exp(-i\rho
a_{2}(T-x))[1]\Big{)}/2,\;x\in[b,T].$

In view of (2.1), these formulas yield

$\Delta(\lambda)=(-i\rho)\Big{(}\Big{(}\omega_{+}\exp(i\rho
a_{1}l_{1})[1]+\omega_{-}\exp(-i\rho a_{1}l_{1})[1]\Big{)}\exp(i\rho
a_{2}l_{2})[1]-$ $\Big{(}\omega_{-}\exp(i\rho
a_{1}l_{1})[1]+\omega_{+}\exp(-i\rho a_{1}l_{1})[1]\Big{)}\exp(-i\rho
a_{2}l_{2})[1]\Big{)}/4,\;|\rho|\to\infty,$ $None$ $M(\lambda)=\pm(i\rho
a_{1})^{-1}[1],\quad\rho\in\Pi_{1}^{\pm}.$ $None$

Using (2.4) by the known technique (see [1, Ch.1]) we obtain that the spectrum
$\Lambda$ of $L$ consists of two subsequences
$\Lambda=\\{\lambda_{k}\\}=\\{\lambda_{k1}\\}\cup\\{\lambda_{k2}\\},$ and

$\rho_{kj}=\sqrt{\lambda_{kj}}=\frac{k\pi}{r_{j}l_{j}}\exp(i\theta_{3-j})+C_{j}+O(1/k),\;k\to\infty,$
$None$

where $C_{1}=-(2ia_{1}l_{1})^{-1}\ln(-\omega_{-}/\omega_{+}),$
$C_{2}=(2ia_{2}l_{2})^{-1}\ln(\omega_{+}/\omega_{-}).$ Moreover,

$|\Delta(\lambda)|\geq C|\rho{\cal E}_{1}(\rho l_{1}){\cal E}_{2}(\rho
l_{2})|,\;|M(\lambda)|\leq C/|\rho|,\;\lambda\in G_{\delta},$ $None$
$|\varphi(x,\lambda)|\leq C|{\cal E}_{1}(\rho
x)|,\;x\in(0,b),\quad\forall\,\lambda,$ $|\varphi(x,\lambda)|\leq C|{\cal
E}_{1}(\rho x){\cal E}_{2}(\rho(x-b))|,\;x\in(b,T),\quad\forall\,\lambda,$
$|\Phi(x,\lambda)|\leq C|\rho{\cal E}_{1}(\rho
x)|^{-1},\;x\in(0,b),\quad\lambda\in G_{\delta},$ $|\Phi(x,\lambda)|\leq
C|\rho{\cal E}_{1}(\rho x){\cal
E}_{2}(\rho(x-b))|^{-1},\;x\in(b,T),\quad\lambda\in G_{\delta},$

where $G_{\delta}:=\\{\rho:\;|\rho-\rho_{k}|\\}\geq\delta,$ ${\cal E}_{k}(\rho
x):=\exp(\pm i\rho a_{k}x)$ for $\rho\in\Pi_{k}^{\pm},$ $x\in l_{k}$. Let
$m_{k}$ be the multiplicity of the eigenvalue $\lambda_{k}$
($\lambda_{k}=\lambda_{k+1}=\ldots=\lambda_{k+m_{k}-1}$), and put $S:=\\{k\geq
1:\;\lambda_{k-1}\neq\lambda_{k}\\}\cup\\{0\\}.$ It follows from (2.6) that
for sufficiently large $k$ ($k>k^{*}$) all eigenvalues are simple, i.e.
$m_{k}=1$ for $k>k^{*}.$ Similar to [18] one gets

$M(\lambda)=\sum_{k\in
S}\sum_{\nu=0}^{m_{k}-1}\frac{M_{k+\nu}}{(\lambda-\lambda_{k})^{\nu+1}}\,,$
$None$

where
$\displaystyle\sum_{\nu}\displaystyle\frac{M_{k+\nu}}{(\lambda-\lambda_{k})^{\nu+1}}$
is the principal part of $M(\lambda)$ in a neighborhood of $\lambda_{k}$. The
sequence ${\cal M}=\\{M_{k}\\}_{k\geq 0}$ is called the Weyl sequence of $L,$
and the data $W=\\{\lambda_{k},M_{k}\\}_{k\geq 0}$ are called the spectral
data of $L.$ Similar to (2.6) we calculate ${\cal
M}=\\{M_{k1}\\}\cup\\{M_{k2}\\},$ and

$M_{k1}=\frac{2}{a_{1}^{2}l_{1}}\Big{(}1+O\Big{(}\frac{1}{k}\Big{)}\Big{)},\;M_{k2}=\frac{8}{\omega_{-}\omega_{+}l_{2}}\exp\Big{(}\frac{2k\pi
r_{1}l_{1}}{r_{2}l_{2}}(\cos\alpha+i\sin\alpha)\Big{)}\Big{(}1+O\Big{(}\frac{1}{k}\Big{)}\Big{)},$
$None$

as $k\to\infty.$ Here $\alpha:=\varphi_{1}-\varphi_{2}+\pi/2.$ Note that
$\cos\alpha<0,$ since $\alpha\in(\pi/2,3\pi/2).$ Using (2.4), (2.6), (2.7),
(2.9) and the asymptotical formulas for $\varphi(x,\lambda)$ and
$\psi(x,\lambda),$ we obtain the estimates

$|\varphi(x,\lambda_{k1})|\leq C,\quad|\varphi(x,\lambda_{k2})|\leq
C\exp\Big{(}\frac{-k\pi r_{1}l_{1}\cos\alpha}{r_{2}l_{2}}\Big{)},\quad
x\in[0,T].$ $None$

It follows from (2.5) and (2.6) that

$a_{1}=\lim_{|\rho|\to\infty}(i\rho M(\lambda))^{-1},\quad\rho\in\Pi_{1}^{+},$
$None$ $l_{1}=b=-\lim_{k\to\infty}(k\pi/(a_{1}\rho_{k1})),\quad
l_{2}=T-l_{1},\quad a_{2}=\lim_{k\to\infty}(k\pi/(l_{2}\rho_{k2}),$ $None$
$A:=\omega_{+}/\omega_{-}=\lim_{k\to\infty}\exp(2i\rho_{k2}a_{2}l_{2}),\quad
d_{1}=\sqrt{(a_{1}(A+1))/(a_{2}(A-1))}.$ $None$

3\. Inverse problem

In this paper we consider the following inverse problem.

Inverse problem 3.1.  Given the Weyl function $M(\lambda)$ (or the spectral
data $W$), construct $L.$

According to (2.8) the specification of the Weyl function is equivalent to the
specification of the spectral data.

Firstly, we will prove the uniqueness theorem. For this purpose, together with
$L$ we consider a boundary value problem $\tilde{L}$ of the same form but with
$\tilde{q}(x),\tilde{b},\tilde{r}(x),\tilde{h},\tilde{H},\tilde{d}_{1},\tilde{d}_{2}$
instead of $q(x),b,r(x),h,H,d_{1},d_{2}$. We agree that if a certain symbol
$\chi$ denotes an object related to $L,$ then $\tilde{\chi}$ will denote an
analogous object related to $\tilde{L}.$

Theorem 3.1. If $M(\lambda)\equiv\tilde{M}(\lambda)$ (or $W=\tilde{W}$), then
$L=\tilde{L}.$ Thus, the specification of the Weyl function (or the spectral
data) uniquely determines the functions $q(x),r(x)$ and the parameters
$b,h,H,d_{1},d_{2}$.

Proof.  It follows form (2.11)-(2.13) that $b=\tilde{b},$
$a_{k}=\tilde{a}_{k},$ $d_{1}=\tilde{d}_{1}.$ We construct the functions

${\cal P}_{0}=\Phi\tilde{\varphi}-\varphi\tilde{\Phi},\quad{\cal
P}_{1}=\varphi\tilde{\Phi}^{\prime}-\Phi\tilde{\varphi}^{\prime}.$ $None$

In view of (2.3), this yields

$\varphi={\cal P}_{1}\tilde{\varphi}+{\cal
P}_{0}\tilde{\varphi}^{\prime},\quad\Phi={\cal P}_{1}\tilde{\Phi}+{\cal
P}_{0}\tilde{\Phi}^{\prime},\quad{\cal
P}_{1}-1=\varphi(\tilde{\Phi}^{\prime}-\Phi^{\prime})-\Phi(\tilde{\varphi}^{\prime}-\varphi^{\prime}).$
$None$

Using (2.2), (3.1), (3.2) and the asymptotical formulas for $\varphi$ and
$\psi,$ we infer

$|{\cal P}_{1}(x,\lambda)-1|\leq C/|\rho|,\quad|{\cal P}_{0}(x,\lambda)|\leq
C/|\rho|,\quad\rho\in G_{\delta}\cap\tilde{G}_{\delta}.$ $None$

Taking (2.2), (3.1) and the assumption of the theorem into account, we
conclude that the functions ${\cal P}_{k}(x,\lambda)$ are entire in $\lambda$
for each $x.$ Together with (3.3) this yields ${\cal P}_{1}(x,\lambda)\equiv
1,$ ${\cal P}_{0}(x,\lambda)\equiv 0.$ Using (3.2) we calculate
$\varphi(x,\lambda)\equiv\tilde{\varphi}(x,\lambda),$
$\Phi(x,\lambda)\equiv\tilde{\Phi}(x,\lambda),$ hence $L=\tilde{L}.$ Theorem
3.1 is proved.

Let us go on to deriving a constructive solution of the inverse problem. For
this purpose we will use ideas of the method of spectral mappings [2]. We will
reduce our nonlinear inverse problem to the solution of the so-called main
equation, which is a linear equation in a corresponding Banach space of
sequences. We give a derivation of the main equation, and prove its unique
solvability. Using the solution of the main equation we provide an algorithm
for the solution of the inverse problem considered. For simplicity, in the
sequel we confine ourselves to the case when the function $\Delta(\lambda)$
has only simple zeros (the general case requires minor technical
modifications).

Let the Weyl function $M(\lambda)$ and the spectral data $W$ be given. Using
(2.15)-(2.17) we compute $b,a_{k}$ and $d_{1}$. Then we choose a model
boundary value problem $\tilde{L}$ such that
$\tilde{b}=b,\tilde{a}_{k}=a_{k},\tilde{d}_{1}=d_{1}$, and arbitrary in the
rest (for example, we can take $\tilde{q}=0$). Let $\theta_{k}:=1$ if
$\lambda_{k}=\lambda_{k1}$, and $\theta_{k}:=\exp(-k\pi
r_{1}l_{1}(r_{2}l_{2})^{-1}\cos\alpha)$ if $\lambda_{k}=\lambda_{k2}$. Denote

$\xi_{k}:=|\rho_{k}-\tilde{\rho}_{k}|+|M_{k}-\tilde{M}_{k}|\theta_{k}^{2},\;z_{k0}:=\lambda_{k},\;z_{k1}:=\tilde{\lambda}_{k},\;\beta_{k0}:=M_{k},\;\beta_{k1}:=\tilde{M}_{k}.$

By virtue of (2.6) and (2.9) one has $\xi_{k}=O(1/k).$ Consider the functions

$\varphi_{kj}(x):=\varphi(x,z_{kj}),\;\tilde{\varphi}_{kj}(x):=\tilde{\varphi}(x,z_{kj}),\;j=0,1,$
$B_{ni,kj}(x):=D(x,z_{ni},z_{kj})\beta_{kj},\;\tilde{B}_{ni,kj}(x):=\tilde{D}(x,z_{ni},z_{kj})\beta_{kj},\;i,j=0,1,$
$f_{k0}(x):=(\varphi_{k0}(x)-\varphi_{k1}(x))/(\xi_{k}\theta_{k}),\;f_{k1}(x):=\varphi_{k1}(x)/\theta_{k},$
$A_{n0,k0}(x):=(B_{n0,k0}(x)-B_{n1,k0}(x))\xi_{k}\theta_{k}/(\xi_{n}\theta_{n}),$
$A_{n1,k1}(x):=(B_{n1,k0}(x)-B_{n1,k1}(x))\theta_{k}/\theta_{n},\;A_{n1,k0}(x):=B_{n1,k0}(x)\xi_{k}\theta_{k}/\theta_{n},$
$A_{n0,k1}(x):=(B_{n0,k0}(x)-B_{n1,k0}(x)-B_{n0,k1}(x)+B_{n1,k1}(x))\theta_{k}/(\xi_{n}\theta_{n}).$

Similarly $\tilde{f}_{kj}(x)$ and $\tilde{A}_{ni,kj}(x)$ are defined. Using
(2.6), (2.9), (2.10) and the asymptotical formulas for $\varphi(x,\lambda)$ we
get

$|f_{kj}(x)|,\;|\tilde{f}_{kj}(x)|\leq
C,\quad|A_{ni,kj}(x)|,|\tilde{A}_{ni,kj}(x)|\leq C\xi_{k}(|n-k|+1)^{-1}.$
$None$

Denote by $V$ the set of indices $u=(n,i),$ where $n\geq 0,\,i=0.1.$

Theorem 3.2. The following relation holds

$\tilde{f}_{ni}(x)=f_{ni}(x)+\sum_{(k,j)\in
V}\tilde{A}_{ni,kj}(x)f_{kj}(x),\;(n,i)\in V,$ $None$

where the series converge absolutely and uniformly on $x\in[0,T]$ and
$\lambda$ on compact sets.

Proof.  Consider the contours $\Gamma_{N}:=\\{\lambda:\,|\lambda|=R_{N}\\},$
where $R_{N}\to\infty$ such that $\Gamma_{N}\subset G_{\delta}$. Denote ${\cal
S}_{k}:=\\{\rho:\,Im(\rho a_{k})=0\\},$ ${\cal S}_{0}:={\cal S}_{1}\cup{\cal
S}_{2}$, ${\cal S}:=\\{\rho:\,dist({\cal S}_{0},\rho)=\delta\\},$ where
$\delta>0$ is such that $\Lambda\cup\tilde{\Lambda}\subset\,int\,{\cal S}.$
Let $\gamma$ be the image of ${\cal S}$ in the $\lambda$\- plane, and
$\Gamma^{\prime}_{N}:=\Gamma_{N}\cap int\,\gamma,$
$\Gamma^{\prime\prime}_{N}:=\Gamma_{N}\setminus\Gamma^{\prime}_{N}$,
$\gamma^{*}_{N}:=\gamma\cap int\,\Gamma_{N}$. Denote by
$\gamma_{N}:=\gamma^{*}_{N}\cup\Gamma^{\prime}_{N}$ and
$\gamma^{0}_{N}:=\gamma^{*}_{N}\cup\Gamma^{\prime\prime}_{N}$ the closed
contours with counterclockwise circuit. Applying Cauchy’s integral formula we
get

${\cal P}_{k}(x,\lambda)-\delta_{1k}=\frac{1}{2\pi
i}\int_{\gamma^{0}_{N}}\frac{{\cal
P}_{k}(x,\mu)-\delta_{1k}}{\lambda-\mu}d\mu=\frac{1}{2\pi
i}\int_{\gamma_{N}}\frac{{\cal P}_{k}(x,\mu)}{\lambda-\mu}d\mu-\frac{1}{2\pi
i}\int_{\Gamma_{N}}\frac{{\cal P}_{k}(x,\mu)-\delta_{1k}}{\lambda-\mu}d\mu,$

where $k=0,1,$ $\lambda\in int\,\gamma^{0}_{N}$, and $\delta_{jk}$ is the
Kronecker delta. Taking (3.2) into account we calculate

$\varphi(x,\lambda)=\tilde{\varphi}(x,\lambda)+\frac{1}{2\pi
i}\int_{\gamma_{N}}\Big{(}\tilde{\varphi}(x,\lambda){\cal
P}_{1}(x,\mu)+\tilde{\varphi}^{\prime}(x,\lambda){\cal
P}_{0}(x,\mu)\Big{)}\frac{d\mu}{\lambda-\mu}+\varepsilon_{N}(x,\lambda).$

In view of (3.3), one has
$\displaystyle\lim_{N\to\infty}\varepsilon_{N}(x,\lambda)=0$ uniformly in
$x\in[0,T]$ and $\lambda$ on compact sets. Taking (3.1) and (2.2) into account
we obtain

$\tilde{\varphi}(x,\lambda)=\varphi(x,\lambda)+\frac{1}{2\pi
i}\int_{\gamma_{N}}\tilde{D}(x,\lambda,\mu)(M(\mu)-\tilde{M}(\mu))\varphi(x,\mu)d\mu+\varepsilon_{N}(x,\lambda).$

Note that the terms with $S(x,\lambda)$ and $\tilde{S}(x,\lambda)$ are zero
because of Cauchy’s theorem. Using the residue theorem we get the relation

$\tilde{\varphi}_{ni}(x)=\varphi_{ni}(x)+\sum_{k=0}^{\infty}\Big{(}\tilde{B}_{ni,k0}(x)\varphi_{k0}(x)-\tilde{B}_{ni,k1}(x)\varphi_{k1}(x)\Big{)},$

which is equivalent to (3.5). Theorem 3.2 is proved.

By similar arguments we calculate

$A_{ni,kj}(x)-\tilde{A}_{ni,kj}(x)+\sum_{(l,s)\in
V}\tilde{A}_{ni,ls}(x)A_{ls,kj}(x)=0,\;(n,i),(k,j)\in V.$ $None$

Let $f(x)=[f_{u}(x)]_{u\in V}$, $A(x)=[A_{u,v}(x)]_{u,v\in V}$,
$\tilde{f}(x)=[\tilde{f}_{u}(x)]_{u\in V}$,
$\tilde{A}(x)=[\tilde{A}_{u,v}(x)]_{u,v\in V}$. We denote by $m$ the Banach
space of bounded sequences $\chi=[\chi_{u}]_{u\in V}$ with the norm
$\|\chi\|=\sup_{u\in V}|\chi_{u}|.$ According to (3.4), one has that for each
fixed $x,$ the operators $I+\tilde{A}(x)$ and $I-A(x),$ acting from $m$ to
$m,$ are linear bounded operators. Relations (3.5) and (3.6) can be written as
follows

$\tilde{f}(x)=(I+\tilde{A}(x))f(x),\quad(I+\tilde{A}(x))(I-A(x))=I.$

Symmetrically one has
$f(x)=(I-A(x))\tilde{f}(x),\quad(I-A(x))(I+\tilde{A}(x))=I.$ Thus, for each
fixed $x,$ the operator $I+\tilde{A}(x)$ has a bounded inverse operator, hence
the linear equation $\tilde{f}(x)=(I+\tilde{A}(x))f(x)$ is uniquely solvable.
This equation is called the main equation of the inverse problem. Solving the
main equation we find the vector $f(x),$ and also the solutions
$\varphi_{ni}(x)=\varphi(x,\lambda_{ni})$ of Eq. (1.1), hence we can construct
$q(x),h,H$ and $d_{2}$. Thus, the solution of the inverse problem can be found
by the following algorithm.

Algorithm 3.1. Given the Weyl function $M(\lambda)$ and the spectral data $W.$
1) Calculate $b,a_{k}$ and $d_{1}$ via (2.11)-(2.13).
2) Choose a model boundary value problem $\tilde{L}$ such that
$\tilde{b}=b,\tilde{a}_{k}=a_{k},\tilde{d}_{1}=d_{1}$.
3) Construct $\tilde{f}(x)$ and $\tilde{A}(x)$ (see above).
4) Find $f(x)=[f_{u}]_{u\in V}$ by solving the main equation
$\tilde{f}(x)=(I+\tilde{A}(x))f(x).$
5) Calculate
$\varphi_{n1}(x)=f_{n1}(x)\theta_{n},\varphi_{n0}=\varphi_{n1}(x)+f_{n0}(x)\xi_{n}\theta_{n}$.
6) Find $q(x),h,H$ and $d_{2}$ using (1.1)-(1.3).

Remark 3.1.  We can also calculate $q(x)$ by the formula
$q(x)=\tilde{q}(x)-2{\cal F}(x),$ where

${\cal
F}(x)=\frac{d}{dx}\sum_{k=0}^{\infty}\Big{(}M_{k0}\tilde{\varphi}_{k0}(x)\varphi_{k0}(x)-M_{k1}\tilde{\varphi}_{k1}(x)\varphi_{k1}(x)\Big{)}.$
$None$

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Vjacheslav A. Yurko<EMAIL_ADDRESS>https://orcid.org/0000-0002-4853-0102