\documentclass[12pt,a4paper]{article}
\usepackage[cp1251]{inputenc} % cp866 - DOS, cp1251 - Windows
%\usepackage[russian]{babel}
\usepackage{amsfonts}

%\documentstyle[12pt]{article}
\pagestyle{myheadings}
\textwidth=16.7cm
\textheight=26.0cm
\oddsidemargin=0.0cm
\mathsurround=3pt
\topmargin=-1.5cm

\newcommand{\di}{\displaystyle}
\newcommand{\B}{$\hfill\Box$}
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\de}{\delta}
\newcommand{\la}{\lambda}
\newcommand{\om}{\omega}
\newcommand{\ee}{\varepsilon}
\newcommand{\vv}{\varphi}
\newcommand{\iy}{\infty}

\begin{document}

\begin{center}
{\large\bf
Recovering Differential Pencils on Graphs with a Cycle from Spectra}\\[0.2cm]
{\bf V.\,Yurko} \\[0.2cm]
\end{center}

\thispagestyle{empty}

\noindent {\bf Abstract.} We study boundary value problems on compact graphs with a cycle 
for second-order ordinary differential equations with nonlinear dependence on the 
spectral parameter. We establish properties of the spectral characteristics and 
investigate inverse spectral problems of recovering coefficients of the differential 
equation from spectra. For these inverse problems we prove uniqueness theorems and provide 
procedures for constructing their solutions.

\smallskip 
\noindent Key words:  geometrical graphs with a cycle, differential pencils, inverse spectral problems 

\smallskip
\noindent AMS Classification:  34A55  34B45 34B07 47E05 \\

{\bf 1. Introduction}\\

In this paper we study inverse spectral problems for second-order differential pencils on
compact graphs with a cycle. Inverse spectral problems consist in recovering coefficients
of differential equations from their spectral characteristics. The main results on inverse 
spectral problems for ordinary differential operators on {\it an interval} are presented
in the monographs [1-5]. 
Differential operators on graphs (spatial networks) often appear in mathematics, mechanics, 
physics, geophysics, physical chemistry, biology, electronics, nanoscale technology and other 
branches of natural sciences and engineering (see [6-7] and the references therein). 
Inverse spectral problems for {\it Sturm-Liouville operators} on compact graphs have been studied
fairly completely in [8]-[13] and other works. Differential pencils (when differential equations 
depend nonlinearly on the spectral parameter) produce serious qualitative changes in the spectral 
theory. In particular, there are only a few works on inverse spectral problems for differential 
pencils on graphs. In [14] an inverse problem have been solved for differential pencils on trees 
(graphs without cycles). Inverse problems for differential pencils on graphs with cycles have not 
been studied yet.

In this paper we investigate inverse spectral problems for second-order differential pencils on 
compact graphs having a cycle under generalized matching conditions in interior vertices and 
boundary conditions in boundary vertices. For these inverse problems we prove uniqueness theorems 
and provide procedures for constructing their solutions.
 
The paper is organized as follows:
Properties of spectral characteristics are established in Section 2.
In section 3 algorithms for the solutions of the inverse problems considered are provided
and the corresponding uniqueness theorems are proved.

\smallskip
Consider a compact graph $T$ in ${\bf R^m}$ with the set of vertices $V=\{v_0,\ldots, v_r\}$, 
$r\ge 1,$ and the set of edges ${\cal E}=\{e_0,\ldots, e_r\},$ where  $v_1,\ldots, v_r$ are 
the boundary vertices, $v_0$ is the internal vertex, $e_j=[v_j,v_0],$ $j=\overline{1,r},$ 
$\di\bigcap_{j=0}^{r}\,e_j=\{v_0\}$, and $e_0$ is a cycle. Thus, the graph $T$ has one cycle 
$e_0$ and one internal vertex $v_0$. Let $T_j$, $j=\overline{0,r},$ be the length of the edge 
$e_j$. Each edge $e_j\in {\cal E}$ is parameterized by the parameter $x_j\in [0,T_j].$ 
It is convenient for us to choose the following orientation: for $j=\overline{1,r},$ the vertex 
$v_j$ corresponds to $x_j=0,$ and the vertex $v_0$ corresponds to $x_j=T_j$; for $j=0,$ both 
ends $x_0=+0$ and $x_0=T_0-0$ corespond to $v_0$.

An integrable function $Y$ on $T$ may be represented as $Y=\{y_j\}_{j=\overline{0,r}}$, where 
the function $y_j(x_j),$ $x_j\in [0,T_j],$ is defined on the edge $e_j$. Let $q=\{q_j\}_{j=
\overline{0,r}}$ and $p=\{p_j\}_{j=\overline{0,r}}$ be complex-valued functions on $T$; the 
pair $(q,p)$ is called the potential. Assume that $q_j(x_j)\in L(0,T_j),$ and $p_j(x_j)$ is 
absolutely continuous on $[0, T_j].$ Consider the following differential equation on $T$:
$$
y''_j(x_j)+(\rho^2+\rho p_j(x_j)+q_j(x_j))y_j(x_j)=0,\quad x_j\in (0,T_j),                  \eqno(1)
$$
where $\rho$ is the spectral parameter, $j=\overline{0,r}$, the functions $y_j(x_j),\, 
y'_j(x_j)$ are absolutely continuous on $[0,T_j]$ and satisfy the following matching 
conditions in the internal vertex $v_0$:
$$
y_0(0)=\al_{j}y_j(T_j),\; j=\overline{0,r},\qquad 
y'_0(0)-(i\rho h_{01}+h_{00})y_0(0)=\di\sum_{j=0}^r \be_j y'_j(T_j),                        \eqno(2)
$$
where $\al_j$ and $\be_j$ are complex numbers such that $\al_j\be_j\ne 0,\;
1+\al_0\be_0\ne 0.$ Matching conditions (2) are a generalization of Kichhoff's matching 
conditions [9]. Let us consider the boundary value problem $B_0:=B_0(q,p,h_1,h_0)$ on 
$T$ for equation (1) with matching conditions (2) and with the following boundary 
conditions at the boundary vertices $v_1,\ldots, v_r$:
$$
U_j(Y)=0,\quad j=\overline{1,r}.                                                            \eqno(3)
$$
where $U_j(Y):=y'_j(0)-(i\rho h_{j1}+h_{j0})y_j(0),$ $h_{jk}$ are complex numbers, 
$h_k=\{h_{jk}\}_{j=\overline{0,r}},\; k=0,1,$ and $h_{j1}\ne\pm 1$ for $j=\overline{1,r}.$ 
We also consider the boundary value problems $B_k:=B_k(q,p,h_1,h_0),$ $k=\overline{1,r},$ 
for equation (1) with matching conditions (2) and with the boundary conditions
$$
y_k(0)=0,\quad U_j(Y)=0,\quad j=\overline{1,r}\setminus k.     
$$
We denote by $\Lambda_k:=\{\rho_{kn}\}_{n\in{\bf Z}}$ the eigenvalues
(counting with multiplicities) of $B_k(q)$, $k=\overline{0,r}.$
In contrast to the case of trees (see [9, 14]), here the specification of the spectra 
$\Lambda_k$, $k=\overline{0,r}$ does not uniquely determine the potential, and we need an 
additional information. Let $\Lambda_{-1}:=\{\rho_{-1,n}\}_{n\in{\bf Z}}$ be the spectrum of
the boundary value problem $B_{-1}$ for equation (1) under boundary conditions (3) and matching 
conditions of the form (2), but with $\al_{-1}$ instead of $\al_0$ ($\al_{-1}\ne \al_0$).

\smallskip
{\bf Inverse problem 1.} Given $\Lambda_k,\; k=\overline{-1,r},$ construct the potential 
$(q,p)$ on $T$ and the coefficients $h_1,\, h_0.$

\smallskip
Let $\Lambda_{r+1}:=\{\rho_{r+1,n}\}_{n\in{\bf Z}}$ be the spectrum of the boundary value 
problem $B_{r+1}$ for equation (1) under boundary conditions (3) and matching conditions 
of the form (2), but with $\be_{-1}$ instead of $\be_0$ ($\be_{-1}\ne \be_0$).

\smallskip
{\bf Inverse problem 2.} Given $\Lambda_k,\; k=\overline{0,r+1},$ construct the potential 
$(q,p)$ on $T$ and the coefficients $h_1,\, h_0.$

\smallskip
For these inverse problems we provide constructive procedures for their solutions and 
prove their uniqueness. We note that the coefficients $\al_j, \be_j$ from (2) are known 
a priori and fixed. Denote 
$$
z_0^{\pm}=\al_0(1\mp h_{01})+\be_0,\quad z_{k+1}^{\pm}=
\al_{k+1}z_k^{\pm}+\be_{k+1}\prod_{j=0}^k \al_j, \quad k=\overline{0,r-1}.
$$
We assume that $z_0^{\pm}z_r^{\pm}\ne 0.$ This condition is called the regularity condition 
for matching. Differential operators on $T$ which do not satisfy the regularity condition, 
possess qualitatively different properties for formulation and investigation of inverse 
problems, and are not considered in this paper; they require a separate investigation.
We note that for classical Kirchhoff's matching conditions we have $\al_{j}=\be_{j}=1,\,
h_{0k}=0,$ and the regularity condition is satisfied obviously.

Let us formulate uniqueness theorems for the solution of Inverse problems 1 and 2. 
For this purpose together with $B_k$ we consider boundary value problems 
$\tilde B_k=B_k(\tilde q, \tilde p, \tilde h_1, \tilde h_0)$ of the same form but with 
different coefficients. Everywhere below if a symbol $\al$ denotes an object related to 
$B_k$, then $\tilde\al$ will denote the analogous object related to $\tilde B_k$.

\smallskip
{\bf Theorem 1. }{\it If $\Lambda_k=\tilde\Lambda_k,\; k=\overline{-1,r},$ then 
$q=\tilde q,$ $p=\tilde p,$ $h_1=\tilde h_1$, $h_0=\tilde h_0$. Thus, the specification 
of the spectra $\Lambda_k,\; k=\overline{-1,r}$ uniquely determines the potential 
$(q,p)$ on $T$ and the coefficients $h_1,\, h_0.$}

\smallskip
{\bf Theorem 2. }{\it If $\Lambda_k=\tilde\Lambda_k,\; k=\overline{0,r+1},$ then 
$q=\tilde q,$ $p=\tilde p,$ $h_1=\tilde h_1$, $h_0=\tilde h_0$. Thus, the specification 
of the spectra $\Lambda_k,\; k=\overline{0,r+1}$ uniquely determines the potential 
$(q,p)$ on $T$ and the coefficients $h_1,\, h_0.$}

\smallskip
These theorems will be proved below in Section 3. Moreover, we will give constructive 
procedures for the solutions of Inverse problems 1 and 2 (see Algorithms 3 and 5). 
In Section 2 properties of the spectra, characteristic functions and the 
Weyl functions are investigated for boundary value problems on the graph.\\


{\bf 2. Auxiliary propositions}\\

Denote ${\cal E}_k(x_k)=\frac{1}{2}\int_0^{x_k} p_k(t)\,dt,\; \omega_k=T_k^{-1}{\cal E}_k(T_k),
\; E^{\pm}(\rho)=\prod_{j=0}^r \exp(\mp i(\rho+\omega_j)T_j),$\\
$\Pi^{\pm}=\{\rho:\; \pm\mbox{Im}\,\rho\ge 0\},\; \Pi^{+}_\de=\{\rho:\; 
\arg\rho\in[\de,\pi-\de]\},\; \Pi^{-}_\de=\{\rho:\; \arg\rho\in[\pi+\de,2\pi-\de]\}.$
Fix $k=\overline{1,r}.$ Let $\Phi_k=\{\Phi_{kj}\}_{j=\overline{0,r}}$, be 
the solution of equation (1) satisfying (2) and the boundary conditions
$$
U_j(\Phi_{k})=\de_{jk}, \quad j=\overline{1,r},                                             \eqno(4)
$$
where $\de_{jk}$ is the Kronecker symbol. Denote $M_k(\rho):=\Phi_{kk}(0,\rho),$ 
$k=\overline{1,r}.$ The function $M_k(\la)$ is called the {\it Weyl function} with 
respect to the boundary vertex $v_k$. Clearly,
$$
\Phi_{kk}(x_k,\rho)=S_k(x_k,\rho)
+M_k(\rho)\vv_{k}(x_k,\rho),\quad x_k\in[0,T_k],\quad k=\overline{1,r},                     \eqno(5)
$$
where $S_k(x_k,\rho),\; \vv_k(x_k,\rho),\; k=\overline{0,r}$ are solutions of 
equation (1) on the edge $e_k$ with the initial conditions $S_k(0,\rho)=0,\; 
S'_k(0,\rho)=\vv_k(0,\rho)=1,\;\vv'_k(0,\rho)=i\rho h_{k1}+h_{k0}.$ 
For each fixed $x_k\in[0,T_k],$ the functions $S_k^{(\nu)}(x_k,\rho),\,
\vv_k^{(\nu)}(x_k,\rho),\;\nu=0,1,$ are entire in $\rho$ of exponential type, 
and $\langle\vv_{k}(x_k,\rho),\,\Phi_{kk}(x_k,\rho)\rangle\equiv 1,$ 
where $\langle y,z\rangle :=yz'-y'z$ is the Wronskian of $y$ and $z.$ For 
$k=\overline{0,r},$ $\nu=0,1,$ $x_k\in[0,T_k],$ $|\rho|\to\iy,$ one has (see [14]),
$$
\vv_k^{(\nu)}(x_k,\rho)=(-i\rho)^{\nu}\di\frac{1-h_{k1}}{2}\exp(-i(\rho x_k+
{\cal E}_k(x_k)))[1]
$$
$$
+(i\rho)^{\nu}\di\frac{1+h_{k1}}{2}\exp(i(\rho x_k+{\cal E}_k(x_k)))[1].                    \eqno(6)
$$
Similarly, for $k=\overline{1,r},$
$\nu=0,1,$ $x_k\in[0,T_k),$ $\rho\in\Pi_\de^{\pm},\;|\rho|\to\iy$,
$$
\Phi_{kk}^{(\nu)}(x_k,\rho)=\di\frac{1}{(\pm i\rho)^{1-\nu}(1\mp h_{k1})}
\exp(\pm i(\rho x_k+{\cal E}_k(x_k)))[1],                                                   \eqno(7)
$$
$$
M_k(\rho)=\frac{[1]}{(\pm i\rho)(1\mp h_{k1})},\quad k=\overline{1,r}.                      \eqno(8)
$$
Denote $M_{kj}^1(\rho):=\Phi_{kj}(0,\rho),\;
M_{kj}^0(\rho):=\Phi'_{kj}(0,\rho)-(i\rho h_{j1}+h_{j0})\Phi_{kj}(0,\rho).$ Then
$$
\Phi_{kj}(x_j,\rho)=M_{kj}^0(\rho)S_j(x_j,\rho)+M_{kj}^1(\rho)\vv_j(x_j,\rho), 
\quad x_j\in[0,T_j],\; j=\overline{0,r},\; k=\overline{1,r}.                                \eqno(9)
$$
In particular, $M_{kk}^0(\rho)=1$, $M_{kk}^1(\rho)=M_k(\rho),$ and $M_{kj}^0(\rho)=0$ 
for $j=\overline{1,r}\setminus k.$ Substituting (9) into (2) and (4) we obtain a linear 
algebraic system $s_k$ with respect to $M_{kj}^\nu(\rho)$, $\nu=0,1,\; j=\overline{0,r}.$
The determinant $\Delta_0(\rho)$ of $s_k$ does not depend on $k$ and has the form
$$
\Delta_0(\rho)=d(\rho)\prod_{j=1}^r (\al_j\vv_j(T_j,\rho))+d_0(\rho)\sum_{i=1}^{r}
(\be_i\vv'_i(T_i,\rho)) \prod_{j=1,\,j\ne i}^{r}(\al_j\vv_j(T_j,\rho)),                     \eqno(10)
$$
where
$$
d(\rho)=\al_0\vv_0(T_0,\rho)
+\be_0S'_0(T_0,\rho)-(1+\al_0\be_0),\quad d_0(\rho)=\al_0S_0(T_0,\rho).                     \eqno(11)
$$
The function $\Delta_0(\rho)$ is entire in $\rho$ of exponential type, and its zeros 
coincide with the eigenvalues of the boundary value problem $B_0$. Solving the 
algebraic system $s_k$ we get by Cramer's rule: $M_{kj}^s(\rho)=\Delta_{kj}^s(\rho)/
\Delta_0(\rho)$, $s=0,1,\,j=\overline{0,r},$ where the determinant $\Delta_{kj}^s(\rho)$ 
is obtained from $\Delta_0(\rho)$ by the replacement of the column which corresponds 
to $M_{kj}^s(\rho)$ with the column of free terms. In particular,
$$
M_k(\rho)=-\di\frac{\Delta_k(\rho)}{\Delta_0(\rho)},\quad k=\overline{1,r},                 \eqno(12)
$$
where
$$
\Delta_k(\rho)=d(\rho)(\al_k S_k(T_k,\rho))\prod_{j=1,\,j\ne k}^r (\al_j \vv_j(T_j,\rho))
+d_0(\rho)\Big(\be_k S'_k(T_k,\rho)
$$
$$
\times\prod_{j=1,\,j\ne k}^r (\al_j \vv_j(T_j,\rho))+
(\al_k S_k(T_k,\rho))\sum_{i=1,\,i\ne k}^r (\be_i \vv'_i(T_i,\rho))
\prod_{j=1,\,j\ne i,k}^r (\al_j \vv_j(T_j,\rho))\Big),\; k=1,2.                            \eqno(13)
$$
We note that $\Delta_k(\rho)$ in (13) is obtained from $\Delta_0(\rho)$ by the replacement 
of $\vv_k^{(\nu)}(T_k,\rho),\; \nu=0,1,$ with $S_k^{(\nu)}(T_k,\rho),\; \nu=0,1.$ The 
function $\Delta_k(\rho)$ is entire in $\rho$ of exponential type, and its zeros coincide 
with the eigenvalues of the boundary value problem $B_k$. The functions $\Delta_k(\rho)$, 
$k=\overline{0,r},$ are called the characteristic functions for the boundary value 
problems $B_k$.\\


{\bf 3. Solution of Inverse problems 1-2}\\

Fix $k=\overline{1,r},$ and consider the following auxiliary inverse 
problem on the edge $e_k$, which is called IP(k).

\smallskip
{\bf IP(k).} Given the Weyl function $M_k(\rho),$ construct 
$q_k(x_k),\, p_k(x_k),\, x_k\in[0,T_k],\, h_{k1},\, h_{k0}$.

\smallskip
In IP(k) we construct the potential only on the edge $e_k$, but the Weyl function brings a 
global information from the whole graph. In other words, IP(k) is not a local inverse problem 
related to the edge $e_k$. Let us prove the uniqueness theorem for the solution of IP(k). 

\smallskip
{\bf Theorem 3. }{\it Fix $k=\overline{1,r}.$ If $M_k(\rho)=\tilde M_k(\rho),$ then 
$q_k(x_k)=\tilde q_k(x_k)$, $p_k(x_k)=\tilde p_k(x_k)$ a.e. on $[0,T_k],$ and $h_{k\nu}=
\tilde h_{k\nu},\; \nu=0,1$. Thus, the specification of the Weyl function $M_k(\rho)$ uniquely 
determines the potential $(q_k, p_k)$ on the edge $e_k$, and the coefficients $h_{k1}, h_{k0}$.}

\smallskip
{\bf Proof.} We introduce the functions
$$
P^k_{1s}(x_k,\rho)=(-1)^{s-1}\Big(\vv_k(x_k,\rho)\tilde\Phi^{(2-s)}_{kk}(x_k,\rho)
-\tilde\vv^{(2-s)}_k(x_k,\rho)\Phi_{kk}(x_k,\rho)\Big),\quad s=1,2.                         \eqno(14)
$$
By direct calculations we get
$$
\vv_k(x_k,\rho)=P^k_{11}(x_k,\rho)\tilde\vv_k(x_k,\rho)
+P^k_{12}(x_k,\rho)\tilde\vv'_k(x_k,\rho).                                                  \eqno(15)
$$
Denote $\Omega_k(x_k)=\cos\hat{\cal E}_k(x_k),$ where 
$\hat{\cal E}_k(x_k)={\cal E}_k(x_k)-\tilde{\cal E}_k(x_k).$ 
Since $M_k(\rho)=\tilde M_k(\rho),$ it follows from (8) that
$$
h_{k1}=\tilde h_{k1}\,.                                                                     \eqno(16)
$$
Taking (6), (7), (14) and (16) into account we obtain
$$
P^k_{1s}(x_k,\rho)=\de_{1s}\Omega_k(x_k)+O(\rho^{-1}),\quad
\rho\in\Pi_\de^{\pm},\;|\rho|\to\iy,\; x_k\in(0,T_k),\; s=1,2.                              \eqno(17)
$$
According to (5) and (14),
$$
P^k_{1s}(x_k,\rho)=(-1)^{s-1}\Big(\Big(\vv_k(x_k,\rho)\tilde S^{(2-s)}_k(x_k,\rho)
-S_{k}(x_k,\rho)\tilde\vv_k^{(2-s)}(x_k,\rho)\Big)
$$
$$
+(\tilde M_k(\rho)-M_k(\rho))\vv_k(x_k,\rho)\tilde\vv^{(2-s)}_k(x_k,\rho)\Big).
$$
Since $M_k(\rho)=\tilde M_k(\rho),$ it follows that for each fixed $x_k$, the functions 
$P^k_{1s}(x_k,\rho)$ are entire in $\rho$ of exponential type. Together with (17) this 
yields $P^k_{11}(x_k,\rho) \equiv\Omega_k(x_k),$ $P^k_{12}(x_k,\rho)\equiv 0.$ 
Substituting these relations into (14) and (15) we get
$$
(\tilde\vv_k(x_k,\rho))^{-1}\vv_k(x_k,\rho)=
(\tilde\Phi_{kk}(x_k,\rho))^{-1}\Phi_{kk}(x_k,\rho),                                        \eqno(18)
$$
$$
\vv_k(x_k,\rho)=\Omega_k(x_k)\tilde\vv_k(x_k,\rho),
$$
for all $x_k$ and $\rho.$ Using the asymptotical formulae (6) and
(7) we obtain for $|\rho|\to\iy,\;\rho\in\Pi_{\de}^{\pm}$,
$$
(\tilde\vv_k(x_k,\rho))^{-1}\vv_k(x_k,\rho)=\exp(\mp\hat{\cal E}_k(x_k))[1],\quad
(\tilde\Phi_{kk}(x_k,\rho))^{-1}\Phi_{kk}(x_k,\rho)=\exp(\pm\hat{\cal E}_k(x_k))[1].
$$
From this and from (18) we infer $\exp(2\hat{\cal E}_k(x_k))\equiv 1.$ Since 
$\hat{\cal E}_k(0)=0,$ it follows that $\hat{\cal E}_k(x_k)\equiv 0,$ i.e. 
$P_{11}(x_k,\rho)\equiv 1,$ $\vv_k(x_k,\rho)\equiv\tilde\vv_k(x_k,\rho),$ 
$\Phi_{kk}(x_k,\rho)\equiv \tilde\Phi_{kk}(x_k,\rho),$ and consequently, $q_k(x_k)=
\tilde q_k(x_k),$ $p_k(x_k)=\tilde p_k(x_k)$ a.e. on $[0,T_k]$, and $h_{k0}=\tilde h_{k0}$.
\B

\smallskip
Using the method of spectral mappings [5] for equation (1) on the edge $e_k$ one can get 
a constructive procedure for the solution of the local inverse problem IP(k). 
coefficients $q_k(x_k),\,p_k(x_k)$ and $h_{ks}$ (for details see [5], [14]).

\smallskip
Let the spectra $\Lambda_k$, $k=\overline{0,r},$ be given. The characteristic functions 
$\Delta_k(\rho),$ $k=\overline{0,r},$ are entire in $\rho$ of exponential type. By 
Hadamard's factorization theorem,
$$
\Delta_k(\rho)=B_k\exp(A_k\rho)\de_k(\rho),\quad k=\overline{0,r},                         \eqno(19)
$$
$$
\de_k(\rho)=\rho^{\xi_k}\prod_{n\in \Lambda'_k}
\Big(1-\frac{\rho}{\rho_{kn}}\Big)\exp(\rho/\rho_{kn}),\quad k=\overline{0,r},             \eqno(20)
$$
where $\Lambda'_k=\{n:\; \rho_{kn}\ne 0\},$ and $\xi_{k}\ge 0$
is the multiplicity of the zero eigenvalue. In view of (12), we deduce
$$
M_k(\rho)=-b_k\exp(a_k\rho)\frac{\de_k(\rho)}{\de_0(\rho)},\quad k=\overline{1,r},         \eqno(21)
$$
where $b_k=B_k/B_0$, $a_k=A_k-A_0$. Using (8) we calculate for 
$\rho\in\Pi_{\de}^{\pm},$ $|\rho|\to\iy$:
$$
b_k\exp(a_k\rho)=\frac{\de_0(\rho)[1]}{\de_k(\rho)(\mp i\rho)(1\mp h_{k1})},
$$
and consequently,
$$
a_k=\lim_{|\rho|\to\iy}\,\frac{1}{\rho}\ln\Big(\frac{\de_0(\rho)}{\de_k(\rho)}\Big),
\quad \rho\in\Pi_{\de}^{\pm},\quad k=\overline{1,r}.                                       \eqno(22)
$$
Here and below we agree that if $z=|z|e^{i\xi},$ $\xi\in[0,2\pi),$ then 
$\ln z=\ln |z|+i\xi,$ and $\sqrt{z}:=|z|^{1/2}e^{i\xi/2}$. Denote
$$
\mu_k^{\pm}=\lim_{|\rho|\to\iy}\, \frac{\de_0(\rho)\exp(-a_k\rho)}{\de_k(\rho)(\mp i\rho)},
\quad \rho\in\Pi_{\de}^{\pm},\quad k=\overline{1,r}.
$$
Then $b_k(1\mp h_{k1})=\mu_k^{\pm}$, hence
$$
b_k=\frac{\mu_k^{-}+\mu_k^{+}}{2},\quad h_{k1}=
\frac{\mu_k^{-}-\mu_k^{+}}{\mu_k^{-}+\mu_k^{+}},\quad k=\overline{1,r}.                    \eqno(23)
$$
Thus, we have uniquely constructed $M_k(\rho)$ and $h_{k1}$ by (20)-(23). Solving auxiliary 
inverse problems IP(k) for each $k=\overline{1,r},$ we find $q_k(x_k), p_k(x_k), h_{k1}$ 
and $h_{k0}$ for $k=\overline{1,r}.$ In particular, this means that the functions 
$\vv_k^{(\nu)}(T_k,\rho)$ and $S_k^{(\nu)}(T_k,\rho),$ $k=\overline{1,r},\;\nu=0,1,$ are 
known. Denote $\chi:=\exp(2i\om_0 T_0).$ Using (19)-(20), one can uniquely construct the 
functions $d_0(\rho),$ $d(\rho)$ and $\Delta_0(\rho)$ by the following algorithm 
(see [prepr] for details).

\smallskip
{\bf Algorithm 1. }{\it 1) Calculate $A_k$, $k=0,1,$ by 
$$
A_k=-\kappa_k^{\pm}\mp i\sum_{j=0}^r T_j,\quad
\kappa_k^{\pm}:=\lim_{|\rho|\to\iy}\,\frac{\ln\de_k(\rho)}{\rho},\;\rho\in\Pi_{\de}^{\pm},
$$
where $\de_k(\rho)$ is constructed by (20).\\
2) Find $\sigma_k^{\pm},\; k=0,1,$ via 
$$
\sigma_0^{\pm}=\frac{1}{2^{r+1}}\prod_{j=1}^r (1\mp h_{j1})\prod_{j=1}^r
\exp(\mp i\omega_j T_j)\lim_{|\rho|\to\iy}\frac{\exp(\kappa_0^{\pm}\rho)}{\de_0(\rho)},
\quad \rho\in\Pi_{\de}^{\pm},   
$$
$$
\sigma_1^{\pm}=\frac{1}{2^{r+1}}\prod_{j=2}^r (1\mp h_{j1})\prod_{j=1}^r
\exp(\mp i\om_jT_j)\lim_{|\rho|\to\iy}\frac{\exp(\kappa_1^{\pm}\rho)}{\de_1(\rho)(\mp i\rho)},
\quad \rho\in\Pi_{\de}^{\pm}.
$$
3) Construct $\Delta_k^{\pm}(\rho)=\sigma_k^{\pm}\exp(A_k\rho)\de_k(\rho),\; k=0,1.$\\
4) Calculate 
$$
d_0^{\pm}(\rho)=\frac{1}{\be_1}\Big(\prod_{j=2}^r (\al_j\vv_j(T_j,\rho))\Big)^{-1}
\Big(\vv_1(T_1,\rho)\Delta_1^{\pm}(\rho)-S_1(T_1,\rho)\Delta_0^{\pm}(\rho)\Big).
$$
5) Find 
$$
z_r^{\pm}=\frac{\al_0}{2}\lim_{|\rho|\to\iy}\,
\frac{\exp(\mp i\rho T_0)}{d_0^{\pm}(\rho)(\mp i\rho)},\quad \rho\in\Pi_{\de}^{\pm}.
$$
6) Calculate $\chi=(z_r^{+}\sigma_k^{+})/(z_r^{-}\sigma_k^{-}),\; k=0,1.$ \\
7) Construct $\Delta_k^{*}(\rho)=z_r^{\pm}\chi^{\mp 1/2}\Delta_k^{\pm}(\rho),\; k=0,1.$\\
8) Find $F_s^{*}(\rho),\; s=1,2$: 
$$
F_1^{*}(\rho)=\frac{1}{\al_1}\Big(\Delta_0^{*}(\rho) S'_1(T_1,\rho)
-\Delta_1^{*}(\rho)\vv'_1(T_1,\rho)\Big),\;
F_2^{*}(\rho)=\frac{1}{\be_1}\Big(\vv_1(T_1,\rho)\Delta_1^{*}(\rho)
-S_1(T_1,\rho)\Delta_0^{*}(\rho)\Big).
$$
9) Calculate $d_0^{*}(\rho)$ and $d^{*}(\rho)$ by
$$
d_0^{*}(\rho)=\Big(\prod_{j=2}^r (\al_j\vv_j(T_j,\rho))\Big)^{-1}F_2^{*}(\rho),  
$$
$$
d^{*}(\rho)=\Big(\prod_{j=2}^r (\al_j\vv_j(T_j,\rho))\Big)^{-1}
\Big(F_1^{*}(\rho)-d_0^{*}(\rho)\sum_{i=2}^{r}(\be_i\vv'_i(T_i,\rho))
\prod_{j=2,\,j\ne i}^{r}(\al_j\vv_j(T_j,\rho))\Big).  
$$
10) Find $\ee$:
$$
\ee=\frac{1}{1+\al_0\be_0}\lim_{|\rho|\to\iy}\,\Big(\frac{z_0^{-}\sqrt{\chi}}{2}
\exp(i\rho T_0)+\frac{z_0^{+}}{2\sqrt{\chi}}\exp(-i\rho T_0)-d^{*}(\rho)\Big). 
$$
11) Construct $d(\rho)=\ee d^{*}(\rho),\; d_0(\rho)=\ee d_0^{*}(\rho),
\; \Delta_0(\rho)=\ee \Delta_0^{*}(\rho).$}

\medskip
Let $\Delta_{-1}(\rho)$ be the characteristic function of the boundary value problem 
$B_{-1}$. Similarly to (10) we calculate
$$
\Delta_{-1}(\rho)=d_{-1}(\rho)\prod_{j=1}^r(\al_j\vv_j(T_j,\rho))+d_0(\rho)\frac{\al_{-1}}{\al_0}
\sum_{i=1}^{r}(\be_i\vv'_i(T_i,\rho))\prod_{j=1,\,j\ne i}^{r}(\al_j\vv_j(T_j,\rho)),        \eqno(24)
$$
where 
$$
d_{-1}(\rho)=\al_{-1}\vv_0(T_0,\rho)+\be_0S'_0(T_0,\rho)-(1+\al_{-1}\be_0).                  \eqno(25)
$$
Since $\exp(i\om_0 T_0)$ and $h_{01}$ are already found, it follows that the function
$\Delta_{-1}(\rho)$ can be uniquely reconstructed from its zeros by the following algorithm.

\smallskip
{\bf Algorithm 2. }{\it 1) Calculate $A_{-1}$ by 
$$
A_{-1}=-\kappa_{-1}^{\pm}\mp i\sum_{j=0}^r T_j,\quad \kappa_{-1}^{\pm}:=
\lim_{|\rho|\to\iy}\,\frac{\ln\de_{-1}(\rho)}{\rho},\;\rho\in\Pi_{\de}^{\pm},
$$
$$
\de_{-1}(\rho)=\rho^{\xi_{-1}}\prod_{n\in \Lambda'_{-1}}
\Big(1-\frac{\rho}{\rho_{-1,n}}\Big)\exp(\rho/\rho_{-1,n}),
$$
where $\Lambda'_{-1}=\{n:\; \rho_{-1,n}\ne 0\},$ and $\xi_{-1}\ge 0$ 
is the multiplicity of the zero eigenvalue. \\
2) Find $\sigma_{-1}^{\pm}$ via
$$
\sigma_{-1}^{\pm}=\frac{1}{2^{r+1}}\prod_{j=1}^r (1\mp h_{j1})\prod_{j=1}^r
\exp(\mp i\omega_j T_j)\lim_{|\rho|\to\iy}
\frac{\exp(\kappa_{-1}^{\pm}\rho)}{\de_{-1}(\rho)},\quad \rho\in\Pi_{\de}^{\pm}, 
$$
3) Calculate $B_{-1}=\sigma_{-1}^{\pm}z_{r,-1}^{\pm}\exp(\mp i\omega_0 T_0),$
where $z_{r,-1}^{\pm}$ is obtained from $z_{r}^{\pm}$ by the replacement 
of $\al_0$ with $\al_{-1}$. \\
4) Construct} $\Delta_{-1}(\rho)=B_{-1}\exp(A_{-1}\rho)\de_{-1}(\rho).$

\medskip
Using (24) we find the function $d_{-1}(\rho).$  Denote
$$
\mu_0(\rho):=\vv_0(T_0,\rho),\quad \mu_1(\rho):=S_0(T_0,\rho).
$$
Taking (11) and (25) into account, one can calculate $\mu_0(\rho)$ and 
$\mu_1(\rho)$ by the formulae
$$
\mu_0(\rho)=\frac{d(\rho)-d_{-1}(\rho)}{\al_0-\al_{-1}}+\be_0,
\quad \mu_1(\rho)=\frac{d_0(\rho)}{\al_0}.                                                  \eqno(26)
$$
The functions $\mu_0(\rho)$ and $\mu_1(\rho)$ are the characteristic functions 
for the boundary value problems 
$$
-y''_0(x_0)+(\rho^2+\rho p_0(x_0)+q_0(x_0))y_0(x_0)=0,\;x_0\in (0,T_0),\quad
y'_0(0)-(i\rho h_{01}+h_{00})y_0(0)=y(T_0)=0,
$$
and
$$
-y''_0(x_0)+(\rho^2+\rho p_0(x_0)+q_0(x_0))y_0(x_0)=0,\;x_0\in (0,T_0),\quad
y_0(0)=y(T_0)=0,
$$
respectively. It was shown in [15] that the specification of $\mu_0(\rho)$ and $\mu_1(\rho)$
uniquely determines the potential $(q_0, p_0)$ on the edge $e_0$, and the coefficients 
of boundary conditions. Moreover, a constructive procedure for this inverse problem
is given in [15]. Thus, we have obtained a procedure for the solution of Inverse 
problem 1 and proved its uniqueness. In other words, Theorem 1 is proved, and the
solution of Inverse problem 1 can be found by the following algorithm.

\smallskip
{\bf Algorithm 3. }{\it Let $\Lambda_k,\; k=\overline{-1,r}$ be given.\\
1) Construct $\de_k(\rho),\; k=\overline{-1,r}$ by (20).\\
2) Calculate $M_k(\rho)$ and $h_{k1},\; k=\overline{1,r}$ via (21)-(23).\\
3) For each fixed $k=\overline{1,r},$ solve the inverse problem $IP(k)$ and find
the functions $q_k(x_k),$ $p_k(x_k),$ $x_k\in(0, T_k)$ on the edge $e_k$, and the
coefficient $h_{k0}$.\\
4) For each fixed $k=\overline{1,r},$ calculate the functions $\vv_k^{(\nu)}(T_k,\rho),$
$S_k^{(\nu)}(T_k,\rho),\; \nu=0,1.$\\
5) Construct the functions $d(\rho),\; d_0(\rho)$ and $\Delta_0(\rho)$ by Algorithm 1.\\
6) Find the function $\Delta_{-1}(\rho)$ by Algorithm 2.\\
7) Calculate the function $d_{-1}(\rho)$ using (24).\\
8) Construct the functions $\mu_0(\rho)$ and $\mu_1(\rho)$ via (26).\\
9) Find $q_0(x_0),\; p_0(x_0),\; x_0\in (0,T_k)$ and $h_{00},\; h_{01}$ from
$\mu_1(\rho)$ and $\mu_2(\rho),$ using results from [15].}

\medskip
By similar arguments one can solve Inverse problem 2. Indeed, let $\Delta_{r+1}(\rho)$ 
be the characteristic function of the boundary value problem $B_{r+1}$. Then
$$
\Delta_{r+1}(\rho)=d_{r+1}(\rho)\prod_{j=1}^r(\al_j\vv_j(T_j,\rho))+d_0(\rho)
\sum_{i=1}^{r}(\be_i\vv'_i(T_i,\rho)) \prod_{j=1,\,j\ne i}^{r}(\al_j\vv_j(T_j,\rho)),       \eqno(27)
$$
where 
$$
d_{r+1}(\rho)=\al_{0}\vv_0(T_0,\rho)+\be_{-1}S'_0(T_0,\rho)-(1+\al_{0}\be_{-1}).            \eqno(28)
$$
The function $\Delta_{r+1}(\rho)$ can be uniquely reconstructed from its zeros by 
the following algorithm.

\smallskip
{\bf Algorithm 4. }{\it 1) Calculate $A_{r+1}$ by 
$$
A_{r+1}=-\kappa_{r+1}^{\pm}\mp i\sum_{j=0}^r T_j,\quad \kappa_{r+1}^{\pm}:=
\lim_{|\rho|\to\iy}\,\frac{\ln\de_{r+1}(\rho)}{\rho},\;\rho\in\Pi_{\de}^{\pm},
$$
$$
\de_{r+1}(\rho)=\rho^{\xi_{r+1}}\prod_{n\in \Lambda'_{r+1}}
\Big(1-\frac{\rho}{\rho_{r+1,n}}\Big)\exp(\rho/\rho_{r+1,n}),
$$
where $\Lambda'_{r+1}=\{n:\; \rho_{r+1,n}\ne 0\},$ and $\xi_{r+1}\ge 0$ 
is the multiplicity of the zero eigenvalue. \\
2) Find $\sigma_{r+1}^{\pm}$ via
$$
\sigma_{r+1}^{\pm}=\frac{1}{2^{r+1}}\prod_{j=1}^r (1\mp h_{j1})\prod_{j=1}^r
\exp(\mp i\omega_j T_j)\lim_{|\rho|\to\iy}
\frac{\exp(\kappa_{r+1}^{\pm}\rho)}{\de_{r+1}(\rho)},\quad \rho\in\Pi_{\de}^{\pm}, 
$$
3) Calculate $B_{r+1}=\sigma_{r+1}^{\pm}z_{r,r+1}^{\pm}\exp(\mp i\omega_0 T_0),$
where $z_{r,r+1}^{\pm}$ is obtained from $z_{r}^{\pm}$ by the replacement 
of $\be_0$ with $\be_{-1}$.\\
4) Construct} $\Delta_{r+1}(\rho)=B_{r+1}\exp(A_{r+1}\rho)\de_{r+1}(\rho).$

\medskip
Using (27) we find the function $d_{r+1}(\rho).$  Denote $\mu_2(\rho):=S'_0(T_0,\rho).$
Taking (11) and (28) into account, one can calculate $\mu_1(\rho)$ and $\mu_2(\rho)$ 
by the formulae
$$
\mu_2(\rho)=\frac{d(\rho)-d_{r+1}(\rho)}{\be_0-\be_{-1}}+\al_0,
\quad \mu_1(\rho)=\frac{d_0(\rho)}{\al_0}.                                                 \eqno(29)
$$
The function $\mu_2(\rho)$ is the characteristic functions for the boundary value problems 
$$
-y''_0(x_0)+(\rho^2+\rho p_0(x_0)+q_0(x_0))y_0(x_0)=0,\;x_0\in (0,T_0),\quad
y_0(0)=y'(T_0)=0.
$$
It was shown in [15] that the specification of $\mu_1(\rho)$ and $\mu_2(\rho)$
uniquely determines the potential $(q_0, p_0)$ on the edge $e_0$, and the coefficients 
of boundary conditions. Moreover, a constructive procedure for this inverse problem
is given in [15]. Thus, we have obtained a procedure for the solution of Inverse 
problem 1 and proved its uniqueness. In other words, Theorem 1 is proved, and the
solution of Inverse problem 1 can be found by the following algorithm.

\smallskip
{\bf Algorithm 5. }{\it Let $\Lambda_k,\; k=\overline{0,r+1}$ be given.\\
1) Construct $\de_k(\rho),\; k=\overline{0,r+1}$ by (20).\\
2) Calculate $M_k(\rho)$ and $h_{k1},\; k=\overline{1,r}$ via (21)-(23).\\
3) For each fixed $k=\overline{1,r},$ solve the inverse problem $IP(k)$ and find
the functions $q_k(x_k),$ $p_k(x_k),$ $x_k\in(0, T_k)$ on the edge $e_k$, and the
coefficient $h_{k0}$.\\
4) For each fixed $k=\overline{1,r},$ calculate the functions $\vv_k^{(\nu)}(T_k,\rho),$
$S_k^{(\nu)}(T_k,\rho),\; \nu=0,1.$\\
5) Construct the functions $d(\rho),\; d_0(\rho)$ and $\Delta_0(\rho)$ by Algorithm 1.\\
6) Find the function $\Delta_{r+1}(\rho)$ by Algorithm 4.\\
7) Calculate the function $d_{r+1}(\rho)$ using (27).\\
8) Construct the functions $\mu_1(\rho)$ and $\mu_2(\rho)$ via (29).\\
9) Find $q_0(x_0),\; p_0(x_0),\; x_0\in (0,T_k)$ and $h_{00},\; h_{01}$ from
$\mu_1(\rho)$ and $\mu_2(\rho),$ using results from [15].}

\medskip
Denote by $B$ the boundary value problem on the edge $e_0$ for equation (1) with $j=0,$ 
under the conditions $y_0(0)=\al_{0}y_0(T_0),\; U_0(y_0)=\be_0y'_0(T_0).$ Let 
$\Omega=\{\omega_n\}$ be the $\Omega$-- sequence for $B$ (see [15]).

\smallskip
{\bf Inverse problem 3. } Given $\Lambda_k,\; k=\overline{0,r},$ and $\Omega,$ construct 
the potential $(q,p)$ on $T$ and the coefficients $h_1,\, h_0.$

\smallskip
By similar arguments as above one can prove the uniqueness theorem for Inverse problem 3,
and provide an algorithm for its solution (see [17] for details).

\bigskip
{\bf Acknowledgment.}  This research was supported in part by Grant 
13-01-00134 of Russian Foundation for Basic Research.

\begin{center}
{\bf REFERENCES}
\end{center}
\begin{enumerate}
\item[{[1]}] Marchenko V.A., Sturm-Liouville operators and their
     applications. "Naukova Dumka",  Kiev, 1977;  English  transl., Birkh\"auser, 1986.
\item[{[2]}] Levitan B.M., Inverse Sturm-Liouville problems. Nauka,
     Moscow, 1984; English transl., VNU Sci.Press, Utrecht, 1987.
\item[{[3]}] Beals R., Deift P. and Tomei C.,  Direct and Inverse Scattering
     on the Line, Math. Surveys and Monographs, v.28. Amer. Math. Soc.
     Providence: RI, 1988.
\item[{[4]}] Freiling G. and Yurko V.A., Inverse Sturm-Liouville
     Problems and their Applications. NOVA Science Publishers, New York, 2001.
\item[{[5]}] Yurko V.A., Method of Spectral Mappings in the Inverse
     Problem Theory, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002.
\item[{[6]}] Pokornyi Yu.V. and Borovskikh A.V., Differential equations on networks 
     (geometric graphs). J. Math. Sci. (N.Y.) 119, no.6 (2004), 691-718.
\item[{[7]}] Pokornyi Yu. and Pryadiev V., The qualitative Sturm-Liouville
     theory on spatial networks. J. Math. Sci. (N.Y.) 119 (2004), no.6, 788-835.
\item[{[8]}] Belishev M.I., Boundary spectral inverse problem on a class
     of graphs (trees) by the BC method. Inverse Problems 20 (2004), 647-672.
\item[{[9]}] Yurko V.A., Inverse spectral problems for Sturm-Liouville
     operators on graphs. Inverse Problems 21 (2005), 1075-1086.
\item[{[10]}] Brown B.M. and Weikard R., A Borg-Levinson theorem for trees.
     Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461, no.2062 (2005), 3231-3243.
\item[{[11]}] Yurko V.A. Inverse problems for Sturm-Liouville operators
     on bush-type graphs. Inverse Problems 25, no.10 (2009), 105008, 14pp.
\item[{[12]}] Yurko V.A. An inverse problem for Sturm-Liouville operators on A-graphs. 
     Applied Math. Letters 23, no.8 (2010), 875-879.
\item[{[13]}] Yurko V.A. Inverse spectral problems for differential operators on arbitrary 
     compact graphs. Journal of Inverse and Ill-Posed Proplems 18, no.3 (2010), 245-261.
\item[{[14]}] Yurko V.A. Recovering differential pencils on compact 
     graphs. J. Differ. Equations 244 (2008), 431-443. 
\item[{[15]}] Yurko V.A. Inverse problems for non-selfadjoint quasi-periodic
     differential pencils. Analysis and Math. Physics, vol.2, no.3 (2012), 215-230.
\item[{[16]}] Naimark M.A. Linear differential operators. 2nd ed., Nauka, Moscow, 1969; 
     English transl. of 1st ed., Parts I,II, Ungar, New York, 1967, 1968.
\item[{[17]}] Yurko V.A. Differential pensils on graphs with a cycle. Schriftenreiche des 
     Fachbereichs Mathematik, SM-DUE-758, Universitaet Duisburg-Essen, 2013, 12pp. 
\end{enumerate}


\begin{tabular}{ll}
Name:             &   Yurko, Vjacheslav Anatoljevich  \\
Place of work:    &   Department of Mathematics, Saratov University \\
{}                &   Astrakhanskaya 83, Saratov 410012, Russia \\
Present Position: &   Professor of Mathematics,      \\
{}                &   Head of the Faculty of Mathematical Physics \\
E-mail:           &   yurkova@info.sgu.ru  \\
Phone:            &   (8452)275526  \\
URL:              &   mexmat.sgu.ru/yurko\\
\end{tabular}


\end{document}