$w(u,v)=\left\langle{\mathcal{S}u,v}\right\rangle=\left\langle{Su_{1},v_{1}}\right\rangle-\left\langle{Su_{2},v_{2}}\right\rangle,\qquad
u=\operatorname{col}(x,t),\quad v=\operatorname{col}(x^{\prime},t^{\prime}).$

Then condition (8.8) means that the subspace $\operatorname{Ker}(C_{*}\
D_{*})$ is the right $w$-orthogonal complement to $\operatorname{Ker}(C\ D)$
in $\mathbb{C}^{n}\oplus\mathbb{C}^{n}$.

(ii) This statement was proved in [43, Corollary 3.3] in the case of constant
matrix $B(x)=B=\operatorname{const}$. The proof remains the same in the case
of non-constant matrix $B(x)$. ∎

Next, assuming boundary conditions (5.2) to be regular, we find an explicit
form of the matrices $C_{*}$ and $D_{*}$. The proof substantially relies on
the canonical form (5.57) of regular boundary conditions.

######  Lemma 8.2.

Let the pair of matrices $\\{C,D\\}$ from regular boundary conditions $Uy=0$
be of the canonical form (5.57). Then the matrices $C_{*}$ and $D_{*}$ from
boundary conditions (8.2) of the adjoint operator $L_{0,U}^{*}$ admit the
following triangular block-matrix representation:

$C_{*}=\begin{pmatrix}D_{11}^{*}&\mathbb{O}\\\
C_{12}^{*}&I_{n_{+}}\end{pmatrix}\qquad
D_{*}=\begin{pmatrix}I_{n_{-}}&D_{21}^{*}\\\
\mathbb{O}&C_{22}^{*}\end{pmatrix}.$ (8.9)

###### Proof.

Recall that $S=\operatorname{diag}(-I_{n_{-}},I_{n_{+}})$ is a canonical form
of the signature matrix of $B(x)$, where $n_{-}\in\\{0,1,\ldots,n\\}$ and
$n_{+}=n-n_{-}$.

(i) First assume that $n_{-}=0$, and so $n_{+}=n$. Then $D=S=I_{n}$ and
$f(\ell)=-Cf(0)$ for $f\in\operatorname{dom}(L_{0,U})$. Hence (8.8) turns into

$\left\langle{-Cf(0),g(\ell)}\right\rangle=\left\langle{f(0),g(0)}\right\rangle,\qquad
f\in\operatorname{dom}(L_{0,U}),\quad g\in\operatorname{dom}(L_{0,U}^{*}),$

or

$\left\langle{u,g(0)+C^{*}g(\ell)}\right\rangle=0,\qquad
u\in\mathbb{C}^{n},\quad g\in\operatorname{dom}(L_{0,U}^{*}).$

Hence $g\in\operatorname{dom}{L_{0,U}^{*}}$ if and only if
$g(0)+C^{*}g(\ell)=0$, which leads to the desired formulas for $C_{*}$ and
$D_{*}$. The case $n_{+}=0$ is treated similarly.

(ii) Now assume that $n_{-},n_{+}>0$ in the representation (5.57). For any
vector function $f=f(x)$ let $f=:\operatorname{col}(f_{-},f_{+})$, where
$f_{\pm}(x)\in\mathbb{C}^{n_{\pm}}$, $x\in[0,\ell]$, be its decomposition with
respect to the decomposition
$\mathbb{C}^{n}=\mathbb{C}^{n_{-}}\oplus\mathbb{C}^{n_{+}}$. With account of
this notation we have for $f\in\operatorname{dom}(L_{0,U})$:

$\displaystyle 0=Cf(0)+Df(\ell)$
$\displaystyle=\begin{pmatrix}I_{n_{-}}&C_{12}\\\
\mathbb{O}&C_{22}\end{pmatrix}\binom{f_{-}(0)}{f_{+}(0)}+\begin{pmatrix}D_{11}&\mathbb{O}\\\
D_{21}&I_{n_{+}}\end{pmatrix}\binom{f_{-}(\ell)}{f_{+}(\ell)}$
$\displaystyle=\binom{f_{-}(0)+C_{12}f_{+}(0)+D_{11}f_{-}(\ell)}{C_{22}f_{+}(0)+D_{21}f_{-}(\ell)+f_{+}(\ell)}$
$\displaystyle=\binom{f_{-}(0)}{f_{+}(\ell)}+\begin{pmatrix}D_{11}&C_{12}\\\
D_{21}&C_{22}\end{pmatrix}\binom{f_{-}(\ell)}{f_{+}(0)}=f_{0,\ell}+Mf_{\ell,0},$
(8.10)

where

$f_{0,\ell}:=\binom{f_{-}(0)}{f_{+}(\ell)},\quad
f_{\ell,0}:=\binom{f_{-}(\ell)}{f_{+}(0)},\quad f\in
C([0,\ell];\mathbb{C}^{n}),\qquad M:=\begin{pmatrix}D_{11}&C_{12}\\\
D_{21}&C_{22}\end{pmatrix}.$ (8.11)

Recall that $S=\operatorname{diag}(-I_{n_{-}},I_{n_{+}})$,
$f=\operatorname{col}(f_{-},f_{+})$ and $g=\operatorname{col}(g_{-},g_{+})$
are decompositions of the matrix $S$ and the vector functions $f$ and $g$ with
respect to the decomposition
$\mathbb{C}^{n}=\mathbb{C}^{n_{-}}\oplus\mathbb{C}^{n_{+}}$. Taking into
account notation (8.11) and explicit formula (8.10) for the domain
$\operatorname{dom}(L_{0,U})$, we have from (8.8):

$\displaystyle 0$
$\displaystyle=\left\langle{Sf(0),g(0)}\right\rangle-\left\langle{Sf(\ell),g(\ell)}\right\rangle$
$\displaystyle=\left\langle{\binom{-f_{-}(0)}{f_{+}(0)},\binom{g_{-}(0)}{g_{+}(0)}}\right\rangle-\left\langle{\binom{-f_{-}(\ell)}{f_{+}(\ell)},\binom{g_{-}(\ell)}{g_{+}(\ell)}}\right\rangle$
$\displaystyle=\left\langle{\binom{f_{-}(\ell)}{f_{+}(0)},\binom{g_{-}(\ell)}{g_{+}(0)}}\right\rangle-\left\langle{\binom{f_{-}(0)}{f_{+}(\ell)},\binom{g_{-}(0)}{g_{+}(\ell)}}\right\rangle$
$\displaystyle=\left\langle{f_{\ell,0},g_{\ell,0}}\right\rangle-\left\langle{f_{0,\ell},g_{0,\ell}}\right\rangle=\left\langle{f_{\ell,0},g_{\ell,0}}\right\rangle+\left\langle{Mf_{\ell,0},g_{0,\ell}}\right\rangle$
$\displaystyle=\left\langle{f_{\ell,0},g_{\ell,0}+M^{*}g_{0,\ell}}\right\rangle,\qquad
f\in\operatorname{dom}(L_{0,U}),\quad g\in\operatorname{dom}(L_{0,U}^{*}).$
(8.12)

This implies that $g\in\operatorname{dom}(L_{0,U}^{*})$ if and only if

$\displaystyle 0=g_{\ell,0}+M^{*}g_{0,\ell}$
$\displaystyle=\binom{g_{-}(\ell)}{g_{+}(0)}+\begin{pmatrix}D_{11}^{*}&D_{21}^{*}\\\
C_{12}^{*}&C_{22}^{*}\end{pmatrix}\binom{g_{-}(0)}{g_{+}(\ell)}$
$\displaystyle=\binom{g_{-}(\ell)+D_{11}^{*}g_{-}(0)+D_{21}^{*}g_{+}(\ell)}{g_{+}(0)+C_{12}^{*}g_{-}(0)+C_{22}^{*}g_{+}(\ell)}$
$\displaystyle=\begin{pmatrix}D_{11}^{*}&\mathbb{O}\\\
C_{12}^{*}&I_{n_{+}}\end{pmatrix}\binom{g_{-}(0)}{g_{+}(0)}+\begin{pmatrix}I_{n_{-}}&D_{21}^{*}\\\
\mathbb{O}&C_{22}^{*}\end{pmatrix}\binom{g_{-}(\ell)}{g_{+}(\ell)}$ (8.13)
$\displaystyle=C_{*}g(0)+D_{*}g(\ell),$ (8.14)

where $C_{*}$ and $D_{*}$ are given by (8.9). This finishes the proof. ∎

######  Remark 8.3.

(i) If either $n_{-}=0$ or $n_{-}=n$, the canonical form (8.9) can be
simplified. Indeed, if $n_{-}=n$, in which case $n_{+}=0$ and $D=S=I_{n}$,
then $C_{*}=I_{n}$ and $D_{*}=C^{*}$. And if $n_{-}=0$, in which case
$n_{+}=n$ and $C=-S=I_{n}$, then $C_{*}=D^{*}$ and $D_{*}=I_{n}$.

(ii) Let $D_{11}$ be invertible. It is clear that $C_{*}$ is also invertible.
Set $X:=C_{*}^{-1}D_{*}$. It is interesting to mention that the matrices
$C_{*}^{-1}$ and $D_{*}$ serve the triangular factorization of the matrix $X$.

Next, we indicate an explicit form of eigenvectors of the operators
$L_{U}^{*}(Q)$ and $L_{U}^{*}(0)$ corresponding to their simple eigenvalues.
To this end let

$\Phi_{*}(x,\mu):=\Phi_{Q_{*}}(x,\mu)=:\begin{pmatrix}\Phi_{*1}(x,\mu)&\ldots&\Phi_{*n}(x,\mu)\end{pmatrix}=:\left(\varphi_{*jk}(x,\mu)\right)_{j,k=1}^{n}$

be a fundamental matrix solution of the system $\mathcal{L}(Q_{*})Y=\mu Y$
that corresponds to the adjoint operator $L_{U}^{*}(Q)$ according to Lemma
8.1. Recall, that $\Phi_{0}(x,\mu)$ is a fundamental matrix solution of
$\mathcal{L}_{0}^{*}Y=\mathcal{L}_{0}Y=\mu Y$. Set

$\displaystyle\Delta_{*}(\mu)$ $\displaystyle:=\det(A_{*}(\mu)),\qquad
A_{*}(\mu):=C_{*}+D_{*}\Phi_{*}(\ell,\mu)=:(a_{*jk}(\mu))_{j,k=1}^{n},$ (8.15)
$\displaystyle\Delta_{0*}(\mu)$ $\displaystyle:=\det(A_{0*}(\mu)),\qquad
A_{0*}(\mu):=C_{*}+D_{*}\Phi_{0}(\ell,\mu)=:(a_{*jk}^{0}(\mu))_{j,k=1}^{n}.$
(8.16)

Recall that $A_{*}^{a}(\mu)$ ($A_{0*}^{a}(\mu)$) denotes the adjugate matrix
of $A_{*}(\mu)$ (resp. $A_{0*}(\mu)$). Set

$A^{a}_{*}(\mu)=:(A_{*jk}(\mu))_{j,k=1}^{n},\qquad
A^{a}_{0*}(\mu)=:(A_{*jk}^{0}(\mu))_{j,k=1}^{n}.$

Lemma 5.1 implies that for any simple eigenvalue $\mu$ of
$L_{U}^{*}(Q)=L_{U_{*}}(Q_{*})$ (simple zero of $\Delta_{*}(\cdot)$) there
exists $q\in\\{1,\ldots,{n}\\}$ such that the vector function

$y_{*}(x,\mu):=Y_{*q}(x,\mu):=\sum_{k=1}^{n}A_{*kq}(\mu)\Phi_{*k}(x,\mu)$
(8.17)

(see (5.18)) is a non-trivial eigenvector of $L_{U}^{*}(Q)=L_{U_{*}}(Q_{*})$
corresponding to its simple eigenvalue $\mu$. Similarly,

$y_{0*}(x,\mu_{0}):=Y_{*q}^{0}(x,\mu_{0}):=\operatorname{col}\left(A_{*1q}^{0}(\mu_{0})e^{i\mu_{0}\rho_{1}(x)},\ldots,A_{*nq}^{0}(\mu_{0})e^{i\mu_{0}\rho_{n}(x)}\right)$
(8.18)

is a non-trivial eigenvector of $L_{0,U}^{*}(=L_{0,U_{*}})$ corresponding to
its simple eigenvalue $\mu_{0}$.

Let us formulate Lemma 5.3 for adjoint operator $L_{U}^{*}(Q)$.

######  Lemma 8.4.

Let $\lambda$ be an algebraically simple eigenvalue of the operator
$L_{U}(Q)$. Then $\overline{\lambda}$ is an algebraically simple eigenvalue of
the operator $L_{U}^{*}(Q)$. Let $g$ be any eigenvector of $L_{U}^{*}(Q)$ in
$\mathfrak{H}$ corresponding to $\overline{\lambda}$. Then, there exists
$q=q_{\overline{\lambda}}\in\\{1,\ldots,{n}\\}$ and
$\gamma_{*q}\in\mathbb{C}$, such that

$g(\cdot)=\gamma_{*q}Y_{*q}(\cdot,\overline{\lambda})=\gamma_{*q}\sum_{k=1}^{n}A_{*kq}(\overline{\lambda})\Phi_{k}(\cdot,\overline{\lambda}),\qquad|\gamma_{*q}|=\|g\|_{\mathfrak{H}}/\|Y_{*q}(\cdot,\overline{\lambda})\|_{\mathfrak{H}}.$
(8.19)

Morever, this is valid for any $q\in\\{1,\ldots,{n}\\}$, for which
$Y_{*q}(\cdot,\overline{\lambda})\not\equiv 0$.

###  8.2 Key identity for scalar product of eigenvectors

The following result plays a crucial role in proving the Riesz basis property.
To state it we set

$\displaystyle\mathcal{E}(\lambda)$
$\displaystyle:=\det(\Phi_{0}(\ell,-\lambda))=\exp(-i(b_{1}+\ldots+b_{n})\lambda).$
(8.20) $\displaystyle b_{k}^{-}$ $\displaystyle:=\min\\{b_{k},0\\}\leqslant
0,\qquad b_{k}^{+}:=\max\\{b_{k},0\\}\geqslant 0,\qquad
k\in\\{1,\ldots,{n}\\}.$ (8.21)

######  Proposition 8.5.

Let diagonal matrix function $B(\cdot)$ satisfies relaxed conditions (5.9) and
let regular boundary conditions (5.2) be of canonical form (5.57). Further,
let $\lambda\in\mathbb{C}$ be a simple zero of $\Delta_{0}(\cdot)$ and let
$y_{0}(\cdot,\lambda)$ and $y_{0*}(\cdot,\overline{\lambda})$ be the
corresponding eigenvectors given by (5.18) and (8.18), respectively. Then the
following identity holds

$\left(y_{0}(\cdot,\lambda),y_{0*}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}=\left(Y_{p}^{0}(\cdot,\lambda),Y_{*q}^{0}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}=-i\mathcal{E}(\lambda)\exp\left(ib_{q}^{-}\lambda\right)\cdot
A_{qp}^{0}(\lambda)\Delta_{0}^{\prime}(\lambda),$ (8.22)

where $A_{qp}^{0}(\lambda)\neq 0$.

###### Proof.

First we note that combining formulas for the eigenvectors (5.18) and (8.18)
of the operators $L_{0,U}$ and $L_{0,U_{*}}$ corresponding to the eigenvalues
$\lambda$ and $\overline{\lambda}$, respectively, yields

$\left(Y_{p}^{0}(\cdot,\lambda),Y_{*q}^{0}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}=\int_{0}^{\ell}\sum_{k=1}^{n}A_{kp}^{0}(\lambda)e^{i\lambda\rho_{k}(x)}\overline{A_{*kq}^{0}(\overline{\lambda})e^{i\overline{\lambda}\rho_{k}(x)}}|\beta_{k}(x)|dx\\\
=\sum_{k=1}^{n}\overline{A_{*kq}^{0}(\overline{\lambda})}\cdot|b_{k}|\cdot
A_{kp}^{0}(\lambda),\qquad
p,q\in\\{1,\ldots,{n}\\},\quad\lambda\in\mathbb{C}.$ (8.23)

Our purpose is to transform identity (8.23) into (8.22). To this end we divide
the proof in three steps.

(i) First we consider the case $n_{-}=0$, i.e. $D=S=I_{n}$. Let us start with
some general facts valid without the assumption $\Delta_{0}(\lambda)=0$. Since
$D=I_{n}$, one has $A_{0}(\lambda)=C+\Phi_{0}(\ell,\lambda)$ and

$A_{0}^{\prime}(\lambda)=\frac{d}{d\lambda}\Phi_{0}(\ell,\lambda)=\operatorname{diag}\left(ib_{1}e^{i\lambda
b_{1}},\ldots,ib_{n}e^{i\lambda b_{n}}\right),\qquad\lambda\in\mathbb{C}.$
(8.24)

Further, Lemma 8.2 implies that $C_{*}=I_{n}$ and $D_{*}=C^{*}$. Since
$[\Phi_{0}(\ell,\overline{\lambda})]^{-1}=\Phi_{0}(\ell,-\overline{\lambda})=[\Phi_{0}(\ell,\lambda)]^{*}$
then

$A_{0*}(\overline{\lambda})=I_{n}+C^{*}\Phi_{0}(\ell,\overline{\lambda})=(\Phi_{0}(\ell,-\overline{\lambda})+C^{*})\Phi_{0}(\ell,\overline{\lambda})\\\
=[C+\Phi_{0}(\ell,\lambda)]^{*}\Phi_{0}(\ell,\overline{\lambda})=[A_{0}(\lambda)]^{*}\Phi_{0}(\ell,\overline{\lambda}).$
(8.25)

It follows from (8.20) that

$\det(\Phi_{0}(\ell,\overline{\lambda}))=\exp(i(b_{1}+\ldots+b_{n})\overline{\lambda})=\overline{\mathcal{E}(\lambda)}.$
(8.26)

Combining this identity with (8.25) and taking into account properties (2.18)
and (2.20) of adjugate matrices and identity
$[\Phi_{0}(\ell,\overline{\lambda})]^{-1}=\Phi_{0}(\ell,-\overline{\lambda})$,
imply

$\Delta_{0*}(\overline{\lambda})=\overline{\mathcal{E}(\lambda)\cdot\Delta_{0}(\lambda)},\qquad
A^{a}_{0*}(\overline{\lambda})=\overline{\mathcal{E}(\lambda)}\cdot\Phi_{0}(\ell,-\overline{\lambda})\cdot[A_{0}^{a}(\lambda)]^{*}.$
(8.27)

Taking adjoint in (8.27) we arrive at

$[A^{a}_{0*}(\overline{\lambda})]^{*}=\mathcal{E}(\lambda)A_{0}^{a}(\lambda)\Phi_{0}(\ell,\lambda),\quad\text{i.e.}\quad\overline{A_{*kj}^{0}(\overline{\lambda})}=\mathcal{E}(\lambda)e^{i\lambda
b_{k}}A_{jk}^{0}(\lambda),$ (8.28)

for $j,k\in\\{1,\ldots,{n}\\}$. Note that since $n_{-}=0$ then $b_{k}>0$,
$k\in\\{1,\ldots,{n}\\}$. Taking this into account and inserting (8.28) into
(8.23), we get

$\left(Y_{p}^{0}(\cdot,\lambda),Y_{*q}^{0}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}=\mathcal{E}(\lambda)\sum_{k=1}^{n}A_{qk}^{0}(\lambda)b_{k}e^{i\lambda
b_{k}}A_{kp}^{0}(\lambda),\qquad p,q\in\\{1,\ldots,{n}\\}.$ (8.29)

Since $\operatorname{rank}(A_{0}^{a}(\lambda))=1$, one has

$0=\det\begin{pmatrix}A_{qk}^{0}(\lambda)&A_{kk}^{0}(\lambda)\\\
A_{qp}^{0}(\lambda)&A_{kp}^{0}(\lambda)\end{pmatrix}=A_{qk}^{0}(\lambda)A_{kp}^{0}(\lambda)-A_{qp}^{0}(\lambda)A_{kk}^{0}(\lambda),\qquad
k,p,q\in\\{1,\ldots,{n}\\}.$ (8.30)

Inserting this identity into (8.29), we derive

$\left(Y_{p}^{0}(\cdot,\lambda),Y_{*q}^{0}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}=\mathcal{E}(\lambda)\sum_{k=1}^{n}b_{k}e^{i\lambda
b_{k}}A_{qk}^{0}(\lambda)A_{kp}^{0}(\lambda)\\\
=\mathcal{E}(\lambda)A_{qp}^{0}(\lambda)\sum_{k=1}^{n}b_{k}e^{i\lambda
b_{k}}A_{kk}^{0}(\lambda).$ (8.31)

Inserting (8.24) in formula (5.66) implies

$\Delta_{0}^{\prime}(\lambda)=\operatorname{tr}\left(A_{0}^{a}(\lambda)A_{0}^{\prime}(\lambda)\right)=i\sum_{k=1}^{n}A_{kk}^{0}(\lambda)b_{k}e^{i\lambda
b_{k}}=i\sum_{k=1}^{n}b_{k}e^{i\lambda b_{k}}A_{kk}^{0}(\lambda).$ (8.32)

Inserting (8.32) into (8.31) and taking into account that $b_{q}^{-}=0$ when
$n_{-}=0$, we arrive at (8.22), which finishes the proof in the case
$n_{-}=0$.

(ii) In this step we consider the general case $n_{-},n_{+}>0$. Our first goal
is to obtain formula for $\Delta_{0}^{\prime}(\lambda)$ similar to (8.32).
Since $n_{-}>0$, the matrix $A_{0}^{a}(\lambda)$ has more complicated
structure. Applying Jacobi formula to it directly will lead to more
complicated unusable formula for $\Delta_{0}^{\prime}(\lambda)$. Hence, we
need to do some preparations first. With account of definition (8.21), let us
decompose the diagonal matrix function
$\Phi_{0}(\ell,\lambda)=\operatorname{diag}\left(e^{i\lambda
b_{1}},\ldots,e^{i\lambda b_{n}}\right)$ with respect to the orthogonal
decomposition $\mathbb{C}^{n}=\mathbb{C}^{n_{-}}\oplus\mathbb{C}^{n_{+}}$,

$\displaystyle\Phi_{0}(\ell,\lambda)$
$\displaystyle=:\operatorname{diag}(\Phi^{0}_{-}(\ell,\lambda),\Phi^{0}_{+}(\ell,\lambda)),\qquad\Phi^{0}_{\pm}(\ell,\lambda)\in\mathbb{C}^{n_{\pm}\times
n_{\pm}},\quad\lambda\in\mathbb{C},$ (8.33)
$\displaystyle\widehat{\Phi}^{0}_{-}(\ell,\lambda)$
$\displaystyle:=\operatorname{diag}(\Phi^{0}_{-}(\ell,\lambda),I_{n_{+}})=\operatorname{diag}\bigl{(}e^{i\lambda
b_{k}^{-}}\bigr{)}_{k=1}^{n},$ (8.34)
$\displaystyle\widehat{\Phi}^{0}_{+}(\ell,\lambda)$
$\displaystyle:=\operatorname{diag}(I_{n_{-}},\Phi^{0}_{+}(\ell,\lambda))=\operatorname{diag}\bigl{(}e^{i\lambda
b_{k}^{+}}\bigr{)}_{k=1}^{n}.$ (8.35)

Using formulas (8.34)–(8.35) and (canonical) triangular block-matrix
representation (5.57) of matrices $C$ and $D$ from the boundary conditions
$Uy=0$, we obtain

$\displaystyle A_{0}(\lambda)$ $\displaystyle=C+D\Phi_{0}(\ell,\lambda)$
$\displaystyle=\begin{pmatrix}I_{n_{-}}&C_{12}\\\
0&C_{22}\end{pmatrix}+\begin{pmatrix}D_{11}&0\\\
D_{21}&I_{n_{+}}\end{pmatrix}\begin{pmatrix}\Phi^{0}_{-}(\ell,\lambda)&0\\\
0&\Phi^{0}_{+}(\ell,\lambda)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}I_{n_{-}}+D_{11}\Phi^{0}_{-}(\ell,\lambda)&C_{12}\\\
D_{21}\Phi^{0}_{-}(\ell,\lambda)&C_{22}+\Phi^{0}_{+}(\ell,\lambda)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\Phi^{0}_{-}(\ell,-\lambda)+D_{11}&C_{12}\\\
D_{21}&C_{22}+\Phi^{0}_{+}(\ell,\lambda)\end{pmatrix}\begin{pmatrix}\Phi^{0}_{-}(\ell,\lambda)&0\\\
0&I_{n_{+}}\end{pmatrix}$
$\displaystyle=\widehat{A}_{0}(\lambda)\widehat{\Phi}^{0}_{-}(\ell,\lambda),\qquad$
(8.36)

where

$\widehat{A}_{0}(\lambda):=\begin{pmatrix}\Phi^{0}_{-}(\ell,-\lambda)+D_{11}&C_{12}\\\
D_{21}&C_{22}+\Phi^{0}_{+}(\ell,\lambda)\end{pmatrix}=A_{0}(\lambda)\widehat{\Phi}^{0}_{-}(\ell,-\lambda).$
(8.37)

It is clear that the derivative of the matrix $\widehat{A}_{0}(\lambda)$ is
diagonal, which makes it more suitable for applying Jacobi’s formula. To this
end we note that formulas (8.34), (8.35) and (8.21) imply

$\operatorname{diag}\left(\Phi^{0}_{-}(\ell,-\lambda),\Phi^{0}_{+}(\ell,\lambda)\right)=[\widehat{\Phi}^{0}_{-}(\ell,\lambda)]^{-1}\widehat{\Phi}^{0}_{+}(\ell,\lambda)=\operatorname{diag}\bigl{(}e^{-i\lambda
b_{k}^{-}}\cdot e^{i\lambda
b_{k}^{+}}\bigr{)}_{k=1}^{n}=\operatorname{diag}\left(e^{i\lambda|b_{k}|}\right)_{k=1}^{n},$
(8.38)

where we used trivial identity $|b_{k}|=-b_{k}^{-}+b_{k}^{+}$,
$k\in\\{1,\ldots,{n}\\}$. Let also

$e_{-}(\lambda):=\det(\Phi^{0}_{-}(\ell,-\lambda))=e^{-i\lambda
b_{-}},\quad\lambda\in\mathbb{C},\qquad b_{-}:=b_{1}+\ldots+b_{n_{-}}<0.$
(8.39)

Definition (8.37), properties (2.19) and (2.17) of adjugate matrices, and
relation (8.38) imply that

$\displaystyle\widehat{\Delta}_{0}(\lambda):=\det(\widehat{A}_{0}(\lambda))=\det(A_{0}(\lambda))\det(\Phi^{0}_{-}(\ell,-\lambda))=e_{-}(\lambda)\Delta_{0}(\lambda),$
(8.40)
$\displaystyle\widehat{A}_{0}^{a}(\lambda)=[A_{0}(\lambda)\widehat{\Phi}^{0}_{-}(\ell,-\lambda)]^{a}=e_{-}(\lambda)\widehat{\Phi}^{0}_{-}(\ell,\lambda)A_{0}^{a}(\lambda)=\bigl{(}e_{-}(\lambda)e^{i\lambda
b_{j}^{-}}A_{jk}^{0}(\lambda)\bigr{)}_{j,k=1}^{n},$ (8.41)
$\displaystyle\widehat{A}_{0}^{\prime}(\lambda)=\frac{d}{d\lambda}\operatorname{diag}\left(\Phi^{0}_{-}(\ell,-\lambda),\Phi^{0}_{+}(\ell,\lambda)\right)=\operatorname{diag}\left(i|b_{k}|e^{i\lambda|b_{k}|}\right)_{k=1}^{n}.$
(8.42)

Now Jacobi’s formula (2.21) applies to
$\widehat{\Delta}_{0}(\cdot)=e_{-}(\cdot){\Delta}_{0}(\cdot)$ and gives with
account of (8.41) and (8.42)

$e_{-}(\lambda)\Delta^{\prime}_{0}(\lambda)+e_{-}^{\prime}(\lambda)\Delta_{0}(\lambda)=\widehat{\Delta}^{\prime}_{0}(\lambda)=\operatorname{tr}\bigl{(}\widehat{A}_{0}^{a}(\lambda)\widehat{A}_{0}^{\prime}(\lambda)\bigr{)}\\\
=\sum_{k=1}^{n}e_{-}(\lambda)e^{i\lambda b_{k}^{-}}A_{kk}^{0}(\lambda)\cdot
i|b_{k}|e^{i\lambda|b_{k}|}=ie_{-}(\lambda)\sum_{k=1}^{n}|b_{k}|e^{i\lambda
b_{k}^{+}}A_{kk}^{0}(\lambda).$ (8.43)

Given that $e_{-}^{\prime}(\lambda)=-ib_{-}e_{-}(\lambda)$, it follows from
(8.43) that

$\Delta^{\prime}_{0}(\lambda)=ib_{-}\Delta_{0}(\lambda)+i\sum_{k=1}^{n}|b_{k}|e^{i\lambda
b_{k}^{+}}A_{kk}^{0}(\lambda),\qquad\lambda\in\mathbb{C}.$ (8.44)

Now let us find an explicit form of the adjugate matrix
$A^{a}_{0*}(\overline{\lambda})$. Since

$[\Phi^{0}_{\pm}(\ell,\overline{\lambda})]^{-1}=\Phi^{0}_{\pm}(\ell,-\overline{\lambda})=[\Phi^{0}_{\pm}(\ell,\lambda)]^{*},$
(8.45)

then

$\displaystyle A_{0*}(\overline{\lambda})$
$\displaystyle=C_{*}+D_{*}\Phi_{0}(\ell,\overline{\lambda})$
$\displaystyle=\begin{pmatrix}D_{11}^{*}&0\\\
C_{12}^{*}&I_{n_{+}}\end{pmatrix}+\begin{pmatrix}I_{n_{-}}&D_{21}^{*}\\\
0&C_{22}^{*}\end{pmatrix}\begin{pmatrix}\Phi^{0}_{-}(\ell,\overline{\lambda})&0\\\
0&\Phi^{0}_{+}(\ell,\overline{\lambda})\end{pmatrix}$
$\displaystyle=\begin{pmatrix}D_{11}^{*}+\Phi^{0}_{-}(\ell,\overline{\lambda})&D_{21}^{*}\Phi^{0}_{+}(\ell,\overline{\lambda})\\\
C_{12}^{*}&I_{n_{+}}+C_{22}^{*}\Phi^{0}_{+}(\ell,\overline{\lambda})\\\
\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\Phi^{0}_{-}(\ell,\overline{\lambda})&0\\\
0&I_{n_{+}}\end{pmatrix}\begin{pmatrix}[\Phi^{0}_{-}(\ell,\lambda)]^{*}D_{11}^{*}+I_{n_{-}}&[\Phi^{0}_{-}(\ell,\lambda)]^{*}D_{21}^{*}\\\
C_{12}^{*}&[\Phi^{0}_{+}(\ell,\lambda)]^{*}+C_{22}^{*}\\\
\end{pmatrix}\begin{pmatrix}I_{n_{-}}&0\\\
0&\Phi^{0}_{+}(\ell,\overline{\lambda})\end{pmatrix}$
$\displaystyle=\widehat{\Phi}^{0}_{-}(\ell,\overline{\lambda})\cdot[A_{0}(\lambda)]^{*}\cdot\widehat{\Phi}^{0}_{+}(\ell,\overline{\lambda}).$

Further, it follows from (8.34), (8.35) and (8.20) that

$\det\bigl{(}\widehat{\Phi}^{0}_{-}(\ell,\overline{\lambda})\cdot\widehat{\Phi}^{0}_{+}(\ell,\overline{\lambda})\bigr{)}=\det(\Phi_{0}(\ell,\overline{\lambda}))=\exp(i(b_{1}+\ldots+b_{n})\overline{\lambda})=\overline{\mathcal{E}(\lambda)}.$
(8.46)

Hence due to (8.26) and properties (2.18) and (2.20) of adjugate matrix, we
have

$\displaystyle\Delta_{0*}(\overline{\lambda})$
$\displaystyle=\det(A_{0*}(\overline{\lambda}))=\det\left(\widehat{\Phi}^{0}_{-}(\ell,\overline{\lambda})\cdot\widehat{\Phi}^{0}_{+}(\ell,\overline{\lambda})\right)\cdot\overline{\det(A_{0}(\lambda))}=\overline{\mathcal{E}(\lambda)\cdot\Delta_{0}(\lambda)},$
$\displaystyle A^{a}_{0*}(\lambda)$
$\displaystyle=\overline{\mathcal{E}(\lambda)}\cdot\widehat{\Phi}^{0}_{+}(\ell,-\overline{\lambda})\cdot[A_{0}^{a}(\lambda)]^{*}\cdot\widehat{\Phi}^{0}_{-}(\ell,-\overline{\lambda}).$
(8.47)

Taking adjoint we get

$[A^{a}_{0*}(\overline{\lambda})]^{*}=\mathcal{E}(\lambda)\cdot\widehat{\Phi}^{0}_{-}(\ell,\lambda)\cdot
A_{0}^{a}(\lambda)\cdot\widehat{\Phi}^{0}_{+}(\ell,\lambda).$ (8.48)

Inserting (8.34)–(8.35) into (8.48) we arrive at the key identity

$\overline{A_{*kj}^{0}(\overline{\lambda})}=\mathcal{E}(\lambda)e^{i\lambda
b_{j}^{-}}e^{i\lambda b_{k}^{+}}A_{jk}^{0}(\lambda),\qquad
j,k\in\\{1,\ldots,{n}\\},\quad\lambda\in\mathbb{C}.$ (8.49)

Inserting (8.49) into (8.23) and taking into account formula (8.44) for
$\Delta^{\prime}_{0}(\cdot)$, identity (8.30), one deduces

$\displaystyle\left(Y_{p}^{0}(\cdot,\lambda),Y_{*q}^{0}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}$
$\displaystyle=\mathcal{E}(\lambda)e^{i\lambda
b_{q}^{-}}\sum_{k=1}^{n}A_{qk}^{0}(\lambda)|b_{k}|e^{i\lambda
b_{k}^{+}}A_{kp}^{0}(\lambda)$ $\displaystyle=\mathcal{E}(\lambda)e^{i\lambda
b_{q}^{-}}A_{qp}^{0}(\lambda)\sum_{k=1}^{n}|b_{k}|e^{i\lambda
b_{k}^{+}}A_{kk}^{0}(\lambda)$
$\displaystyle=-i\mathcal{E}(\lambda)e^{i\lambda
b_{q}^{-}}A_{qp}^{0}(\lambda)\left(\Delta^{\prime}_{0}(\lambda)-ib_{-}\Delta_{0}(\lambda)\right),\quad
p,q\in\\{1,\ldots,{n}\\}.$ (8.50)

Since $\lambda$ is a simple root of $\Delta_{0}(\cdot)$ then (8.50) implies
(8.22).

The case $n_{+}=0$ can be treated similarly.

(iii) Assuming $\lambda$ to be a simple eigenvalue of the operator $L_{0,U}$,
let us show that $A_{qp}^{0}(\lambda)\neq 0$ whenever,
$Y_{p}^{0}(\cdot,\lambda)\not\equiv 0$ and
$Y_{*q}^{0}(\cdot,\overline{\lambda})\not\equiv 0$. According to Lemma 5.1, we
have

$\Delta_{0}(\lambda)=0,\qquad\Delta_{0}^{\prime}(\lambda)\neq
0,\qquad\operatorname{rank}(A_{0}^{a}(\lambda))=1.$ (8.51)

Since $A_{0}^{a}(\lambda)$ is non-zero matrix, there exist
$j,k\in\\{1,\ldots,{n}\\}$ such that $A_{kj}^{0}(\lambda)\neq 0$. Formula
(8.22) implies that

$\left(Y_{j}^{0}(\cdot,\lambda),Y_{*k}^{0}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}=-i\mathcal{E}(\lambda)\exp\left(ib_{k}^{-}\lambda\right)\cdot
A_{kj}^{0}(\lambda)\Delta_{0}^{\prime}(\lambda)\neq 0,$ (8.52)

since $A_{jk}^{0}(\lambda)\neq 0$ and $\Delta_{0}^{\prime}(\lambda)\neq 0$.

Now let $p,q\in\\{1,\ldots,{n}\\}$ be such that
$Y_{p}^{0}(\cdot,\lambda)\not\equiv 0$ and
$Y_{*q}^{0}(\cdot,\overline{\lambda})\not\equiv 0$. Since
$\operatorname{rank}(A_{0}^{a}(\lambda))=1$, then $Y_{p}^{0}(\cdot,\lambda)$
is proportional to $Y_{j}^{0}(\cdot,\lambda)$. Formulas (8.28) and (8.47)
imply that $\operatorname{rank}(A_{*0}^{a}(\overline{\lambda}))=1$. Hence
$Y_{*q}^{0}(\cdot,\overline{\lambda})$ is proportional to
$Y_{*k}^{0}(\cdot,\overline{\lambda})$. Therefore,

$Y_{p}^{0}(\cdot,\lambda)=\alpha_{pj}Y_{j}^{0}(\cdot,\lambda),\qquad
Y_{*q}^{0}(\cdot,\overline{\lambda})=\alpha_{*qk}Y_{*k}^{0}(\cdot,\overline{\lambda}),\qquad\alpha_{pj},\alpha_{*qk}\in\mathbb{C}\setminus\\{0\\}.$
(8.53)

Combining formulas (8.22), (8.52) and (8.53) we arrive at

$\displaystyle-i\mathcal{E}(\lambda)\exp\left(ib_{q}^{-}\lambda\right)\cdot
A_{qp}^{0}(\lambda)\Delta_{0}^{\prime}(\lambda)=\left(Y_{p}^{0}(\cdot,\lambda),Y_{*q}^{0}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}=\alpha_{pj}\overline{\alpha_{*qk}}\left(Y_{j}^{0}(\cdot,\lambda),Y_{*k}^{0}(\cdot,\overline{\lambda})\right)_{\mathfrak{H}}\neq
0,$ (8.54)

which implies $A_{qp}^{0}(\lambda)\neq 0$ and finishes the proof. ∎

######  Remark 8.6.

The proof remains valid for non-regular boundary conditions provided that
$J_{P}(C,D)\neq 0$ for some $P\in\mathcal{P}_{n}$. Indeed, we can use
alternative canonical form of boundary conditions (5.2) outlined in Remark
5.12.

##  9 Uniform minimality and Riesz basis property

###  9.1 Uniform minimality

Here we apply results of the previous section to show an important property of
the system of root vectors of the operator $L_{U}(Q)$: uniform minimality.

######  Definition 9.1.

A sequence $\\{\varphi_{m}\\}_{m\in\mathbb{Z}}$ in a Banach space $X$ is
called minimal if

$\varphi_{m}\notin\operatorname{span}\\{\varphi_{k}:k\neq m\\}\qquad\text{for
any}\quad m\in\mathbb{Z}.$ (9.1)

It is called uniformly minimal if

$\inf_{m\in\mathbb{Z}}{\operatorname{dist}}\bigl{(}\|\varphi_{m}\|^{-1}\varphi_{m},{\operatorname{span}}\\{\varphi_{k}:\
k\neq m\\}\bigr{)}>0.$ (9.2)

The following statement is well known.

######  Lemma 9.2.

(i) The sequence $\\{\varphi_{m}\\}_{m\in\mathbb{Z}}\subset X$ is minimal if
and only if there exists a biorthogonal system
$\\{\varphi_{*m}\\}_{m\in\mathbb{Z}}\subset X^{*}$, i.e. a system satisfying
$(f_{k},\varphi_{*m})=\delta_{km}$, $k,m\in\mathbb{Z}$.

(ii) The sequence $\\{\varphi_{m}\\}_{m\in\mathbb{Z}}\subset X$ is uniformly
minimal if and only if it admits a biorthogonal system
$\\{\varphi_{*m}\\}_{m\in\mathbb{Z}}\subset X^{*}$ satisfying

$\sup_{m\in\mathbb{Z}}\|\varphi_{m}\|\cdot\|\varphi_{*m}\|<\infty.$ (9.3)

Now we are ready to prove uniform minimality of the strictly regular BVP
(5.1)–(5.2). If $Q=0$, this can be done under relaxed conditions on the matrix
function $B(\cdot)$.

######  Proposition 9.3.

Let self-adjoint invertible diagonal matrix function $B(\cdot)$ satisfy
relaxed conditions (5.9). Namely, $B\in L^{1}([0,\ell];\mathbb{R}^{n\times
n})$ and every its entry does not change sign on $[0,\ell]$. Let boundary
conditions of the boundary value problem (5.8), (5.2) be strictly regular.
Then any system of root vectors of the operator $L_{0,U}=L_{U}(0)$ is
uniformly minimal in $\mathfrak{H}$.

###### Proof.

Since boundary conditions are strictly regular then $\Delta_{0}(\cdot)$ has a
countable asymptotically separated sequence of zeros
$\Lambda_{0}:=\\{\lambda_{m}^{0}\\}_{m\in\mathbb{Z}}$ (counting multiplicity),
satisfying (5.61) with certain $\delta,m_{0}>0$ and lying in the strip
$\Pi_{h}$. Clearly, $\Lambda_{0}$ is a sequence of eigenvalues of $L_{0,U}$
(counting multiplicity) and
$\overline{\Lambda_{0}}:=\\{\overline{\lambda_{m}^{0}}\\}_{m\in\mathbb{Z}}$ is
a sequence of eigenvalues of $L_{0,U}^{*}$ (counting multiplicity). Moreover,
each eigenvalue of $L_{0,U}$ has finite multiplicity.

Let $\mathcal{F}_{0}:=\\{f_{m}^{0}\\}_{m\in\mathbb{Z}}$ be any system of root
vectors of the operator $L_{0,U}$. Since operator $L_{0,U}$ has discrete
spectrum and each its eigenvalue has finite multiplicity, we can choose system
of root vectors $\mathcal{F}_{*0}:=\\{f_{*m}^{0}\\}_{m\in\mathbb{Z}}$ of the
operator $L_{0,U}^{*}$ in such a way that $\mathcal{F}_{0}$ and
$\mathcal{F}_{*0}$ are biorthogonal systems. i.e.
$(f_{j}^{0},f_{*k}^{0})=\delta_{jk}$, $j,k\in\mathbb{Z}$. This, implies
minimality of the system $\mathcal{F}_{0}$. In accordance with Lemma 9.2(ii),
to prove uniform minimality it is sufficient to show that

$\sup_{|m|>m_{0}}\|f_{m}^{0}\|_{\mathfrak{H}}\cdot\|f_{*m}^{0}\|_{\mathfrak{H}}<\infty,$
(9.4)

where $m_{0}$ is from Definition 5.13(iii) of strict regularity.

Let $|m|>m_{0}$. Then $\lambda_{m}^{0}$ and $\overline{\lambda_{m}^{0}}$ are
algebraically simple eigenvalues of the operators $L_{0,U}$ and $L_{0,U}^{*}$,
respectively. Moreover, by Lemma 5.15, there exist indices
$p=p_{m}\in\\{1,\ldots,{n}\\}$ and $q=q_{m}\in\\{1,\ldots,{n}\\}$, and a
constant $C_{2}>0$ such that estimate (5.65) holds, i.e.
$|A_{qp}^{0}(\lambda_{m}^{0})|\geqslant C_{2}$, $|m|>m_{0}$. Emphasize, that
although $p$ and $q$ depend on $m$, the constant $C_{2}$ in the above estimate
does not. Starting with this $A_{qp}^{0}(\lambda_{m}^{0})(\not=0)$ we define
the vector functions

$\widetilde{f}_{m}^{0}(\cdot):=Y_{p}^{0}(\cdot,\lambda_{m}^{0})\quad\text{and}\quad\widetilde{f}_{*m}^{0}(\cdot):=Y_{*q}^{0}(\cdot,\overline{\lambda_{m}^{0}})$
(9.5)

by formulas (5.19) and (8.18), respectively.

Since boundary conditions (5.2) are regular then by Lemmas 5.11 and 8.2 we can
assume boundary conditions of the operators $L_{0,U}$ and
$L_{0,U}^{*}=L_{0,U_{*}}$ to be of canonical forms (5.57) and (8.9),
respectively. Hence Proposition 8.5 can be used. Besides, in accordance with
Lemma 5.14, $|\Delta_{0}^{\prime}(\lambda_{m}^{0})|\geqslant C_{0}$,
$|m|>m_{0}$, where a constant $C_{0}>0$ is independent on $m$. Combining
Proposition 8.5 with this estimate and the above estimate on
$|A_{qp}^{0}(\lambda_{m}^{0})|$ yields

$\displaystyle\bigl{|}{\bigl{(}\widetilde{f}_{m}^{0},\widetilde{f}_{*m}^{0}\bigr{)}_{\mathfrak{H}}}\bigr{|}$
$\displaystyle=\bigl{|}{\bigl{(}Y_{p}^{0}(\cdot,\lambda_{m}^{0}),Y_{*q}^{0}(\cdot,\overline{\lambda_{m}^{0}})\bigr{)}_{\mathfrak{H}}}\bigr{|}$
$\displaystyle=\bigl{|}{\mathcal{E}(\lambda_{m}^{0})\exp\left(ib_{q}^{-}\lambda_{m}^{0}\right)}\bigr{|}\cdot\bigl{|}{A_{qp}^{0}(\lambda_{m}^{0})\cdot\Delta_{0}^{\prime}(\lambda_{m}^{0})}\bigr{|}$
$\displaystyle\geqslant\bigl{|}{\mathcal{E}(\lambda_{m}^{0})\exp\left(ib_{q}^{-}\lambda_{m}^{0}\right)}\bigr{|}C_{2}C_{0}\geqslant
C_{3},$ (9.6)

with some $C_{3}>0$ that does not depend on $m$. Inequality (9.6) in
particular implies that both vector functions $\widetilde{f}_{m}^{0}$ and
$\widetilde{f}_{*m}^{0}$ are non-zero.

Applying Lemma 5.4 to the “adjoint” BVP (8.1)–(8.2), we arrive at the estimate

$\left\|{Y_{*q}^{0}(\cdot,\overline{\lambda})}\right\|_{\mathfrak{H}}\leqslant
M_{*h},\qquad\lambda\in\Pi_{h},$ (9.7)

with some different constant $M_{*h}$. Inclusion $\lambda_{m}^{0}\in\Pi_{h}$
and estimates (5.29) and (9.7) imply that

$\|\widetilde{f}_{m}^{0}\|_{\mathfrak{H}}=\|Y_{p}^{0}(\cdot,\lambda_{m}^{0})\|_{\mathfrak{H}}\leqslant
M_{h},\qquad\|\widetilde{f}_{*m}^{0}\|_{\mathfrak{H}}=\|Y_{*q}^{0}(\cdot,\overline{\lambda_{m}^{0}})\|_{\mathfrak{H}}\leqslant
M_{*h},\qquad|m|>m_{0}.$ (9.8)

Since $\lambda_{m}^{0}$ is algebraically simple eigenvalue of the operator
$L_{0,U}$, and $\widetilde{f}_{m}^{0}\neq 0$ and $\widetilde{f}_{*m}^{0}\neq
0$, Lemmas 5.3 and 8.4 ensure that

$f_{m}^{0}(\cdot)=\gamma_{m}^{0}\widetilde{f}_{m}^{0}(\cdot)=\gamma_{m}^{0}Y_{p}^{0}(\cdot,\lambda_{m}^{0}),\qquad
f_{*m}^{0}(\cdot)=\gamma_{*m}^{0}\widetilde{f}_{*m}^{0}(\cdot)=\gamma_{*m}^{0}Y_{*q}^{0}(\cdot,\overline{\lambda_{m}^{0}}),\qquad|m|>m_{1},$
(9.9)

with some $\gamma_{m}^{0},\gamma_{*m}^{0}\in\mathbb{C}\setminus\\{0\\}$. Since
vector systems $\mathcal{F}_{0}$ and $\mathcal{F}_{*0}$ are biorthogonal, it
follows that

$1=(f_{m}^{0},f_{*m}^{0})_{\mathfrak{H}}=\gamma_{m}^{0}\overline{\gamma_{*m}^{0}}\cdot(\widetilde{f}_{m}^{0},\widetilde{f}_{*m}^{0})_{\mathfrak{H}}.$
(9.10)

Combining estimates (9.8) with estimate (9.6), relations (9.9) and equality
(9.10) yields

$\|f_{m}^{0}\|_{\mathfrak{H}}\cdot\|f_{*m}^{0}\|_{\mathfrak{H}}=|\gamma_{m}^{0}\gamma_{*m}^{0}|\cdot\|\widetilde{f}_{m}^{0}\|_{\mathfrak{H}}\cdot\|\widetilde{f}_{*m}^{0}\|_{\mathfrak{H}}\leqslant\frac{M_{h}M_{*h}}{\bigl{|}{(\widetilde{f}_{m}^{0},\widetilde{f}_{*m}^{0})_{\mathfrak{H}}}\bigr{|}}\leqslant\frac{M_{h}M_{*h}}{C_{3}}=:C_{4},\qquad|m|>m_{0}.$
(9.11)

Since $C_{4}$ does not depend on $m$, estimate (9.11) implies estimate (9.4),
which completes the proof. ∎

In general case we need more strict conditions on the matrix function
$B(\cdot)$

######  Theorem 9.4.

Let matrix function $B(\cdot)$ given by (4.3) satisfy conditions (4.5)–(4.7),
let $Q\in L^{1}([0,\ell];\mathbb{C}^{n\times n})$ and let BVP (5.1)–(5.2) be
strictly regular according to Definition 6.13. Then any system of root vectors
of the operator $L_{U}(Q)$ is uniformly minimal in $\mathfrak{H}$.

###### Proof.

As usual, applying gauge transform from Lemma 6.11 we can reduce general case
to the case of $Q$ satisfying “zero block diagonality” condition (4.8). Since
operators $L_{U}(Q)$ and $L_{\widetilde{U}}(\widetilde{Q})$ are similar this
transform preserves uniform minimality as explained in Remark 6.12. Hence,
without loss of generality we can assume that original $Q$ satisfies “zero
block diagonality” condition (4.8).

Due to assumptions on matrix functions $B(\cdot)$ and $Q(\cdot)$, Proposition
6.5 and Theorem 6.6 imply that characteristic determinant $\Delta_{Q}(\cdot)$
has a countable sequence of eigenvalues
$\Lambda:=\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ (counting multiplicity) with
asymptotic behavior (6.31) and lying in the strip $\Pi_{h}$ (we can assume
that both $\Lambda_{0}$ and $\Lambda$ lie in the same strip $\Pi_{h}$ by
increasing $h$ if needed). By Lemma 5.1, $\Lambda$ is a sequence of
eigenvalues of $L_{U}(Q)$ (counting multiplicity) and
$\overline{\Lambda}:=\\{\overline{\lambda_{m}}\\}_{m\in\mathbb{Z}}$ is a
sequence of eigenvalues of $L_{U}^{*}(Q)$ (counting multiplicity). Moreover,
each eigenvalue of $L_{U}(Q)$ has finite multiplicity. Combining asymptotic
formula (6.31) and separation condition (5.61) on $\Lambda_{0}$ imply that for
some $m_{1}\geqslant m_{0}$ we have

$|\lambda_{j}-\lambda_{k}|>\delta,\qquad j\neq k,\quad|j|,|k|>m_{1}.$ (9.12)

Let $\mathcal{F}:=\\{f_{m}\\}_{m\in\mathbb{Z}}$ be any system of root vectors
of the operator $L_{U}(Q)$. Since operator $L_{U}(Q)$ has discrete spectrum
and each its eigenvalue has finite multiplicity, we can choose system of root
vectors $\mathcal{F}_{*}:=\\{f_{*m}\\}_{m\in\mathbb{Z}}$ of the operator
$L_{U}^{*}(Q)$ in such a way that $\mathcal{F}$ and $\mathcal{F}_{*}$ are
biorthogonal systems. i.e. $(f_{j},f_{*k})=\delta_{jk}$, $j,k\in\mathbb{Z}$.
This, implies minimality of the system $\mathcal{F}$. In accordance with Lemma
9.2(ii), to prove uniform minimality it is sufficient to show that

$\sup_{|m|>m_{2}}\|f_{m}\|_{\mathfrak{H}}\cdot\|f_{*m}\|_{\mathfrak{H}}<\infty,$
(9.13)

for some $m_{2}\geqslant m_{1}$ that we will choose later. Here
$m_{1}\geqslant m_{0}$ is from the separation condition (9.12) on
$\lambda_{m}$, while $m_{0}$ is from separation condition (5.61) on
$\lambda_{m}^{0}$.

Let $|m|>m_{1}\geqslant m_{0}$. Then $\lambda_{m}$ and
$\overline{\lambda_{m}}$ are algebraically simple eigenvalues of the operators
$L_{U}(Q)$ and $L_{U}^{*}(Q)$, respectively. Let
$p=p_{m}\in\\{1,\ldots,{n}\\}$ and $q=q_{m}\in\\{1,\ldots,{n}\\}$ be indices
chosen in the part (i) of the proof for which estimate (9.6) holds. Following
part (i) we define similar vector functions

$\widetilde{f}_{m}(\cdot):=Y_{p}(\cdot,\lambda_{m})\quad\text{and}\quad\widetilde{f}_{*m}(\cdot):=Y_{*q}(\cdot,\overline{\lambda_{m}})$
(9.14)

by formulas (5.18) and (8.17), respectively. Set

$\displaystyle\widetilde{F}_{m}(\cdot):=\widetilde{f}_{m}(\cdot)-\widetilde{f}_{m}^{0}(\cdot)=Y_{p}(\cdot,\lambda_{m})-Y_{p}^{0}(\cdot,\lambda_{m}^{0}),\qquad|m|\geqslant
m_{1},$ (9.15)
$\displaystyle\widetilde{F}_{*m}(\cdot):=\widetilde{f}_{*m}(\cdot)-\widetilde{f}_{*m}^{0}(\cdot)=Y_{*q}(\cdot,\overline{\lambda_{m}})-Y_{*q}^{0}(\cdot,\overline{\lambda_{m}^{0}}),\qquad|m|\geqslant
m_{1}.$ (9.16)

It follows from Schwarz inequality that

$\bigl{|}{\bigl{(}\widetilde{f}_{m},\widetilde{f}_{*m}\bigr{)}_{\mathfrak{H}}}\bigr{|}\geqslant\bigl{|}{\bigl{(}\widetilde{f}_{m}^{0},\widetilde{f}_{*m}^{0}\bigr{)}_{\mathfrak{H}}}\bigr{|}-\|\widetilde{f}_{m}^{0}\|_{\mathfrak{H}}\|\widetilde{F}_{*m}\|_{\mathfrak{H}}-\|\widetilde{F}_{m}\|_{\mathfrak{H}}\|\widetilde{f}_{*m}^{0}\|_{\mathfrak{H}}-\|\widetilde{F}_{m}\|_{\mathfrak{H}}\|\widetilde{F}_{*m}\|_{\mathfrak{H}}.$
(9.17)

It follows from Theorem 6.9, that $\|\widetilde{F}_{m}\|_{\mathfrak{H}}\to 0$
and $\|\widetilde{F}_{*m}\|_{\mathfrak{H}}\to 0$ as $|m|\to\infty$. Combining
this observation with estimates (9.17), (9.6), and estimates (5.29) and (9.7)
on
$\|\widetilde{f}_{m}^{0}\|_{\mathfrak{H}}=\|Y_{p}^{0}(\cdot,\lambda_{m}^{0})\|_{\mathfrak{H}}$
and
$\|\widetilde{f}_{*m}^{0}\|_{\mathfrak{H}}=\|Y_{*q}^{0}(\cdot,\overline{\lambda_{m}^{0})}\|_{\mathfrak{H}}$,
we see that

$\displaystyle\bigl{|}{\bigl{(}\widetilde{f}_{m},\widetilde{f}_{*m}\bigr{)}_{\mathfrak{H}}}\bigr{|}\geqslant
C_{3}/2,\qquad|m|>m_{2},$ (9.18)
$\displaystyle\|\widetilde{f}_{m}\|_{\mathfrak{H}}\leqslant
2M_{h},\qquad\|\widetilde{f}_{*m}\|_{\mathfrak{H}}\leqslant
2M_{h},\qquad|m|>m_{2},$ (9.19)

for some $m_{2}\geqslant m_{1}$. Inequality (9.18) implies that both vector
functions $\widetilde{f}_{m}$ and $\widetilde{f}_{*m}$ are non-zero. Since
$\lambda_{m}$ is algebraically simple eigenvalue of the operator $L_{U}(Q)$,
and $\widetilde{f}_{m}\neq 0$ and $\widetilde{f}_{*m}\neq 0$, Lemmas 5.3 and
8.4 ensure that

$f_{m}(\cdot)=\gamma_{m}\widetilde{f}_{m}(\cdot)=\gamma_{m}Y_{p}(\cdot,\lambda_{m}),\qquad
f_{*m}(\cdot)=\gamma_{*m}\widetilde{f}_{*m}(\cdot)=\gamma_{*m}Y_{*q}(\cdot,\overline{\lambda_{m}}),\qquad|m|>m_{2},$
(9.20)

with some $\gamma_{m},\gamma_{*m}\neq 0$. The proof of estimate (9.13) is
finished the same way as in part (i) by using estimates (9.18)–(9.19). ∎

###  9.2 Riesz basis property

First, let us recall some definitions.

######  Definition 9.5.

(i) A sequence $\\{\varphi_{m}\\}_{m\in\mathbb{Z}}$ of vectors in
$\mathfrak{H}$ is called a Riesz basis if it admits a representation
$\varphi_{m}=Te_{m}$, $m\in\mathbb{Z}$, where $\\{e_{m}\\}_{m\in\mathbb{Z}}$
is an orthonormal basis in $\mathfrak{H}$ and $T:\mathfrak{H}\to\mathfrak{H}$
is a bounded operator with a bounded inverse.

(ii) A sequence $\\{\varphi_{m}\\}_{m\in\mathbb{Z}}$ of vectors in
$\mathfrak{H}$ is called Besselian if

$\sum_{m\in\mathbb{Z}}\left|{(f,\varphi_{m})_{\mathfrak{H}}}\right|^{2}<\infty,\qquad
f\in\mathfrak{H}.$ (9.21)

######  Remark 9.6.

In accordance with closed graph theorem, inequality (9.21) is equivalent to

$\sum_{m\in\mathbb{Z}}\left|{(f,\varphi_{m})_{\mathfrak{H}}}\right|^{2}\leqslant\gamma^{2}\|f\|_{\mathfrak{H}}^{2},\qquad
f\in\mathfrak{H},$ (9.22)

where $\gamma>0$ does not depend on $f$. Putting in (9.22) $f=\varphi_{m}$
implies $\|\varphi_{m}\|_{\mathfrak{H}}\leqslant\gamma$, $m\in\mathbb{Z}$.

Our investigation of the Riesz basis property of the system of root vectors of
the operator $L_{U}(Q)$ heavily relies on the following well-known Bari
criterion.

######  Theorem 9.7.

[18, Theorem VI.2.1] Let $\mathfrak{H}$ be a separable Hilbert space. The
vectors system $\\{\varphi_{m}\\}_{m\in\mathbb{Z}}\subset\mathfrak{H}$ forms a
Riesz basis in $\mathfrak{H}$ if and only if it is complete and Besselian in
$\mathfrak{H}$, and there exists a biorthogonal system
$\\{\varphi_{*m}\\}_{m\in\mathbb{Z}}$ that is also complete and Besselian.

First, we establish a result that implies Besselian property for eigenvectors
of the unperturbed operator $L_{U}(0)$.

######  Lemma 9.8.

Let entries of $B(\cdot)$ satisfy condition (4.5). Let
$\\{\mu_{m}\\}_{m\in\mathbb{Z}}$ be an incompressible sequence lying in the
strip $\Pi_{h}$ (see Definition 5.7). Then the following statements hold:

(i) For any $k\in\\{1,\ldots,{n}\\}$ the sequence
$\\{\Phi^{0}_{k}(\cdot,\mu_{m})\\}_{m\in\mathbb{Z}}$ defined in (4.20) is
Besselian in $\mathfrak{H}$.

(ii) For any $p\in\\{1,\ldots,{n}\\}$ the sequence
$\\{Y_{p}^{0}(\cdot,\mu_{m})\\}_{m\in\mathbb{Z}}$ defined in (5.19) is
Besselian in $\mathfrak{H}$.

###### Proof.

(i) Let $k\in\\{1,\ldots,{n}\\}$ be fixed and let

$f=\operatorname{col}(f_{1},\ldots,f_{n})\in\mathfrak{H},\qquad\text{i.e.}\qquad
f_{j}\in L^{2}_{|\beta_{j}|}[0,\ell],\quad j\in\\{1,\ldots,{n}\\}.$

It follows from definition (4.20) of $\Phi_{k}^{0}(\cdot,\lambda)$ that

$(f,\Phi^{0}_{k}(\cdot,\mu_{m}))_{\mathfrak{H}}=\int_{0}^{\ell}f_{k}(x)\cdot\overline{e^{i\mu_{m}\rho_{k}(x)}}|\beta_{k}(x)|\,dx.$
(9.23)

Condition (4.5) implies conditions (4.18) on $\rho_{k}$ and the inverse
function $\rho_{k}^{-1}$. Let
$x_{k}:=|\rho_{k}^{-1}|\in\operatorname{Lip}[0,\ell_{k}]$, where
$\ell_{k}:=|b_{k}|$. Since $\beta_{k}$ and $\rho_{k}$ do not change sign on
the segment $[0,\ell]$, then $x_{k}$ is the inverse function of $|\rho_{k}|$.
Consider the function $f_{k}\circ x_{k}$ defined on $[0,\ell_{k}]$. Since
$1/\beta_{k}\in L^{\infty}[0,\ell]$, it follows from definition of
$L^{2}_{|\beta_{k}|}[0,\ell]$ that $f_{k}\in L^{2}[0,\ell]$. Since
$x_{k}\in\operatorname{Lip}[0,\ell]$ and strictly monotonous, and $f_{k}\in
L^{2}[0,\ell]$ it follows that $f_{k}\circ x_{k}\in L^{2}[0,\ell_{k}]\subset
L^{1}[0,\ell_{k}]$. Finally, recall that
$s_{k}=\operatorname{sign}(\beta_{k}(\cdot))=\operatorname{sign}(\rho_{k}(\cdot))=\operatorname{const}$.

Taking observations of the previous paragraph into account and making a change
of variable $x=x_{k}(u)$ in (9.23) (and so $u=|\rho_{k}(x)|=s_{k}\rho_{k}(x)$
and $du=|\beta_{k}(x)|dx$), we get

$(f,\Phi^{0}_{k}(\cdot,\mu_{m}))_{\mathfrak{H}}=\int_{0}^{\ell_{k}}f_{k}(x_{k}(u))\cdot\overline{e^{is_{k}\mu_{m}u}}\,du.$
(9.24)

Since sequence $\\{s_{k}\mu_{m}\\}_{m\in\mathbb{Z}}$ is incompressible, then
[22, Lemma 2.2] implies that the sequence
$\\{e^{is_{k}\mu_{m}u}\\}_{m\in\mathbb{Z}}$ is Besselian in
$L^{2}[0,\ell_{k}]$ (see also the proof of Lemma 6.4 in [33]). With account of
this observation, it follows from (9.24) and inclusion $f_{k}\circ x_{k}\in
L^{2}[0,\ell_{k}]$, that the sequence
$\\{\Phi^{0}_{k}(\cdot,\mu_{m})\\}_{m\in\mathbb{Z}}$ is Besselian in
$\mathfrak{H}$.

(ii) Combining formula (5.19) with Schwarz inequality and estimate (5.28)
(applicable since $\mu_{m}\in\Pi_{h}$), we arrive at

$\bigl{|}{\left(f,Y_{p}^{0}(\cdot,\mu_{m})\right)_{\mathfrak{H}}}\bigr{|}^{2}=\Bigl{|}\sum_{k=1}^{n}A_{kp}^{0}(\mu_{m})\cdot\left(f,\Phi_{k}^{0}(\cdot,\mu_{m})\right)_{\mathfrak{H}}\Bigr{|}^{2}\leqslant\sum_{k=1}^{n}|A_{kp}^{0}(\mu_{m})|^{2}\cdot\sum_{k=1}^{n}\bigl{|}{\left(f,\Phi_{k}^{0}(\cdot,\mu_{m})\right)_{\mathfrak{H}}}\bigr{|}^{2}\\\
\leqslant
nM_{h}^{2}\sum_{k=1}^{n}\bigl{|}{\left(f,\Phi_{k}^{0}(\cdot,\mu_{m})\right)_{\mathfrak{H}}}\bigr{|}^{2},\qquad
m\in\mathbb{Z},\quad p\in\\{1,\ldots,{n}\\}.$ (9.25)

Estimate (9.25) and part (i) of Lemma now finish the proof. ∎

Now, using integral representation (4.23) we can extend the previous result to
vector functions $\Phi_{k}(\cdot,\lambda)$ and $Y_{p}(\cdot,\lambda)$, which
will imply Besselian property for eigenvectors of the operator $L_{U}(Q)$.

######  Proposition 9.9.

Let matrix functions $B(\cdot)$ and $Q(\cdot)$ satisfy conditions (4.3)–(4.8).
Let $\\{\mu_{m}\\}_{m\in\mathbb{Z}}$ be an incompressible sequence lying in
the strip $\Pi_{h}$. Then the following statements hold:

(i) For any $k\in\\{1,\ldots,{n}\\}$ the sequence
$\\{\Phi_{k}(\cdot,\mu_{m})\\}_{m\in\mathbb{Z}}$ defined in (4.19) is
Besselian in $\mathfrak{H}$.

(ii) For any $p\in\\{1,\ldots,{n}\\}$ the sequence
$\\{Y_{p}(\cdot,\mu_{m})\\}_{m\in\mathbb{Z}}$ defined in (5.18) is Besselian
in $\mathfrak{H}$.

###### Proof.

(i) Let $k\in\\{1,\ldots,{n}\\}$ be fixed and let

$f=\operatorname{col}(f_{1},\ldots,f_{n})\in\mathfrak{H},\qquad\text{i.e.}\qquad
f_{j}\in L^{2}_{|\beta_{j}|}[0,\ell],\quad j\in\\{1,\ldots,{n}\\}.$

By Proposition 4.2, representation (4.23) holds with $p=k$, where vector
kernel $R_{q}^{[k]}$ is defined via (4.31)–(4.32) and satisfies inclusion
(4.22). Moreover, by definition of signature matrix $S$ we have,

$\beta_{q}(x)=s_{q}|\beta_{q}(x)|,\qquad q\in\\{1,\ldots,{n}\\}.$

Taking these observations and formula (9.24) into account, we get by changing
order of integration

$\displaystyle(f,\Phi_{k}(\cdot,\lambda))_{\mathfrak{H}}$
$\displaystyle=(f,\Phi_{k}^{0}(x,\lambda))_{\mathfrak{H}}+\sum_{j,q=1}^{n}\int_{0}^{\ell}f_{j}(x)\left(\int_{0}^{x}\overline{R_{jq}^{[k]}(x,t)e^{i\lambda\rho_{q}(t)}}\beta_{q}(t)\,dt\right)\cdot|\beta_{j}(x)|\,dx$
$\displaystyle=(f,\Phi_{k}^{0}(x,\lambda))_{\mathfrak{H}}+\sum_{j,q=1}^{n}\int_{0}^{\ell}\left(\int_{t}^{\ell}f_{j}(x)\overline{R_{jq}^{[k]}(x,t)}|\beta_{j}(x)|\,dx\right)\overline{e^{i\lambda\rho_{q}(t)}}\beta_{q}(t)\,dt$
$\displaystyle=\int_{0}^{\ell}f_{k}(t)\cdot\overline{e^{i\lambda\rho_{k}(t)}}|\beta_{k}(t)|\,dx+\sum_{j,q=1}^{n}\int_{0}^{\ell}f_{kjq}(t)\cdot\overline{e^{i\lambda\rho_{q}(t)}}|\beta_{q}(t)|\,dt,$
(9.26)

where

$f_{kjq}(t):=\int_{t}^{\ell}\overline{R_{jq}^{[k]}(x,t)}\cdot
s_{q}f_{j}(x)|\beta_{j}(x)|\,dx,\qquad t\in[0,\ell],\quad
j,q\in\\{1,\ldots,{n}\\}.$ (9.27)

Since $\beta_{j}\in L^{\infty}[0,\ell]$ and $f_{j}\in
L^{2}_{|\beta_{j}|}[0,\ell]$, it follows that

$f_{j},\ s_{q}f_{j}|\beta_{j}|\in L^{2}[0,\ell],\qquad
j,q\in\\{1,\ldots,{n}\\}.$ (9.28)

Consider an operator $\mathcal{R}_{kjq}$ generated by the kernel
$R_{jq}^{[k]}$ by formula (2.13) in $L^{2}[0,\ell]$. Lemma 2.3 and inclusion
(4.22) imply that operator $\mathcal{R}_{kjq}$ is bounded in $L^{2}[0,\ell]$
(and even Volterra operator). It is clear that operator $\mathcal{R}_{kjq}$
and its adjoint operator $\mathcal{R}_{kjq}^{*}$ are of the form

$\displaystyle(\mathcal{R}_{kjq}f)(x)$
$\displaystyle=\int_{0}^{x}R_{jq}^{[k]}(x,t)f(t)dt,\qquad f\in L^{2}[0,\ell],$
(9.29) $\displaystyle(\mathcal{R}_{kjq}^{*}g)(t)$
$\displaystyle=\int_{t}^{\ell}\overline{R_{jq}^{[k]}(x,t)}g(x)dx,\qquad g\in
L^{2}[0,\ell].$ (9.30)

Formulas (9.27), (9.30), inclusion (9.28) and boundedness of the operator
$\mathcal{R}_{kjq}^{*}$ imply that ${f_{kjq}\in L^{2}[0,\ell]}$. Recall, that
$k\in\\{1,\ldots,{n}\\}$ is fixed. Let us set

$F_{q}:=F_{kq}\operatorname{col}(\delta_{1q},\ldots,\delta_{nq}),\qquad
F_{kq}:=\delta_{kq}f_{k}+\sum_{j=1}^{n}f_{kjq}\in L^{2}[0,\ell],\qquad
q\in\\{1,\ldots,{n}\\}.$ (9.31)

Since $\beta_{q}\in L^{\infty}[0,\ell]$, $q\in\\{1,\ldots,{n}\\}$, it is clear
that $F_{q}\in\mathfrak{H}$. With account of notation (9.31) and formula
(9.23), we get by setting $\lambda=\mu_{m}$ in (9.26),

$(f,\Phi_{k}(\cdot,\mu_{m}))_{\mathfrak{H}}=\sum_{q=1}^{n}\int_{0}^{\ell}F_{kq}(t)\cdot\overline{e^{i\mu_{m}\rho_{q}(t)}}|\beta_{q}(t)|\,dt=\sum_{q=1}^{n}(F_{q},\Phi_{q}(\cdot,\mu_{m}))_{\mathfrak{H}},\qquad
m\in\mathbb{Z}.$ (9.32)

Besselian property of the sequence
$\\{\Phi_{k}(\cdot,\mu_{m})\\}_{m\in\mathbb{Z}}$ is now implied by Besselian
property of the sequences
$\\{\Phi_{q}^{0}(\cdot,\mu_{m})\\}_{m\in\mathbb{Z}}$,
$q\in\\{1,\ldots,{n}\\}$, established in Lemma 9.8(i), and inclusion
$F_{q}\in\mathfrak{H}$, $q\in\\{1,\ldots,{n}\\}$, which finishes the proof.

(ii) Let $p\in\\{1,\ldots,{n}\\}$ be fixed. It follows from representation
(6.2) for $A_{kp}(\lambda)$, inclusion $g_{kp}\in L^{1}[b_{-},b_{+}]$ and
estimate (5.28) on $A_{kp}^{0}(\lambda)$ that for all $k\in\\{1,\ldots,{n}\\}$
the following estimate holds,

$|A_{kp}(\lambda)|\leqslant|A_{kp}^{0}(\lambda)|+\left|{\int_{b_{-}}^{b_{+}}g_{kp}(u)e^{i\lambda
u}\,du}\right|\leqslant
M_{h}+\|g_{kp}\|_{1}(e^{-b_{-}h}+e^{b_{+}h}):=M_{h,kp},\qquad|\lambda|\leqslant\Pi_{h}.$
(9.33)

As in the proof of Lemma 9.8(ii), combining formula (5.18) with Schwarz
inequality and estimate (9.33) (applicable since $\mu_{m}\in\Pi_{h}$,
$m\in\mathbb{Z}$), we arrive at

$\bigl{|}{\left(f,Y_{p}(\cdot,\mu_{m})\right)_{\mathfrak{H}}}\bigr{|}^{2}=\Bigl{|}\sum_{k=1}^{n}A_{kp}(\mu_{m})\cdot\left(f,\Phi_{k}(\cdot,\mu_{m})\right)_{\mathfrak{H}}\Bigr{|}^{2}\leqslant\sum_{k=1}^{n}M_{h,kp}^{2}\cdot\sum_{k=1}^{n}\bigl{|}{\left(f,\Phi_{k}(\cdot,\mu_{m})\right)_{\mathfrak{H}}}\bigr{|}^{2},\quad
m\in\mathbb{Z}.$ (9.34)

Estimate (9.34) and part (i) of Proposition finish the proof. ∎

######  Proposition 9.10.

Let entries of matrix function $B(\cdot)$ satisfy condition (4.5), i.e.

$\beta_{k},\ 1/\beta_{k}\ \in
L^{\infty}([0,\ell];\mathbb{R}),\qquad\operatorname{sign}(\beta_{k}(\cdot))\equiv\operatorname{const},\qquad
k\in\\{1,\ldots,{n}\\},$ (9.35)

and let boundary conditions (5.2) be strictly regular. Then any normalized
system of root vectors of the operator $L_{0,U}=L_{U}(0)$ forms a Riesz basis
in $\mathfrak{H}$.

###### Proof.

As in the proof of Proposition 9.3, operator $L_{0,U}$ has countable
asymptotically separated sequence of eigenvalues
$\Lambda_{0}:=\\{\lambda_{m}^{0}\\}_{m\in\mathbb{Z}}$. Let
$\mathcal{F}_{0}:=\\{f_{m}^{0}\\}_{m\in\mathbb{Z}}$ be some normalized system
of root vectors of the operator $L_{0,U}$, where $f_{m}^{0}$ is a root vector
corresponding to $\lambda_{m}^{0}$, $\|f_{m}^{0}\|_{\mathfrak{H}}=1$,
$m\in\mathbb{Z}$. As in the proof of Proposition 9.3, we can choose system of
root vectors $\mathcal{F}_{*0}:=\\{f_{*m}^{0}\\}_{m\in\mathbb{Z}}$ of the
operator $L_{0,U}^{*}$ in such a way that $\mathcal{F}_{0}$ and
$\mathcal{F}_{*0}$ are biorthogonal systems.

Since boundary conditions (5.2) are regular then by Lemma 8.1 boundary
conditions (8.2) of $L_{0,U}^{*}$ are also regular. Hence Theorem 7.4(i)
ensures the completeness property of both systems $\mathcal{F}_{0}$ and
$\mathcal{F}_{*0}$.

Let $|m|>m_{0}$, where $m_{0}$ is from Definition 5.13(iii) of strict
regularity. Based on the proof of Proposition 9.3, all of relations
(9.5)–(9.10) hold. Since $\|f_{m}^{0}\|=1$, it is clear from (9.9)–(9.10) that

$|\gamma_{m}^{0}|=\frac{1}{\|\widetilde{f}_{m}^{0}\|_{\mathfrak{H}}},\qquad|\gamma_{*m}^{0}|=\frac{\|\widetilde{f}_{m}^{0}\|_{\mathfrak{H}}}{\bigl{|}{(\widetilde{f}_{m}^{0},\widetilde{f}_{*m}^{0})_{\mathfrak{H}}}\bigr{|}}.$
(9.36)

Combining estimates (9.6) and (9.8) with the Schwartz inequality yields

$C_{3}\leqslant|(\widetilde{f}_{m}^{0},\widetilde{f}_{*m}^{0})|\leqslant\|\widetilde{f}_{m}^{0}\|\cdot\|\widetilde{f}_{*m}^{0}\|\leqslant
M_{h}\|\widetilde{f}_{m}^{0}\|\leqslant M_{h}^{2}.$ (9.37)

Inserting this estimate into (9.36) implies

$|\gamma_{m}^{0}|\leqslant C_{5},\qquad|\gamma_{*m}^{0}|\leqslant C_{5},\qquad
C_{5}:=M_{h}/C_{3}.$ (9.38)

With account of definitions (9.5) and assumption (9.35), Lemma 9.8(ii) implies
that the sequences $\\{\widetilde{f}_{m}^{0}\\}_{|m|>m_{0}}$ and
$\\{\widetilde{f}_{*m}^{0}\\}_{|m|>m_{0}}$ are Besselian in $\mathfrak{H}$.
Since $f_{m}^{0}=\gamma_{m}^{0}\widetilde{f}_{m}^{0}$,
$f_{*m}^{0}=\gamma_{*m}^{0}\widetilde{f}_{*m}^{0}$, $|m|>m_{0}$, inequality
(9.38) implies that the sequences $\\{f_{m}^{0}\\}_{|m|>m_{0}}$ and
$\\{f_{*m}^{0}\\}_{|m|>m_{0}}$ are also Besselian. And hence so are
$\mathcal{F}_{0}=\\{f_{m}^{0}\\}_{m\in\mathbb{Z}}$ and
$\mathcal{F}_{*0}=\\{f_{*m}^{0}\\}_{m\in\mathbb{Z}}$. Theorem 9.7 now finishes
the proof. ∎

######  Theorem 9.11.

Let matrix function $B(\cdot)=\operatorname{diag}(\beta_{1},\ldots,\beta_{n})$
satisfy conditions (4.5)–(4.7), i.e. for some $n_{-}\in\\{0,1,\ldots,n\\}$,

$\displaystyle\beta_{k},1/\beta_{k}\in L^{\infty}[0,\ell],\qquad
s_{k}:=\operatorname{sign}(\beta_{k}(\cdot))\equiv\operatorname{const}\neq
0,\qquad k\in\\{1,\ldots,{n}\\},$ (9.39)
$\displaystyle\beta_{1}(x)\leqslant\ldots\leqslant\beta_{n_{-}}(x)<0<\beta_{n_{-}+1}(x)\leqslant\ldots\leqslant\beta_{n}(x),\qquad
x\in[0,\ell],$ (9.40)

and there exists $\theta>0$ such that for each $k\in\\{1,\ldots,{n-1}\\}$

$\text{either}\quad\beta_{k}\equiv\beta_{k+1}\quad\text{or}\quad\beta_{k}(x)+\theta<\beta_{k+1}(x),\quad
x\in[0,\ell].$ (9.41)

Further, let $Q\in L^{1}([0,\ell];\mathbb{C}^{n\times n})$ and let BVP
(5.1)–(5.2) be strictly regular according to Definition 6.13. Then any
normalized system of root vectors of the operator $L_{U}(Q)$ forms a Riesz
basis in $\mathfrak{H}$.

###### Proof.

As in the proof of Theorem 9.4, applying gauge transform from Lemma 6.11, we
can assumed that $Q$ satisfy “zero block diagonality” condition (4.8). Hence,
as in the proof of Theorem 9.4, operator $L_{U}(Q)$ has countable
asymptotically separated sequence of eigenvalues
$\Lambda:=\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$.

Let $\mathcal{F}:=\\{f_{m}\\}_{m\in\mathbb{Z}}$ be some normalized system of
root vectors of the operator $L_{U}(Q)$, where $f_{m}$ is a root vector
corresponding to $\lambda_{m}$, $\|f_{m}\|_{\mathfrak{H}}=1$,
$m\in\mathbb{Z}$. As in the proof of Theorem 9.4, we can choose system of root
vectors $\mathcal{F}_{*}:=\\{f_{*m}\\}_{m\in\mathbb{Z}}$ of the operator
$L_{U}^{*}(Q)$ in such a way that $\mathcal{F}$ and $\mathcal{F}_{*}$ are
biorthogonal systems.

Since boundary conditions (5.2) are regular then by Lemma 8.1 boundary
conditions (8.2) of $L_{U}^{*}(Q)$ are also regular. Hence assumptions
(4.5)–(4.7), (4.8) and Theorem 7.4(ii) ensures the completeness property of
both systems $\mathcal{F}$ and $\mathcal{F}_{*}$.

Let $|m|>m_{2}$, where $m_{2}$ was chosen in the proof of Theorem 9.4 to
satisfy relations (9.18)–(9.19). Based on the proof of Theorem 9.4, all of
relations (9.14)–(9.20) hold. With account of definitions (9.14), Proposition
9.9(ii) implies that the sequences $\\{\widetilde{f}_{m}\\}_{|m|>m_{2}}$ and
$\\{\widetilde{f}_{*m}\\}_{|m|>m_{2}}$ are Besselian in $\mathfrak{H}$. From
this point the proof is finished the same way as in part (i) by using key
estimates (9.18)–(9.19). ∎

######  Remark 9.12.

Let us provide brief history of results on Riesz basis property for BVP
(5.1)–(5.2) with constant $2\times 2$ matrix $B(\cdot)\equiv
B=\operatorname{diag}(b_{1},b_{2})=B^{*}$.

(i) The Riesz basis property for $2\times 2$ Dirac type operators $L_{U}(Q)$
and separated boundary conditions was established earlier than for the
operators with general regular boundary conditions. Namely, this property was
proved firstly in [68, 69] by I. Trooshin and M. Yamamoto for
$B=\operatorname{diag}(-1,1)$ and $Q\in C^{1}([0,1];\mathbb{C}^{2\times 2})$.
Later, P. Djakov and B. Mityagin in [8], and A.G. Baskakov, A.V. Derbushev and
A.O. Shcherbakov in [6] relaxed smoothness assumption on potential to $Q\in
L^{2}([0,1];\mathbb{C}^{2\times 2})$. Independently, in [20] S. Hassi and L.
Oridoroga established this property for Dirac type operator with
$B=\operatorname{diag}(b_{1},b_{2})=B^{*}$ and $Q\in
C^{1}([0,1];\mathbb{C}^{2\times 2})$.

(ii) The Bari-Markus property of the Riesz projectors of unperturbed and
perturbed $2\times 2$ Dirac operators with separated, periodic and
antiperiodic boundary conditions was established in [50, 8] and reproved by
another method in [6]. In [11] similar results have been obtained for general
regular boundary conditions.

(iii) The most complete result on the Riesz basis property for $2\times 2$
Dirac and Dirac-type systems with $Q\in L^{1}$ and strictly regular boundary
conditions was obtained independently by different methods and at the same
time by A.M. Savchuk and A.A. Shkalikov [60] and by the authors [31, 33]. The
case of regular boundary conditions and $Q\in L^{1}$ is treated in [60] for
the first time. Other proofs were obtained later in [57], [58] (see also their
recent survey [59] and references therein).

(iv) Periodic and antiperiodic (necessarily non-strictly regular) BVP for
$2\times 2$ Dirac equation have attracted certain attention during the last
decade. In [10, Theorem 13], [9, Theorem 19] and [13], P. Djakov and B.
Mityagin established a _criterion_ for the system of root vectors to contain a
Riesz basis for periodic (resp., antiperiodic) $2\times 2$ Dirac operator in
terms of the Fourier coefficients of $Q$ as well as in terms of periodic
(resp., antiperiodic) and Dirichlet spectra. A. Makin [35, 36] established
Riesz basis property for periodic $2\times 2$ Dirac operator under certain
explicit algebraic assumptions on a potential matrix. See also recent survey
[14] by P. Djakov and B. Mityagin, survey [38] by A.S. Makin, and the
references therein.

######  Remark 9.13.

In this remark we go over known results on Riesz basis property for BVP
(5.1)–(5.2) with constant $n\times n$ matrix $B(\cdot)\equiv B=B^{*}$, when
$n>2$.

(i) In [51] the results of [8] regarding the Bari-Markus property in
$L^{2}([0,1];\mathbb{C}^{2})$ were extended to the case of the Dirichlet BVP
for $2m\times 2m$ Dirac equation with $Q\in L^{2}([0,1];\mathbb{C}^{2m\times
2m})$.

(ii) To the best of our knowledge the first result on the Riesz basis property
for BVP (5.1)–(5.2) generated by general $n\times n$ system (5.1) with
$B(x)=B=\operatorname{diag}(b_{1},\ldots,b_{n})\in\mathbb{C}^{n\times n}\neq
B^{*}$ and bounded $Q\in L^{\infty}([0,1];\mathbb{C}^{n\times n})$ was
obtained by the authors in [32]. Treated boundary conditions form rather broad
class that covers, in particular, periodic, antiperiodic, and regular
separated (not necessarily self-adjoint) boundary conditions.

(iii) In [24, 25], Bessel and Riesz basis properties on abstract level were
established, i.e. the operator $L_{U}(Q)$ was studied without explicit
boundary conditions.

###  9.3 Riesz basis property with parentheses

First, let us recall a corresponding definition.

######  Definition 9.14.

(i) A sequence of subspaces $\\{\mathcal{H}_{m}\\}_{m=1}^{\infty}$ is called a
Riesz basis of subspaces in a separable Hilbert space $\mathcal{H}$ if there
exists a complete sequence of mutually orthogonal subspaces
$\\{\mathcal{H}^{\prime}_{m}\\}_{m=1}^{\infty}$ and a bounded operator $T$ in
$\mathcal{H}$ with a bounded inverse such that
$\mathcal{H}_{m}=T\mathcal{H}^{\prime}_{m}$, $m\in\mathbb{N}$.

(ii) A sequence $\\{\varphi_{m}\\}_{m=1}^{\infty}$ of vectors in $\mathcal{H}$
is called a Riesz basis with parentheses if each its finite subsequence is
linearly independent, and there exists an increasing sequence
$\\{m_{k}\\}_{k=0}^{\infty}\subset\mathbb{N}$ such that $m_{0}=1$ and the
sequence
$\mathcal{H}_{k}:=\operatorname{span}\\{\varphi_{j}\\}_{j=m_{k-1}}^{m_{k}-1}$,
forms a Riesz basis of subspaces in $\mathcal{H}$. Subspaces $\mathcal{H}_{k}$
are called blocks.

In [61], A.A. Shkalikov established Riesz basis property with parentheses for
BVP for ODE of $n$-th order with coefficients $q_{2},\ldots,q_{n}\in L^{1}$
and regular boundary conditions $V(y)=0$. Denote corresponding operator as
$\mathcal{L}(V,q)$. The idea was to present the operator $\mathcal{L}(V,q)$ as
a bounded perturbation of the operator similar to
$\mathcal{L}(\widetilde{V},\widetilde{q})$, where boundary conditions
$\widetilde{V}(y)=0$ are already strictly regular and the operator is known to
have a Riesz basis property (without parentheses). Then abstract result of
Katsnel’son-Markus-Matsaev (see [21, Theorem 3.1] and also [1], [46], [47,
Theorem 6.12]) implies desired Riesz basis property with parentheses for the
original operator $\mathcal{L}(V,q)$.

This idea later was used in [60] to establish Riesz basis property with
parentheses for $2\times 2$ Dirac operator with regular boundary conditions,
i.e. operator $L_{U}(Q)$ with $B(\cdot)\equiv\operatorname{diag}(-1,1)$. Later
we also used this idea in [33], to establish this result for $2\times 2$ Dirac
type operator, i.e. when $B(\cdot)\equiv\operatorname{diag}(b_{1},b_{2})$,
$b_{1}<0<b_{2}$.

In this subsection we use the same idea, and, following [33, Section 7], show
that the system of root vectors of BVP (5.1)–(5.2) with regular boundary
conditions forms a Riesz basis with parentheses under assumptions (4.3)–(4.7).

As in [33, Section 7] we start with certain properties of zeros of exponential
polynomials. The $n\times n$ case is more difficult and requires more advanced
properties. We start with the following simple lemma.

######  Lemma 9.15.

Let $f\not\equiv 0$ and $g\not\equiv 0$ be entire functions.

(i) Let $g$ have simple zeros (possibly empty set), i.e.
$|g(z)|+|g^{\prime}(z)|>0$, $z\in\mathbb{C}$. Then $f+wg$ has simple zeros for
all but countable number of values of $w\in\mathbb{C}$.

(ii) Let $f$, $f^{\prime}$, $g$, $g^{\prime}$ be sine-type functions with
separated (possible empty) sets of zeros. Then for all $w$ outside of a
certain strip $\Pi_{h}$, $h=h_{f,g}$, function $f+wg$ is a sine-type function
with separated (possible empty) set of zeros.

###### Proof.

(i) If $f$ and $g$ are proportional, i.e. $f\equiv\alpha g$ for some
$\alpha\in\mathbb{C}$, it is clear that $f+wg$ has simple zeros for all
$w\neq-\alpha$. Let $f$ and $g$ be non-proportional. Since $f$ and $g$ are
non-zero entire functions, it follows that each of them has at most countable
set of zeros. Hence, ratio $f/g$ is a non-constant analytic function with at
most countable set of poles. This implies that $(f/g)^{\prime}$ is non-zero
analytic function with at most countable set of poles, which implies that
$fg^{\prime}-f^{\prime}g$ is non-zero entire function and has at most
countable set of zeros $\\{\mu_{m}\\}_{m=1}^{N}$, $0\leqslant
N\leqslant\infty$ (it can be empty as well).

Let $w\in\mathbb{C}$ be such that some $z=z_{w}$ is a multiple zero of $f+wg$.
Then

$f(z)+wg(z)=f^{\prime}(z)+wg^{\prime}(z)=0.$

This implies that $f(z)g^{\prime}(z)-f^{\prime}(z)g(z)=0$, i.e. $z$ is a zero
of the entire function $fg^{\prime}-f^{\prime}g$. Hence $z=z_{w}=\mu_{m}$ for
some $m=m_{w}\in\\{1,\ldots,{N}\\}$. Since function $g$ has simple zeros it
follows that $|g(z)|+|g^{\prime}(z)|>0$. Hence

$\text{either}\quad w=-f(z)/g(z)=-f(\mu_{m})/g(\mu_{m})\quad\text{or}\quad
w=-f^{\prime}(z)/g^{\prime}(z)=-f^{\prime}(\mu_{m})/g^{\prime}(\mu_{m}).$

This implies that the set of values $w$ for which function $f+wg$ has multiple
zeros is at most countable and finishes the proof.

(ii) Since $f$ is a sine-type function it follows from estimate on $f$ from
below outside of zeros and estimate on $f^{\prime}$ near zeros that
$|f(z)|+|f^{\prime}(z)|>\varepsilon$, $z\in\mathbb{Z}$, for some
$\varepsilon>0$ and the same is valid for $g$. It also follows that zeros of
$f$ and $f^{\prime}$ combined are separated. I.e. zeros of $ff^{\prime}$ and
$gg^{\prime}$ are separated. These properties are sufficient to show the
desired property of function $f+wg$ for sufficiently large
$|\operatorname{Im}w|$. As an example, let $f(z)=e^{ibz}g(z)$ for some $b>0$.
Then $f(z)+wg(z)=(e^{ibz}+w)g(z)$. So we need to find $w$ for each arithmetic
progression $\left\\{\frac{-i\ln w+2\pi m}{b}\right\\}_{m\in\mathbb{Z}}$ is
separated from zeros of $g(\cdot)$. It is clear that this is true for $w$ with
sufficiently large $|\operatorname{Im}w|$, which finishes the proof. ∎

Let $S$ be the set of entire functions with simple zeros, i.e. $f\in S$ iff
$|f(z)|+|f^{\prime}(z)|>0$, $z\in\mathbb{C}$. Denote
$\mathcal{S}:=S\cup\\{0\\}$. The following property is a trivial consequence
of Lemma 9.15.

######  Corollary 9.16.

Let $f,g\in\mathcal{S}$. Then $f+wg\in\mathcal{S}$ for all but countable
number of values of $w\in\mathbb{C}$.

###### Proof.

If either $f$ or $g$ is zero the statement is trivial and $f+wg\in\mathcal{S}$
for all $w\in\mathbb{C}$. Otherwise the statement follows from Lemma 9.15. ∎

######  Lemma 9.17.

Let $(f_{P})_{P\in\mathcal{P}_{n}}$, be a sequence of $2^{n}$ functions from
the class $\mathcal{S}$ indexed with diagonal idempotent matrices of size $n$
(see (5.39) for the definition of the class $\mathcal{P}_{n}$). Then there
exists a diagonal matrix $W=\operatorname{diag}(w_{1},\ldots,w_{n})$ with non-
zero entries such that

$\sum_{P\in\mathcal{P}_{n}}\det(PW)\cdot f_{P}\in\mathcal{S}.$ (9.42)

For example, for $n=2$, this function looks like
$f_{\operatorname{diag}(0,0)}+w_{1}f_{\operatorname{diag}(1,0)}+w_{2}f_{\operatorname{diag}(0,1)}+w_{1}w_{2}f_{\operatorname{diag}(1,1)}$.

###### Proof.

Let us prove this via induction by $n$. For $n=1$ the statement trivially
follows from Corollary 9.16. Assume the statement is valid for
$n=m-1\in\mathbb{N}$ and consider $n=m$, i.e. we have a sequence
$(f_{P})_{P\in\mathcal{P}_{m}}$ of $2^{m}$ functions from the class
$\mathcal{S}$. Set $\widetilde{W}:=\operatorname{diag}(w_{1},\ldots,w_{m-1})$
and for each
$\widetilde{P}=\operatorname{diag}(p_{1},\ldots,p_{m-1})\in\mathcal{P}_{m-1}$
denote,

$\widetilde{P}_{0}:=\operatorname{diag}(p_{1},\ldots,p_{m-1},0)\in\mathcal{P}_{m},\qquad\widetilde{P}_{1}:=\operatorname{diag}(p_{1},\ldots,p_{m-1},1)\in\mathcal{P}_{m}.$

We can transform the sum in (9.42) the following way,

$\sum_{P\in\mathcal{P}_{m}}\det(PW)\cdot
f_{P}=\sum_{\widetilde{P}\in\mathcal{P}_{m-1}}\det(\widetilde{P}\cdot\widetilde{W})\cdot\left(f_{\widetilde{P}_{0}}+w_{m}f_{\widetilde{P}_{1}}\right)$
(9.43)

By Corollary 9.16, for each $P\in\mathcal{P}_{m-1}$ the function
$f_{\widetilde{P}_{0}}+w_{m}f_{\widetilde{P}_{1}}\in\mathcal{S}$ for all but
countable number of values of $w_{m}\in\mathbb{C}$. Hence we can choose a
single $w_{m}\neq 0$ that “serves” all $P$, i.e.
$f_{\widetilde{P}_{0}}+w_{m}f_{\widetilde{P}_{1}}\in\mathcal{S}$ for all
$P\in\mathcal{P}_{m-1}$. Now induction hypothesis applied to the r.h.s. of
(9.43) implies existence of the desired non-zero $w_{1},\ldots,w_{m-1}$, which
finishes the proof. ∎

######  Proposition 9.18.

Let $b_{1},\ldots,b_{n}$ given by (4.17) are non-zero and let boundary
conditions (5.2) be regular. Then there exists
$W_{\ell}=\operatorname{diag}(w_{1},\ldots,w_{n})\in\mathbb{C}^{n\times n}$
with non-zero entries such that the sequence of zeros of entire function

$\Delta_{0,W_{\ell}}(\cdot):=\det(C+DW_{\ell}\Phi_{0}(\ell,\cdot))$

is separated, where $\Phi_{0}(\cdot,\lambda)$ is given by (4.20).

###### Proof.

Throughout the proof we will heavily use notations and formulas from
Subsection 5.2: $b_{\pm}$, $P\in\mathcal{P}_{n}$, $b_{P}$, $P_{\pm}$,
$J_{P}(C,D)$, etc. First note that

$J_{P}(C,DW_{\ell})=\det(PW_{\ell})\cdot J_{P}(C,D),\qquad
P\in\mathcal{P}_{n}.$ (9.44)

The proof is very similar to the proof of (6.69). It follows from (5.43) and
(9.44) that

$\Delta_{0,W_{\ell}}(\lambda)=\sum_{P\in\mathcal{P}_{n}}J_{P}(C,DW_{\ell})e^{i\lambda
b_{P}}=\sum_{P\in\mathcal{P}_{n}}\det(PW_{\ell})f_{P}(\lambda),\quad
f_{P}(\lambda):=J_{P}(C,D)e^{i\lambda b_{P}},\quad\lambda\in\mathbb{C}.$
(9.45)

(i) Let us consider the case when $b_{k}=m_{k}b_{0}$,
$k\in\\{1,\ldots,{n}\\}$, for some $b_{0}>0$ and
$m_{1},\ldots,m_{n}\in\mathbb{Z}$. Similar to notations $b_{\pm}$ and $b_{P}$
we denote

$m_{\pm}:=b_{\pm}/b_{0},\quad\text{e.g.}\quad m_{-}:=m_{1}+\ldots+m_{n_{-}},$

and

$m_{P}:=b_{P}/b_{0}=\sum_{k=1}^{n}p_{k}m_{k}\quad\text{for}\quad
P=\operatorname{diag}(p_{1},\ldots,p_{n})\in\mathcal{P}_{n}.$

Let $W_{\ell}$ be any invertible diagonal matrix. It follows from (5.43) and
(9.44) that

$\Delta_{0,W_{\ell}}(\lambda)=\sum_{P\in\mathcal{P}_{n}}\det(PW_{\ell})J_{P}(C,D)(e^{i\lambda
b_{0}})^{m_{P}}=(e^{i\lambda b_{0}})^{m_{-}}F_{W_{\ell}}(e^{i\lambda b_{0}}),$
(9.46)

where $F_{W_{\ell}}(\cdot)$ is some polynomial (since $m_{P}\geqslant m_{-}$,
$P\in\mathcal{P}_{n}$, are integers). Since boundary conditions are regular it
follows that $J_{P_{\pm}}(C,D)\neq 0$. Hence
$F_{W_{\ell}}(0)=J_{P_{-}}(C,D)\neq 0$ and $\deg F_{W_{\ell}}=m_{+}-m_{-}=:N$.
Therefore $F_{W_{\ell}}$ has exactly $N$ non-zero roots $z_{1},\ldots,z_{N}$
(that depend on $W_{\ell}$, $z_{k}=z_{k,W_{\ell}}$, $k\in\\{1,\ldots,{N}\\}$).
It is clear that the sequence of zeros of $\Delta_{0,W_{\ell}}(\cdot)$ is of
the form

$\left\\{\frac{-i\ln z_{k}+2\pi
m}{b_{0}}\right\\}_{m\in\mathbb{Z},k\in\\{1,\ldots,{N}\\}}.$

Hence $\Delta_{0,W_{\ell}}(\cdot)$ has separated zeros if and only if roots
are distinct $z_{1},\ldots,z_{N}$. This also means that if
$\Delta_{0,W_{\ell}}(\cdot)$ has simple zeros then they are separated.

Going back to (9.45) we see that $f_{P}\in\mathcal{S}$, $P\in\mathcal{P}_{n}$.
Indeed, if $J_{P}(C,D)=0$, then $f_{P}\equiv 0\in\mathcal{S}$. Otherwise
$f_{P}$ has no zeros (and thus has simple zeros by definition). Thus, Lemma
9.17 implies existence of invertible
$W=W_{\ell}=\operatorname{diag}(w_{1},\ldots,w_{n})$, for which (9.42) holds.
In view of (9.45) this implies that $\Delta_{0,W_{\ell}}(\cdot)$ has simple
zeros (since it is non-zero function, due to observations above about
polynomial $F_{W_{\ell}}(\cdot)$). Another observation above, implies that
$\Delta_{0,W_{\ell}}(\cdot)$ has separated zeros and finishes the proof.

(ii) The proof is similar by using Lemma 9.15. Namely, this lemma implies
variation of Lemma (9.17) for sine-type functions with separated zeros. Using
this result we can finish the proof by following steps above. ∎

To state the next result we recall that $m_{a}(\mu_{0})$ and $m_{g}(\mu_{0})$
denote the algebraic and geometric multiplicities of $\mu_{0}$, respectively.
Moreover, if $\mu_{0}$ is an isolated eigenvalue, then $m_{a}(\mu_{0})$ equals
to the dimension of the Riesz projection. We need the following known abstract
result (see e.g. [60, 33]) that follows from Katsnel’son-Markus-Matsaev
theorem with $p=1$ (see [21, Theorem 3.1] and also [1], [46], [47, Theorem
6.12]).

######  Proposition 9.19 (Proposition 7.3 in [33]).

Let $L$ be an operator with compact resolvent in a separable Hilbert space
$\mathcal{H}$ and let $\\{\mu_{m}\\}_{m\in\mathbb{Z}}$ be the sequence of its
distinct eigenvalues. Assume that $m_{a}(\mu_{m})<\infty$ for $m\in\mathbb{N}$
and that $A$ has finitely many associative vectors, i.e. there exists
$n_{0}\in\mathbb{N}$ such that $m_{a}(\mu_{m})=m_{g}(\mu_{m})$ for
$|m|>n_{0}.$ Further, assume that

$|\mu_{m}|\geqslant C|m|,\quad|\operatorname{Im}\mu_{m}|\leqslant\tau,\qquad
m\in\mathbb{Z},$ (9.47)

for some $C,\tau>0$. Finally, let the system of root vectors of the operator
$L$ forms a Riesz basis in $\mathfrak{H}$. Then for any bounded operator $T$
in $\mathfrak{H}$ the system of root vectors of the perturbed operator $A=L+T$
forms a Riesz basis with parentheses in $\mathfrak{H}$.

Finally, we are ready to prove the main result of this subsection concerning
Riesz basis property with parentheses.

######  Theorem 9.20.

Let matrix function $B(\cdot)$ given by (4.3) satisfy conditions (4.5)–(4.7)
and let $Q\in L^{1}([0,\ell];\mathbb{C}^{n\times n})$. Let boundary conditions
(5.2) be regular Then any normalized system of root vectors of the operator
$L_{U}(Q)$ forms a Riesz basis with parentheses in $\mathfrak{H}$. Moreover,
block sizes are uniformly bounded by $2^{n}-1$ and each block consists of root
vectors corresponding to the eigenvalues with pairwise close real parts.

###### Proof.

By Lemma 6.11(iv), the regularity of boundary conditions is preserved under
the gauge transformation used in Lemma 6.11. Therefore one can assume that $Q$
is off-diagonal. Now let us consider a perturbation of the operator $L(Q)$ by
a constant diagonal potential matrix
$Q_{0}=\operatorname{diag}(q_{1},\ldots,q_{n})$,
$q_{1},\ldots,q_{n}\in\mathbb{C}$. Applying Lemma 6.11 again, but this time to
the operator $L_{U}(Q+Q_{0})$, and noting that
$(Q+Q_{0})_{\operatorname{diag}}=Q_{0}$, we see that the operator
$L_{U}(Q+Q_{0})$ is similar to the operator $L_{\widetilde{U}}(\widetilde{Q})$
with off-diagonal $\widetilde{Q}$ and with boundary conditions

$\widetilde{U}(y)=Cy(0)+\widetilde{D}y(\ell)=0,\quad\text{where}\quad\widetilde{D}=D\cdot
W_{\ell},$ (9.48)

and

$W_{\ell}=\operatorname{diag}(w_{1},\ldots,w_{n}),\qquad
w_{k}=e^{-ib_{k}q_{k}},\quad k\in\\{1,\ldots,{n}\\}.$ (9.49)

By Proposition 9.18, we can choose
$w_{1},\ldots,w_{n}\in\mathbb{C}\setminus\\{0\\}$ such that the boundary
conditions (9.48) are strictly regular. In turn, setting,
$q_{k}=ib_{k}^{-1}\ln w_{k}$, $k\in\\{1,\ldots,{n}\\}$, provides us with the
desired perturbation $Q_{0}$. Namely, solution $W(\cdot)$ of the equation
(6.60) with $Q_{0}$ in place of $Q_{\operatorname{diag}}$, satisfies condition
$W(\ell)=W_{\ell}$, where $W_{\ell}$ is given by (9.49).

Therefore, by Theorem 6.6 and Definition 5.13(iii), the eigenvalues of
$L(Q+Q_{0})$ are of finite multiplicity, asymptotically simple and separated.
In particular, the operator $L(Q+Q_{0})$ has only finitely many associated
vectors. Moreover, according to Theorem 9.11 the root vectors system of the
operator $L(Q+Q_{0})$ forms a Riesz basis in $\mathfrak{H}$.

To verify that the operator $L(Q+Q_{0})$ satisfies conditions of Proposition
9.19 it suffices to note that inequalities (9.47) are implied by Proposition
6.5(iv). Thus, the operator $L(Q+Q_{0})$ meets the conditions of Proposition
9.19, and hence the root vectors system of the original operator
$L(Q)=L(Q+Q_{0})-Q_{0}$ forms a Riesz basis with parentheses, since operator
$T:y\to Q_{0}y$ is clearly bounded in $\mathfrak{H}$.

Further, note that since $\Delta_{0}(\cdot)$ is exponential polynomial with at
most $2^{n}$ terms, we can find a number $w>0$ such that any rectangle
$[x,x+w]\times[-h,h]$, $x\in\mathbb{R}$, has no more than $2^{n}-1$ zeros of
$\Delta_{0}(\cdot)$ (see [7, pp. 95–97] where this property of exponential
polynomial was proved for $n=2$). The same is valid for $\Delta_{Q}(\cdot)$
due to asymptotic formula $\lambda_{m}=\lambda_{m}^{0}+o(1)$ as $m\to\infty$.
Recall that this formula is valid in the case of any regular boundary
conditions. Hence we can separate sequence
$\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ (ordered in increasing order of real
parts) into blocks

$\\{\lambda_{m}\\}_{m=m_{k}}^{m_{k+1}-1},\quad\text{where}\quad
m_{k}<m_{k+1}<m_{k}+2^{n}\quad\text{and}\quad\operatorname{Re}\lambda_{m_{k}-1}+\varepsilon\leqslant\operatorname{Re}\lambda_{m_{k}},\qquad
k\in\mathbb{Z},$

with $\varepsilon:=\frac{w}{\max\\{2^{n}-2,1\\}}$. From the proof of
Katsnel’son-Markus-Matsaev theorem with $p=1$ (see [21, Theorem 3.1] and also
[1], [46], [47, Theorem 6.12]), it follows that each block consists of root
vectors corresponding to the eigenvalues with pairwise close real parts, which
finishes the proof. ∎

Under certain additional assumptions on boundary conditions we can clarify the
sizes of blocks in Theorem 9.20 even more. We restrict ourselves to a special
type of boundary conditions including periodic and antiperiodic conditions.

######  Corollary 9.21.

Assume the conditions of Theorem 9.20 and let $b_{1}<\ldots<b_{n}$. Assume
also that boundary conditions (5.2) are of the form $Cy(0)-y(\ell)=0$, where
$C=\operatorname{diag}(c_{1},\ldots,c_{n})$ is invertible. Then the blocks of
Riesz basis decomposition with parentheses stated in Theorem 9.20 have sizes
at most $n$ and correspond to eigenvalues of the operator $L_{U}(Q)$ with
mutually close real parts. In particular, this is true for periodic and
antiperiodic boundary conditions.

###### Proof.

Applying the gauge transform from Lemma 6.11 we can assume that $Q$ is off-
diagonal matrix, $Q_{11}\equiv\ldots\equiv Q_{nn}\equiv 0$. Since
$b_{1}<\ldots<b_{n}$ it is clear that the new matrix $D$ is of the same
diagonal form as $C$. Hence multiplying boundary conditions by $-D^{-1}$ from
the left we can assume that again $D=-I_{n}$. Let
$\Lambda:=\\{\lambda_{n}\\}_{n\in\mathbb{Z}}$ be the sequence of eigenvalues
of $A:=L_{U}(Q)$ counting multiplicity, ordered in such a way that
$\operatorname{Re}\lambda_{n}\leqslant\operatorname{Re}\lambda_{n+1}$,
$n\in\mathbb{Z}$. Since $Q$ is off-diagonal, one derives by combining Theorem
6.6 with Lemma 5.18(i) that there exists $\varepsilon>0$ and a sequence of
integers $\\{m_{k}\\}_{k\in\mathbb{Z}}$ such that

$m_{k}<m_{k+1}\leqslant
m_{k}+n,\quad\operatorname{Re}\lambda_{m_{k}}-\operatorname{Re}\lambda_{m_{k}-1}\geqslant\varepsilon,\quad
k\in\mathbb{Z}.$ (9.50)

Let $N(t)$ be the number of eigenvalues (counting multiplicity) of the
operator $A$ belonging to the vertical strip
$\\{\lambda:|\operatorname{Re}\lambda|\leqslant t\\}$. Setting
$t_{k}:=2^{-1}\operatorname{Re}(\lambda_{m_{k}-1}+\lambda_{m_{k}})$ we obtain
from (9.50) that there exists $C>0$ such that

$\sup_{t>0,t\neq t_{k}}\frac{N(t)-N(t_{k})}{t-t_{k}}<C,\quad k\in\mathbb{Z}.$
(9.51)

Let $f_{n}$ be the root vector of $A$ corresponding to $\lambda_{n}$,
$n\in\mathbb{Z}$. Then it follows from the proof of Theorem 3.1 in [21] (see
also Theorem 1.3 in [21]) that under the condition (9.51) the subspaces
$\mathfrak{H}_{k}:=\operatorname{span}\\{f_{j}\\}_{j=m_{k}}^{m_{k+1}-1}$
constitute a Riesz basis of subspaces in $\mathfrak{H}$. To complete the proof
it remains to note that due to (9.50) the sizes $\dim\mathfrak{H}_{k}$ of
blocks $\mathfrak{H}_{k}$ do not exceed $n$. Indeed,
$\dim\mathfrak{H}_{k}=m_{k+1}-m_{k}\leqslant n$, $k\in\mathbb{Z}$. ∎

######  Remark 9.22.

(i) Corollary 9.21 is valid under the following slightly more general
assumption,

$\forall j,k\in\\{1,\ldots,{n}\\}:\quad j\neq k\quad\text{and}\quad
b_{j}=b_{k}\quad\Rightarrow\quad Q_{jk}\equiv 0.$ (9.52)

This is evident from the fact that for such $Q$ its block diagonal
$Q_{\operatorname{diag}}$ with respect to the decomposition
$\mathbb{C}^{n}=\mathbb{C}^{n_{1}}\oplus\ldots\oplus\mathbb{C}^{n_{r}}$ is
actually a regular diagonal matrix,
$Q_{\operatorname{diag}}=\operatorname{diag}(Q_{11},\ldots,Q_{nn})$. Hence,
the solution $W(\cdot)$ of the Cauchy problem (6.60) is a diagonal matrix
function. This in turn imply, that the new matrix $D=-W(\ell)$ after applying
gauge transform is of the same diagonal form as $C$, which allows to finish
the proof the same way.

(ii) Result similar to Corollary 9.21 is also valid for boundary conditions of
the form (5.91) if numbers $b_{1},\ldots,b_{n}$ satisfy condition (5.90). We
just need to apply Lemma 5.20(i) instead of Lemma 5.18(i). In fact, the blocks
of Riesz basis decomposition with parentheses stated in Theorem 9.20 have
sizes at most $n/2$ in this case.

(iii) More generally, if the sequence $\Lambda_{0}$ is a union of $N$
arithmetic progressions that lie on the lines parallel to the real axis, then
the blocks of Riesz basis decomposition with parentheses stated in Theorem
9.20 have sizes at most $N$.

##  10 Application to the Timoshenko beam model

###  10.1 Problem statement and reduction to Dirac type operator

In this section we obtain some important geometric properties of the system of
root vectors of the dynamic generator of the Timoshenko beam model. Consider
the following linear system of two coupled hyperbolic equations for
$t\geqslant 0$

$\displaystyle I_{\rho}(x)\Phi_{tt}$
$\displaystyle=K(x)(W_{x}-\Phi)+(EI(x)\Phi_{x})_{x}-p_{1}(x)\Phi_{t},\quad
x\in[0,\ell],$ (10.1) $\displaystyle\rho(x)W_{tt}$
$\displaystyle=(K(x)(W_{x}-\Phi))_{x}-p_{2}(x)W_{t},\qquad\qquad\qquad
x\in[0,\ell].$ (10.2)

The vibration of the Timoshenko beam of the length $\ell$ clamped at the left
end is governed by the system (10.1)–(10.2) subject to the following boundary
conditions for $t\geqslant 0$ [67]:

$\displaystyle W(0,t)=\Phi(0,t)$ $\displaystyle=0,$ (10.3)
$\displaystyle\bigl{(}EI(x)\Phi_{x}(x,t)+\alpha_{1}\Phi_{t}(x,t)+\gamma_{1}W_{t}(x,t)\bigr{)}\bigr{|}_{x=\ell}$
$\displaystyle=0,$ (10.4)
$\displaystyle\bigl{(}K(x)(W_{x}(x,t)-\Phi(x,t))+\alpha_{2}W_{t}(x,t)+\gamma_{2}\Phi_{t}(x,t)\bigr{)}\bigr{|}_{x=\ell}$
$\displaystyle=0.$ (10.5)

Here $W(x,t)$ is the lateral displacement at a point $x$ and time $t$,
$\Phi(x,t)$ is the bending angle at a point $x$ and time $t$, $\rho(x)$ is a
mass density, $K(x)$ is the shear stiffness of a uniform cross-section,
$I_{\rho}(x)$ is the rotary inertia, $EI(x)$ is the flexural rigidity at a
point $x$, $p_{1}(x)$ and $p_{2}(x)$ are locally distributed feedback
functions, $\alpha_{k},\gamma_{k}\in\mathbb{C}$, $k\in\\{1,2\\}$. Boundary
conditions at the right end contain as partial cases most of the known
boundary conditions if $\alpha_{1},\alpha_{2}$ are allowed to be infinity.

Regarding the coefficients, we assume that they all are measurable functions
satisfying the following general conditions for some $M>1$,

$0<M^{-1}\leqslant\rho(x),I_{\rho}(x),K(x),EI(x)\leqslant M,\quad
x\in[0,\ell],\qquad p_{1},p_{2}\in L^{1}([0,\ell];\mathbb{C}).$ (10.6)

Under these assumptions, the energy space associated with the problem
(10.1)–(10.5) is

$\mathcal{H}:=\widetilde{H}^{1}_{0}[0,\ell]\times
L^{2}[0,\ell]\times\widetilde{H}^{1}_{0}[0,\ell]\times L^{2}[0,\ell],$ (10.7)

where $\widetilde{H}^{1}_{0}[0,\ell]:=\\{f\in W^{1,2}[0,\ell]:f(0)=0\\}$. The
norm in the energy space is defined as follows:

$\|y\|_{\mathcal{H}}^{2}=\int_{0}^{\ell}\bigl{(}EI|y_{1}^{\prime}|^{2}+I_{\rho}|y_{2}|^{2}+K|y_{3}^{\prime}-y_{1}|^{2}+\rho|y_{4}|^{2}\bigr{)}dx,\quad
y=\operatorname{col}(y_{1},y_{2},y_{3},y_{4}).$ (10.8)

The problem (10.1)–(10.5) can be rewritten as

$y_{t}=i\mathcal{L}y,\quad y(x,t)|_{t=0}=y_{0}(x),$ (10.9)

where $y$ and $\mathcal{L}$ are given by

$y=\begin{pmatrix}\Phi(x,t)\\\ \Phi_{t}(x,t)\\\ W(x,t)\\\
W_{t}(x,t)\end{pmatrix},\ \ \mathcal{L}\begin{pmatrix}y_{1}\\\ y_{2}\\\
y_{3}\\\ y_{4}\end{pmatrix}=\frac{1}{i}\begin{pmatrix}y_{2}\\\
\frac{1}{I_{\rho}(x)}\Bigl{(}K(x)(y_{3}^{\prime}-y_{1})+\bigl{(}EI(x)y_{1}^{\prime}\bigr{)}^{\prime}-p_{1}(x)y_{2}\Bigr{)}\\\
y_{4}\\\
\frac{1}{\rho(x)}\Bigl{(}\bigl{(}K(x)(y_{3}^{\prime}-y_{1})\bigl{)}^{\prime}-p_{2}(x)y_{4}\Bigr{)}\end{pmatrix}$
(10.10)

on the domain

$\displaystyle\operatorname{dom}(\mathcal{L})=\left\\{y=\operatorname{col}(y_{1},y_{2},y_{3},y_{4}):\
\ y_{1},y_{2},y_{3},y_{4}\in\widetilde{H}^{1}_{0}[0,\ell]\right.,$
$\displaystyle EI\cdot
y_{1}^{\prime}\in\operatorname{AC}[0,\ell],\qquad(EI\cdot
y_{1}^{\prime})^{\prime}-p_{1}y_{2}\in L^{2}[0,\ell],$ $\displaystyle
K\cdot(y_{3}^{\prime}-y_{1})\in\operatorname{AC}[0,\ell],\qquad(K\cdot(y_{3}^{\prime}-y_{1}))^{\prime}-p_{2}y_{4}\in
L^{2}[0,\ell],$ $\displaystyle\bigl{(}EI\cdot
y_{1}^{\prime}\bigr{)}(\ell)+\alpha_{1}y_{2}(\ell)+\gamma_{1}y_{4}(\ell)=0,$
$\displaystyle\Bigl{.}\bigl{(}K\cdot(y_{3}^{\prime}-y_{1})\bigr{)}(\ell)+\alpha_{2}y_{4}(\ell)+\gamma_{2}y_{2}(\ell)=0\Bigr{\\}}.$
(10.11)

Timoshenko beam model is investigated in numerous papers (see [67, 23, 48, 65,
72, 71, 70, 66, 32, 33, 2] and the references therein). A number of stability,
controllability, and optimization problems were studied. Note also that the
general model (10.1)–(10.5) of spatially non-homogenous Timoshenko beam with
both boundary and locally distributed damping covers the cases studied by many
authors. Geometric properties of the system of root vectors of the operator
$\mathcal{L}$ play important role in investigation of different properties of
the problem (10.1)–(10.5).

Below we continue our investigation started in [32, 33], where we established
completeness and the Riesz basis property with parentheses of the root vectors
system of the operator $\mathcal{L}$, without analyzing its spectrum. In our
previous papers we imposed the following additional algebraic assumption on
$\mathcal{L}$: the ratio of wave speeds $\frac{K(\cdot)}{\rho(\cdot)}$ and
$\frac{EI(\cdot)}{I_{\rho}(\cdot)}$ is constant. This assumption has to be
added since in [32, 33] we treated BVP (1.1)–(1.2) with non-trivial potential
matrix $Q$ and _constant matrix_ $B(x)=B$. Below we will establish
completeness and the Riesz basis property with parentheses of the root vectors
system of the operator $\mathcal{L}$ _without this algebraic assumption_ and
additionally establish asymptotic behavior of its eigenvalues. Moreover, under
additional assumptions ensuring that the eigenvalues of the operator
$\mathcal{L}$ are asymptotically separated, we will show that the system of
root vectors of the operator $\mathcal{L}$ forms a Riesz basis in
$\mathcal{H}$ (_without parentheses_) and establish asymptotic behavior of the
eigenvectors. Riesz basis property is essential for obtaining numerous
stability and controllability properties.

As in our previous papers [32, 33], our approach to the spectral properties of
the operator $\mathcal{L}$ is based on the similarity reduction of
$\mathcal{L}$ to a special $4\times 4$ Dirac-type operator $L_{U}(Q)$
associated with appropriate BVP (1.1)–(1.2). To state the result we need some
additional preparations. Let

$\displaystyle B(x)$
$\displaystyle:=\operatorname{diag}(-\beta_{1}(x),\beta_{1}(x),-\beta_{2}(x),\beta_{2}(x)),\qquad\text{where}$
(10.12) $\displaystyle\beta_{1}(x)$
$\displaystyle:=\sqrt{\frac{I_{\rho}(x)}{EI(x)}},\qquad\beta_{2}(x):=\sqrt{\frac{\rho(x)}{K(x)}},\qquad
x\in[0,\ell].$ (10.13)

Recall that with the matrix function $B(\cdot)$ one associates weighted vector
$L^{2}$-space $\mathfrak{H}$ via formulas (5.3)–(5.5). In the case of matrix
function $B(\cdot)$ given by (10.12), it takes the following form,

$\mathfrak{H}=L^{2}_{\beta_{1}}[0,\ell]\oplus L^{2}_{\beta_{1}}[0,\ell]\oplus
L^{2}_{\beta_{2}}[0,\ell]\oplus L^{2}_{\beta_{2}}[0,\ell].$ (10.14)

It follows from condition (10.6) that identity operator from $\mathfrak{H}$ to
$L^{2}([0,\ell];\mathbb{C}^{4})$ is bounded and has a bounded inverse, i.e.
the Hilbert spaces $\mathfrak{H}$ and $L^{2}([0,\ell];\mathbb{C}^{4})$
coincide algebraically and topologically.

Further, we set,

$\displaystyle\Theta(x)$
$\displaystyle:=2\operatorname{diag}(h_{1}(x),h_{1}(x),h_{2}(x),h_{2}(x)),\qquad\text{where}$
(10.15) $\displaystyle h_{1}(x)$
$\displaystyle:=\sqrt{EI(x)I_{\rho}(x)},\qquad
h_{2}(x):=\sqrt{K(x)\rho(x)},\qquad x\in[0,\ell].$ (10.16)

In the sequel we assume that

$h_{1},h_{2}\in\operatorname{AC}[0,\ell].$ (10.17)

It follows from (10.6), definition (10.13) of $\beta_{1}$, $\beta_{2}$, and
definition (10.16) of $h_{1}$, $h_{2}$ that

$0<M^{-1}\leqslant\beta_{1}(x),\beta_{2}(x),h_{1}(x),h_{2}(x)\leqslant
M,\qquad x\in[0,\ell],$ (10.18)

with the same $M$ as in conditions (10.6).

Under assumptions (10.6) and (10.17) the following matrix function $Q(\cdot)$
is well-defined and summable,

$Q(x):=\Theta^{-1}(x)\begin{pmatrix}p_{1}+h_{1}^{\prime}&p_{1}-h_{1}^{\prime}&h_{2}&-h_{2}\\\
p_{1}+h_{1}^{\prime}&p_{1}-h_{1}^{\prime}&h_{2}&-h_{2}\\\
-h_{2}&-h_{2}&p_{2}+h_{2}^{\prime}&p_{2}-h_{2}^{\prime}\\\
h_{2}&h_{2}&p_{2}+h_{2}^{\prime}&p_{2}-h_{2}^{\prime}\end{pmatrix},\qquad
x\in[0,\ell].$ (10.19)

Finally, let

$C=\begin{pmatrix}1&1&0&0\\\ 0&0&0&0\\\ 0&0&1&1\\\ 0&0&0&0\end{pmatrix},\qquad
D=\begin{pmatrix}0&0&0&0\\\
\alpha_{1}-h_{1}(\ell)&\alpha_{1}+h_{1}(\ell)&\gamma_{1}&\gamma_{1}\\\
0&0&0&0\\\
\gamma_{2}&\gamma_{2}&\alpha_{2}-h_{2}(\ell)&\alpha_{2}+h_{2}(\ell)\\\
\end{pmatrix}.$ (10.20)

######  Proposition 10.1 (cf. Proposition 6.1 in [32]).

Let measurable functions $\rho,I_{\rho},K,EI,p_{1},p_{2},h_{1},h_{2}$ satisfy
conditions (10.6) and (10.17). Then the operator $\mathcal{L}$ acting in the
Hilbert space $\mathcal{H}$ is similar to the $4\times 4$ Dirac-type operator
$L_{U}(Q)$ acting in the Hilbert space $\mathfrak{H}$ given by (10.14), where
matrices $B(\cdot),C,D,Q(\cdot)$ are given by (10.12), (10.20) and (10.19).

###### Proof.

Introduce the following operator

$\mathcal{U}y=\operatorname{col}(EI(x)y_{1}^{\prime},\ y_{2},\
K(x)(y_{3}^{\prime}-y_{1}),\ y_{4}),\qquad
y=\operatorname{col}(y_{1},y_{2},y_{3},y_{4}),$ (10.21)

that maps the Hilbert space $\mathcal{H}$ given by (10.7) into
$L^{2}([0,\ell];\mathbb{C}^{4})$. Since $\frac{d}{dx}$ isometrically maps
$\widetilde{H}_{0}^{1}[0,\ell]=\\{f\in W^{1,2}[0,\ell]:f(0)=0\\}$ onto
$L^{2}[0,\ell]$, it follows from condition (10.6) that the operator
$\mathcal{U}$ is bounded with a bounded inverse. It is easy to check that for
$y=\operatorname{col}(y_{1},y_{2},y_{3},y_{4})$

$\mathcal{L}\,\mathcal{U}^{-1}y=\frac{1}{i}\begin{pmatrix}y_{2}\\\
\frac{1}{I_{\rho}}(y_{1}^{\prime}-p_{1}y_{2}+y_{3})\\\ y_{4}\\\
\frac{1}{\rho}(y_{3}^{\prime}-p_{2}y_{4})\end{pmatrix},\ \
\widetilde{\mathcal{L}}y:=\mathcal{U}\mathcal{L}\,\mathcal{U}^{-1}y=\frac{1}{i}\begin{pmatrix}EI\cdot
y_{2}^{\prime}\\\ \frac{1}{I_{\rho}}(y_{1}^{\prime}-p_{1}y_{2}+y_{3})\\\
K\cdot(y_{4}^{\prime}-y_{2})\\\
\frac{1}{\rho}(y_{3}^{\prime}-p_{2}y_{4})\end{pmatrix},$ (10.22)

and

$\operatorname{dom}(\widetilde{\mathcal{L}})=\mathcal{U}\operatorname{dom}(\mathcal{L})=\bigl{\\{}\bigr{.}y=\operatorname{col}(y_{1},y_{2},y_{3},y_{4})\in\operatorname{AC}([0,\ell];\mathbb{C}^{4}):\
\ \widetilde{\mathcal{L}}y\in L^{2}([0,\ell];\mathbb{C}^{4}),\\\
y_{2}(0)=y_{4}(0)=0,\quad
y_{1}(\ell)+\alpha_{1}y_{2}(\ell)+\gamma_{1}y_{4}(\ell)=0,\quad
y_{3}(\ell)+\alpha_{2}y_{4}(\ell)+\gamma_{2}y_{2}(\ell)=0\bigl{.}\bigr{\\}}.$
(10.23)

Thus, the operator $\mathcal{L}$ is similar to the operator
$\widetilde{\mathcal{L}}$,

$\widetilde{\mathcal{L}}y=-i\widetilde{B}(x)y^{\prime}+\widetilde{Q}(x)y,\qquad
y\in\operatorname{dom}(\widetilde{\mathcal{L}}),$ (10.24)

with the domain $\operatorname{dom}(\widetilde{\mathcal{L}})$ given by
(10.23), and the matrix functions $\widetilde{B}(\cdot)$,
$\widetilde{Q}(\cdot)$, given by

$\widetilde{B}(x):=\begin{pmatrix}0&EI(x)&0&0\\\
\frac{1}{I_{\rho}(x)}&0&0&0\\\ 0&0&0&K(x)\\\
0&0&\frac{1}{\rho(x)}&0\end{pmatrix},\qquad\widetilde{Q}(x):=i\begin{pmatrix}0&0&0&0\\\
0&\frac{p_{1}(x)}{I_{\rho}(x)}&-\frac{1}{I_{\rho}(x)}&0\\\ 0&K(x)&0&0\\\
0&0&0&\frac{p_{2}(x)}{\rho(x)}\end{pmatrix},\quad x\in[0,\ell].$ (10.25)

Note, that $\widetilde{Q}\in L^{1}([0,\ell];\mathbb{C}^{4\times 4})$ in view
of condition (10.6). Next we diagonalize the matrix $\widetilde{B}(\cdot)$.
Namely, setting

$\widetilde{\mathcal{U}}:y\to\widetilde{U}(x)y,\qquad\widetilde{U}(x):=\begin{pmatrix}-h_{1}(x)&h_{1}(x)&0&0\\\
1&1&0&0\\\ 0&0&-h_{2}(x)&h_{2}(x)\\\ 0&0&1&1\end{pmatrix},$ (10.26)

and noting that

$\widetilde{U}^{-1}(x)=\frac{1}{2}\begin{pmatrix}-\frac{1}{h_{1}(x)}&1&0&0\\\
\frac{1}{h_{1}(x)}&1&0&0\\\ 0&0&-\frac{1}{h_{2}(x)}&1\\\
0&0&\frac{1}{h_{2}(x)}&1\end{pmatrix},$ (10.27)

we easily get after straightforward calculations

$\widetilde{U}^{-1}(x)\widetilde{B}(x)\widetilde{U}(x)=\operatorname{diag}\left(-\sqrt{\frac{EI(x)}{I_{\rho}(x)}},\sqrt{\frac{EI(x)}{I_{\rho}(x)}},-\sqrt{\frac{K(x)}{\rho(x)}},\sqrt{\frac{K(x)}{\rho(x)}}\right)=B(x)^{-1},$
(10.28)

Here we have used definition (10.16) of $h_{1}$, $h_{2}$, and definition
(10.13) of $\beta_{1}$, $\beta_{2}$.

Further, note that
$\widetilde{U}\in\operatorname{AC}([0,\ell];\mathbb{C}^{4\times 4})$ since
$h_{1},h_{2}\in\operatorname{AC}[0,\ell]$. Moreover, as noted earlier, $Q\in
L^{1}([0,\ell];\mathbb{C}^{4\times 4})$ under assumptions (10.6) and (10.17),
where $Q(\cdot)$ is given by (10.19) and (10.15). Hence, it is easily seen
that

$\widetilde{U}^{-1}\widetilde{Q}\widetilde{U}-i\widetilde{U}^{-1}\widetilde{B}\widetilde{U}^{\prime}=-iB^{-1}Q.$
(10.29)

We consider the operator $\widetilde{\mathcal{U}}:y\to\widetilde{U}(x)y$
acting from $\mathfrak{H}$ given by (10.14) to
$L^{2}([0,\ell];\mathbb{C}^{4})$. It is clear from condition (10.6), that
$\widetilde{\mathcal{U}}$ is bounded with a bounded inverse. Taking into
account (10.28) and (10.29) we obtain for any
$y\in\operatorname{AC}([0,\ell];\mathbb{C}^{4})$ and satisfying
$\widetilde{U}y\in\operatorname{dom}(\widetilde{\mathcal{L}})$ that

$\widehat{\mathcal{L}}y:=\widetilde{\mathcal{U}}^{-1}\widetilde{\mathcal{L}}\
\widetilde{\mathcal{U}}y=-iB(x)^{-1}(y^{\prime}+Q(x)y).$ (10.30)

Next, taking into account formulas (10.20) and (10.26) for matrices $C$, $D$,
and $\widetilde{U}(\cdot)$, respectively, we derive that

$\operatorname{dom}(\widehat{\mathcal{L}})=\\{y\in\operatorname{AC}([0,\ell];\mathbb{C}^{4}):\widehat{\mathcal{L}}y\in\mathfrak{H},\
Cy(0)+Dy(\ell)=0\\}.$ (10.31)

This directly implies that $L_{U}(Q)=\widehat{\mathcal{L}}$. Combining this
identity with (10.22), one concludes that $\mathcal{L}$ is similar to
$L_{U}(Q)$. ∎

###  10.2 Completeness and Riesz basis property with parentheses

Applying Theorems 7.4 and 9.20 to the operator $L_{U}(Q)$ constructed in
Proposition 10.1 we obtain the following result.

######  Theorem 10.2.

Let measurable functions $\rho,I_{\rho},K,EI,p_{1},p_{2},h_{1},h_{2}$ satisfy
conditions (10.6) and (10.17). Set

$\nu(x):=\frac{K(x)}{\rho(x)}-\frac{E(x)}{I_{\rho}(x)},\qquad x\in[0,\ell].$
(10.32)

Let function $\nu(\cdot)$ satisfy the following condition for some
$\varepsilon>0$,

$\text{either}\quad\nu\equiv 0,\qquad\text{or}\qquad\nu(x)>\varepsilon,\quad
x\in[0,\ell],\qquad\text{or}\qquad\nu(x)<-\varepsilon,\quad x\in[0,\ell].$
(10.33)

Let also

$(\alpha_{1}+h_{1}(\ell))(\alpha_{2}+h_{2}(\ell))\neq\gamma_{1}\gamma_{2}\quad\text{and}\quad(\alpha_{1}-h_{1}(\ell))(\alpha_{2}-h_{2}(\ell))\neq\gamma_{1}\gamma_{2}.$
(10.34)

Then the system of root vectors of $\mathcal{L}$ is complete, minimal, and
forms a Riesz basis with parentheses in the Hilbert space $\mathcal{H}$.

###### Proof.

Consider the operator $L_{U}(Q)$ defined in Proposition 10.1. It is clear from
the form (10.12) of the matrix function $B(\cdot)$ that

$P_{-}=\operatorname{diag}(1,0,1,0)\quad\text{and}\quad
P_{+}=\operatorname{diag}(0,1,0,1),$

where “projectors” $P_{\pm}$ are given by (5.41). Combining this with
expression (10.20) for the matrices $C$ and $D$ and with definition (5.40) of
$J_{P}(C,D)$ yields

$J_{P_{+}}(C,D)=\det\begin{pmatrix}1&0&0&0\\\
0&\alpha_{1}+h_{1}(\ell)&0&\gamma_{1}\\\ 0&0&1&0\\\
0&\gamma_{2}&0&\alpha_{2}+h_{2}(\ell)\end{pmatrix}=(\alpha_{1}+h_{1}(\ell))(\alpha_{2}+h_{2}(\ell))-\gamma_{1}\gamma_{2}.$
(10.35)

Similarly one gets

$J_{P_{-}}(C,D)=(\alpha_{1}-h_{1}(\ell))(\alpha_{2}-h_{2}(\ell))-\gamma_{1}\gamma_{2}.$

Condition (10.34) implies that $J_{\pm}(C,D)\neq 0$. Hence, boundary
conditions $U(y)=Cy(0)+Dy(\ell)=0$ are regular.

It is clear from condition (10.6), definition (10.13) of $\beta_{1}(\cdot)$,
$\beta_{2}(\cdot)$ and definition (10.32) of $\nu(\cdot)$ that

$|\beta_{1}(x)-\beta_{2}(x)|=\frac{|\nu(x)|}{\beta_{1}(x)+\beta_{2}(x)}\in\left[\frac{1}{2M}|\nu(x)|,\
\frac{M}{2}|\nu(x)|\right],\qquad x\in[0,\ell].$

Hence, condition (10.33) implies that either $\beta_{1}\equiv\beta_{2}$ or
$\beta_{1}-\beta_{2}$ does not change sign on $[0,\ell]$ and is separated from
zero. Clearly this is valid for all other entries of the matrix function
$B(\cdot)$. Hence it satisfies conditions (4.5), (4.13)–(4.14) (note, that
notations $\beta_{1}$ and $\beta_{2}$ differ in these conditions). Moreover,
conditons (10.6) and (10.17) imply that $Q\in
L^{1}([0,\ell];\mathbb{C}^{4\times 4})$. Therefore, Remark 4.1(ii), Theorem
7.4(ii) and Theorem 9.20 imply that the system of root vectors of the operator
$L_{U}(Q)$ is complete, minimal, and forms a Riesz basis with parentheses in
the Hilbert space $\mathfrak{H}$. By Proposition 10.1, the operator
$\mathcal{L}$ is similar to the operator $L_{U}(Q)$. Hence the system of root
vectors of the operator $\mathcal{L}$ has the same properties, which finishes
the proof. ∎

######  Remark 10.3.

Theorem 10.2 improves similar result from our previous paper [32] (Theorem
6.3) in several ways:

(i) Main improvement is replacing condition of wave speeds
$\frac{K(\cdot)}{\rho(\cdot)}$ and $\frac{EI(\cdot)}{I_{\rho}(\cdot)}$ being
proportional functions, with much more general condition (10.33) on the
difference of these wave speeds.

(ii) To establish Riesz basis property with parentheses, in [32, Theorem 6.3]
we assumed “smoothness” conditions $p_{1},p_{2}\in L^{\infty}[0,\ell]$,
$h_{1},h_{2}\in\operatorname{Lip}_{1}[0,\ell]$ and were only able to handle
simpler case of boundary conditions when $\gamma_{1}=\gamma_{2}=0$. Here we
handle the most general boundary conditions under the most general conditions
on functions $p_{1},p_{2},h_{1},h_{2}$.

(iii) Finally, we replaced condition $\rho,I_{\rho},K,EI\in C[0,\ell]$ with
more general condition $\rho,I_{\rho},K,EI\in L^{\infty}[0,\ell]$, allowing
parameters of the model to have discontinuities. In fact, considerations in
[32] work under these conditions as well if we note that
$f\in\operatorname{AC}$, $g,g^{-1}\in\operatorname{Lip}_{1}$ implies that
$f\circ g\in\operatorname{AC}$ (here $g^{-1}$ is the inverse of monotonous
function $g$).

If $\gamma_{1}=\gamma_{2}=0$ we can significantly improve Theorem 10.2 by
dropping cumbersome condition (10.33) on separation of wave speeds
$\frac{K(\cdot)}{\rho(\cdot)}$ and $\frac{EI(\cdot)}{I_{\rho}(\cdot)}$.

######  Theorem 10.4.

Let measurable functions $\rho,I_{\rho},K,EI,p_{1},p_{2},h_{1},h_{2}$ satisfy
conditions (10.6) and (10.17). Let also

$\alpha_{1}\neq\pm h_{1}(\ell),\qquad\alpha_{2}\neq\pm
h_{2}(\ell),\qquad\gamma_{1}=\gamma_{2}=0.$ (10.36)

Then the system of root vectors of $\mathcal{L}$ is complete, minimal, and
forms a Riesz basis with parentheses in the Hilbert space $\mathcal{H}$.

###### Proof.

Consider the operator $L_{U}(Q)$ defined in Proposition 10.1. Since
$\gamma_{1}=\gamma_{2}=0$ we can represent it as bounded perturbation of the
direct sum of two $2\times 2$ Dirac type operators:

$\displaystyle L_{U}(Q)$ $\displaystyle=L_{1}\oplus
L_{2}+\widetilde{\mathcal{Q}},\quad\text{where for}\quad k\in\\{1,2\\},$
(10.37) $\displaystyle(L_{k}y)(x)$
$\displaystyle:=-iB_{k}^{-1}(x)\bigl{(}y^{\prime}(x)+Q_{k}(x)y(x)\bigr{)},\quad
x\in[0,\ell],\quad
y=\operatorname{col}(y_{1},y_{2})\in\operatorname{dom}L_{k},$ (10.38)
$\displaystyle\operatorname{dom}L_{k}$
$\displaystyle:=\\{y\in\operatorname{AC}([0,\ell];\mathbb{C}^{2}):L_{k}y\in
L^{2}_{|\beta_{k}|}([0,\ell];\mathbb{C}^{2}),$ $\displaystyle\qquad
y_{1}(0)+y_{2}(0)=(\alpha_{k}-h_{k}(\ell))y_{1}(\ell)+(\alpha_{k}+h_{k}(\ell))y_{2}(\ell)=0\\},$
(10.39) $\displaystyle B_{k}$ $\displaystyle:=\begin{pmatrix}-\beta_{k}&0\\\
0&\beta_{k}\end{pmatrix},\qquad
Q_{k}:=\frac{1}{2h_{k}}\begin{pmatrix}p_{k}+h_{k}^{\prime}&p_{k}-h_{k}^{\prime}\\\
p_{k}+h_{k}^{\prime}&p_{k}-h_{k}^{\prime}\\\ \end{pmatrix},$ (10.40)
$\displaystyle(\widetilde{\mathcal{Q}}y)(x)$
$\displaystyle=\widetilde{Q}(x)y(x),\qquad\widetilde{Q}=\Theta^{-1}\begin{pmatrix}0&0&h_{2}&-h_{2}\\\
0&0&h_{2}&-h_{2}\\\ -h_{2}&-h_{2}&0&0\\\ h_{2}&h_{2}&0&0\end{pmatrix}.$
(10.41)

It follows from (10.6) and (10.17) that $Q_{1},Q_{2}\in
L^{1}([0,\ell];\mathbb{C}^{2\times 2})$ and $\widetilde{Q}\in
L^{\infty}([0,\ell];\mathbb{C}^{2\times 2})$. Due to conditions (10.34), the
operator $L_{k}$ is a $2\times 2$ Dirac type operator with separated regular
boundary conditions. Boundary conditions remain separated and regular after
applying gauge transform from Lemma 6.11. Thus, by Lemma 5.20(ii), the
$2\times 2$ BVP corresponding to the operator $L_{k}$ is strictly regular
according to Definition 6.13. Theorem 9.11 now implies that the system of its
root vectors forms a Riesz basis in
$L^{2}_{|\beta_{k}|}([0,\ell];\mathbb{C}^{2})$ and its eigenvalues have a
proper asymptotic, in particular, inequality (9.47) is satisfied for them. It
is also clear that $L_{k}$ has finitely many associated vectors. Clearly, the
direct sum $L:=L_{1}\oplus L_{2}$ has the same properties. Since
$\widetilde{\mathcal{Q}}$ is a bounded operator, the operator $L_{U}(Q)$ is a
bounded perturbation of “spectral” operator $L$. Hence by Proposition 9.19,
the system of root vectors of the operator $L_{U}(Q)$ forms a Riesz basis with
parentheses in $\mathfrak{H}$. Since, by Proposition 10.1, $\mathcal{L}$ is
similar to the operator $L_{U}(Q)$, the system of root vectors of
$\mathcal{L}$ forms a Riesz basis with parentheses in $\mathcal{H}$. ∎

######  Remark 10.5.

The proof above follows the proof of Theorem 8.2 in [33], where this result
was proved under additional assumption: the ratio of wave speeds
$\frac{K(\cdot)}{\rho(\cdot)}$ and $\frac{EI(\cdot)}{I_{\rho}(\cdot)}$ is
constant. Note that the in the proof of Theorem 10.4 none of results for Dirac
type operators for $n>2$ were used. In fact, it follows from results of [33]
by applying additional similarity transformation to $L_{k}$ that realizes a
special change of variable that makes the matrix function $B_{k}$ above
constant.

###  10.3 Asymptotic behavior of eigenvalues and Riesz basis property

Going over to the asymptotic behavior of the eigenvalues of the operator
$\mathcal{L}$, first we restrict ourselves to the general case
$\beta_{1}\not\equiv\beta_{2}$. Recall that
$\alpha_{1},\alpha_{2},\gamma_{1},\gamma_{2}$ are parameters from boundary
conditions (10.4)–(10.5), functions $\beta_{1}=\sqrt{\frac{I_{\rho}}{EI}}$ and
$\beta_{2}=\sqrt{\frac{\rho}{K}}$ were defined in (10.13), and functions
$h_{1}=\sqrt{EI\cdot I_{\rho}}$ and $h_{2}=\sqrt{K\cdot\rho}$ were defined in
(10.16). Let us introduce some notations:

$b_{k}=\int_{0}^{\ell}\beta_{k}(t)dt,\qquad\alpha_{k}^{\pm}:=\alpha_{k}\pm
h_{k}(\ell),\qquad v_{k}^{\pm}=\left(\frac{h_{k}(\ell)}{h_{k}(0)}\right)^{\pm
1/2},\quad k\in\\{1,2\\}.$ (10.42)

The following exponential polynomial plays a crucial role in establishing the
asymptotic behavior of the eigenvalues of the operator $\mathcal{L}$,

$\Delta^{\operatorname{Tim}}_{0}(\lambda)=(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})v_{1}^{+}v_{2}^{+}\cdot
e^{i\lambda(b_{1}+b_{2})}+(\alpha_{1}^{-}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})v_{1}^{-}v_{2}^{-}\cdot
e^{-i\lambda(b_{1}+b_{2})}\\\
-(\alpha_{1}^{+}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})v_{1}^{+}v_{2}^{-}\cdot
e^{i\lambda(b_{1}-b_{2})}-(\alpha_{1}^{-}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})v_{1}^{-}v_{2}^{+}\cdot
e^{i\lambda(-b_{1}+b_{2})},\quad\ \ \lambda\in\mathbb{C}.$ (10.43)

Now we ready to state our main result on the asymptotic behavior of the
eigenvalues of the operator $\mathcal{L}$. For reader’s convenience we state
all involved conditions on the parameters
$\rho(\cdot),I_{\rho}(\cdot),K(\cdot),EI(\cdot),p_{1}(\cdot),p_{2}(\cdot)$ and
$\alpha_{1},\alpha_{2},\gamma_{1},\gamma_{2}$ of the Timoshenko beam model
(10.1)–(10.5) without appealing to the previous formulas and notations (except
definition (10.43) for $\Delta^{\operatorname{Tim}}_{0}(\cdot))$.

######  Theorem 10.6.

Let parameters $\rho,I_{\rho},K,EI,p_{1},p_{2}$ of the Timoshenko beam model
(10.1)–(10.2) be measurable functions and for some $M>1$ the following
conditions hold,

$\displaystyle 0<M^{-1}\leqslant\rho(x),I_{\rho}(x),K(x),EI(x)\leqslant
M,\qquad x\in[0,\ell],$ (10.44) $\displaystyle p_{1},p_{2}\in
L^{1}([0,\ell];\mathbb{C}),\qquad h_{1}:=\sqrt{EI\cdot
I_{\rho}}\in\operatorname{AC}[0,\ell],\qquad
h_{2}:=\sqrt{K\cdot\rho}\in\operatorname{AC}[0,\ell].$ (10.45)

Let also wave speeds $\frac{K(\cdot)}{\rho(\cdot)}$ and
$\frac{EI(\cdot)}{I_{\rho}(\cdot)}$ be separated from each other. i.e. for
some $\varepsilon>0$ the following condition holds,

$\text{either}\qquad\frac{K(x)}{\rho(x)}-\frac{E(x)}{I_{\rho}(x)}>\varepsilon,\quad
x\in[0,\ell],\qquad\text{or}\qquad\frac{K(x)}{\rho(x)}-\frac{E(x)}{I_{\rho}(x)}<-\varepsilon,\quad
x\in[0,\ell].$ (10.46)

Recall also that

$b_{1}=\int_{0}^{\ell}\sqrt{\frac{I_{\rho}(t)}{EI(t)}}dt=\int_{0}^{\ell}\beta_{1}(t)dt,\qquad
b_{2}=\int_{0}^{\ell}\sqrt{\frac{\rho(t)}{K(t)}}dt=\int_{0}^{\ell}\beta_{2}(t)dt.$
(10.47)

(i) Let parameters $\alpha_{1},\alpha_{2},\gamma_{1},\gamma_{2}\in\mathbb{C}$
from boundary conditions (10.4)–(10.5) satisfy the following condition,

$(\alpha_{1}+h_{1}(\ell))(\alpha_{2}+h_{2}(\ell))\neq\gamma_{1}\gamma_{2}\quad\text{and}\quad(\alpha_{1}-h_{1}(\ell))(\alpha_{2}-h_{2}(\ell))\neq\gamma_{1}\gamma_{2}.$
(10.48)

Then the dynamic generator $\mathcal{L}$ of the general Timoshenko beam model
(10.1)–(10.5) has a countable sequence of eigenvalues
$\Lambda:=\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ counting multiplicity. The
sequence $\Lambda$ is incompressible $($see Definition 5.7$)$ and lies in the
strip $\Pi_{h}=\\{\lambda\in\mathbb{C}:|\operatorname{Im}\lambda|\leqslant
h\\}$ for some $h\geqslant 0$.

Moreover, exponential polynomial $\Delta^{\operatorname{Tim}}_{0}(\cdot)$
given by (10.43) has a countable sequence of zeros
$\Lambda_{0}:=\\{\lambda_{m}^{0}\\}_{m\in\mathbb{Z}}$ counting multiplicity
that satisfies the same properties, and both sequences $\Lambda$ and
$\Lambda_{0}$ can be ordered in such a way that the following sharp
asymptotical formula holds

$\lambda_{m}=\lambda_{m}^{0}+o(1)=\frac{\pi
m}{b_{1}+b_{2}}+o(m)\quad\text{as}\quad m\to\infty.$ (10.49)

(ii) Let the following conditions hold,

$\gamma_{1}\gamma_{2}=0,\qquad\alpha_{1}\neq\pm
h_{1}(\ell),\qquad\alpha_{2}\neq\pm h_{2}(\ell).$ (10.50)

Then the sequence $\Lambda$ of the eigenvalues of the operator $\mathcal{L}$
is the union of two sequences asymptotically close to arithmetic progressions.
Namely, $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, where for $k\in\\{1,2\\}$ we
have,

$\Lambda_{k}:=\\{\lambda_{k,m}\\}_{m\in\mathbb{Z}},\quad\lambda_{k,m}:=\frac{\pi
m}{b_{k}}-\frac{i\ln\tau_{k}}{2b_{k}}+o(1),\qquad\tau_{k}:=\frac{(\alpha_{k}-h_{k}(\ell))h_{k}(0)}{(\alpha_{k}+h_{k}(\ell))h_{k}(\ell)}\neq
0,\qquad m\in\mathbb{Z}.$ (10.51)

Moreover, the sequence $\Lambda$ is asymptotically separated if and only if
the following condition holds:

$\text{either}\quad b_{1}\ln|\tau_{2}|\neq
b_{2}\ln|\tau_{1}|\quad\text{or}\quad\left(\frac{b_{1}}{b_{2}}\in\mathbb{Q}\quad\text{and}\quad\frac{b_{1}\arg(\tau_{2})-b_{2}\arg(\tau_{1})}{2\pi\gcd(b_{1},b_{2})}\not\in\mathbb{Z}\right).$
(10.52)

(iii) Let the following conditions hold,

$\gamma_{1}\gamma_{2}\neq
0,\qquad\alpha_{1}^{2}=h_{1}^{2}(\ell)+\frac{h_{1}(\ell)}{h_{2}(\ell)}\gamma_{1}\gamma_{2},\qquad\alpha_{2}=\frac{h_{2}(\ell)}{h_{1}(\ell)}\alpha_{1}.$
(10.53)

Then the sequence $\Lambda=\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ is
asymptotically separated and the following sharp asymptotical formula holds,

$\lambda_{m}:=\frac{\pi
m}{b_{1}+b_{2}}-\frac{i\ln\tau}{2(b_{1}+b_{2})}+o(1),\qquad\tau:=\frac{(\alpha_{1}-h_{1}(\ell))\cdot
h_{1}(0)h_{2}(0)}{(\alpha_{1}+h_{1}(\ell))\cdot h_{1}(\ell)h_{2}(\ell)}\neq
0,\qquad m\in\mathbb{Z}.$ (10.54)

(iv) Let $b_{1}/b_{2}\in\mathbb{Q}$. Namely, $b_{1}=n_{1}b$, $b_{2}=n_{2}b$
for some $n_{1},n_{2}\in\mathbb{N}$ and $b>0$. Let also condition (10.48)
holds. Then

$\Delta^{\operatorname{Tim}}_{0}(\lambda)=e^{-i\lambda(b_{1}+b_{2})}\mathcal{P}(e^{i\lambda
b}),\qquad\lambda\in\mathbb{C},$

where $\mathcal{P}(\cdot)$ is a polynomial of degree $N:=2(n_{1}+n_{2})$ such
that $\mathcal{P}(0)\neq 0$. Let $z_{1},\ldots,z_{N}\neq 0$ be its roots
(counting multiplicity). Then the sequence $\Lambda$ of the eigenvalues of the
operator $\mathcal{L}$ is the union of $N$ sequences asymptotically close to
arithmetic progressions,

$\Lambda=\\{\widetilde{\Lambda}_{k}\\}_{k=1}^{N},\quad\widetilde{\Lambda}_{k}:=\\{\widetilde{\lambda}_{k,m}\\}_{m\in\mathbb{Z}},\quad\widetilde{\lambda}_{k,m}:=\frac{2\pi
m}{b}-\frac{i\ln z_{k}}{b}+o(1),\qquad m\in\mathbb{Z},\quad
k\in\\{1,\ldots,{N}\\}.$ (10.55)

Moreover, the sequence $\Lambda$ is asymptotically separated if and only if
numbers $z_{1},\ldots,z_{n}$ are distinct.

###### Proof.

(i) It is clear, that conditions (10.44)–(10.45) are the same as conditions
(10.6) and (10.17). Further, condition (10.46) implies condition (10.33) on
the wave speed difference $\nu(\cdot)$ given by (10.32). Finally, condition
(10.48) is the same as (10.34). Therefore, parameters of the Timoshenko beam
model (10.1)–(10.5) satisfy conditions of Proposition 10.1 and Theorem 10.2.

By Proposition 10.1, the operator $\mathcal{L}$ is similar to the $4\times 4$
Dirac-type operator $L_{U}(Q)$ acting in the Hilbert space $\mathfrak{H}$
given by (10.14), where matrix functions $B(\cdot)$ and $Q(\cdot)$ are given
by (10.12) and (10.19), equipped with the boundary conditions (5.2),

$U(y)=Cy(0)+Dy(\ell)=0,$ (10.56)

where matrices $C,D$ are given by (10.20). Hence both operators have the same
spectrum (counting multiplicity).

According to the proof of Theorem 10.2, condition (10.48) implies regularity
of boundary conditions (10.56), while conditions (10.6) and (10.33) imply
conditions (4.5), (4.13)–(4.14) on the matrix function $B(\cdot)$. Moreover,
conditions (10.6) and (10.17) trivially imply that $Q\in
L^{1}([0,\ell];\mathbb{C}^{4\times 4})$. Therefore, Remark 4.1(ii) and Theorem
6.15 imply the desired relation (10.49), and all other desired properties of
the sequence $\Lambda$ with the sequence $\widetilde{\Lambda}_{0}$ in place of
$\Lambda_{0}$, where $\widetilde{\Lambda}_{0}$ is the sequence of zeros of the
modified characteristic determinant $\widetilde{\Delta}_{0}(\cdot)$ given by
(6.70). To finish the proof, it is sufficient to show that
$\widetilde{\Delta}_{0}(\cdot)$ is proportional to
$\Delta^{\operatorname{Tim}}_{0}(\cdot)$ given by (10.43).

Recall that

$\widetilde{\Delta}_{0}(\cdot)=\det(C+DW(\ell)\Phi_{0}(\ell,\cdot)),$ (10.57)

where $W(\cdot)$ is the solution of the Cauchy problem (6.60) that involves
the block diagonal $Q_{\operatorname{diag}}$ of the matrix functions $Q$,

$W^{\prime}(x)+Q_{\operatorname{diag}}(x)W(x)=0,\quad x\in[0,\ell],\qquad
W(0)=I_{4}.$ (10.58)

To this end, note that condition (10.46) implies that
$\beta_{1}\not\equiv\beta_{2}$. Hence matrix function $B(\cdot)$ given by
(10.12) has simple spectrum. Therefore, its block matrix decomposition has all
blocks of size one. This observation and formula (10.19) yield that

$Q_{\operatorname{diag}}=\operatorname{diag}\left(\frac{p_{1}}{2h_{1}}+\frac{h_{1}^{\prime}}{2h_{1}},\
\ \frac{p_{1}}{2h_{1}}-\frac{h_{1}^{\prime}}{2h_{1}},\ \
\frac{p_{2}}{2h_{2}}+\frac{h_{2}^{\prime}}{2h_{2}},\ \
\frac{p_{2}}{2h_{2}}-\frac{h_{2}^{\prime}}{2h_{2}}\right).$ (10.59)

First observe that since functions $h_{1},h_{2}$ are positive and absolutely
continuous, we have

$\exp\left(\int_{0}^{x}\frac{h_{k}^{\prime}(t)}{2h_{k}(t)}dt\right)=\exp\left(\frac{\ln
h_{k}(x)-\ln
h_{k}(0)}{2}\right)=\left(\frac{h_{k}(x)}{h_{k}(0)}\right)^{1/2},\qquad
x\in[0,\ell],\quad k\in\\{1,2\\}.$ (10.60)

Taking into account this observation, solution $W(\cdot)$ of the Cauchy
problem (10.58) has the following explicit form

$W(x)=\operatorname{diag}(W_{1}^{-}(x),W_{1}^{+}(x),W_{2}^{-}(x),W_{2}^{+}(x)),\qquad
x\in[0,\ell].$ (10.61)

where

$W_{k}^{\pm}(x):=\exp\left(-\int_{0}^{x}\left(\frac{p_{k}(t)}{2h_{k}(t)}\mp\frac{h_{k}^{\prime}(t)}{2h_{k}(t)}\right)dt\right)\\\
=\left(\frac{h_{k}(x)}{h_{k}(0)}\right)^{\pm
1/2}\exp\left(-\int_{0}^{x}\frac{p_{k}(t)}{2h_{k}(t)}dt\right),\qquad
x\in[0,\ell],\quad k\in\\{1,2\\}.$ (10.62)

To this end, note the fundamental matrix solution $\Phi_{0}(\cdot,\lambda)$ of
the matrix equation $Y^{\prime}=iB(x)Y$ with the matrix function $B(\cdot)$
given by (10.12) is of the following form,

$\Phi_{0}(x,\lambda)=\operatorname{diag}(e^{-i\lambda\rho_{1}(x)},e^{i\lambda\rho_{1}(x)},e^{-i\lambda\rho_{2}(x)},e^{i\lambda\rho_{2}(x)}),\qquad
x\in[0,\ell],\qquad\lambda\in\mathbb{C},$ (10.63)

where

$\rho_{k}(x):=\int_{0}^{x}\beta_{k}(t)dt,\qquad x\in[0,\ell],\qquad
k\in\\{1,2\\}.$ (10.64)

It follows from definition (10.13) of $\beta_{1}$, $\beta_{2}$ and definition
(10.47) of $b_{1}$, $b_{2}$, that $b_{k}=\rho_{k}(\ell)$, $k\in\\{1,2\\}$.

Further, for brevity we set,

$w_{k}^{\pm}:=W_{k}^{\pm}(\ell),\qquad\mathcal{E}_{k}:=\exp\left(-\int_{0}^{\ell}\frac{p_{k}(t)}{2h_{k}(t)}dt\right),\qquad
k\in\\{1,2\\},$ (10.65)

It is clear from definition (10.42) of $v_{k}^{\pm}$ that
$w_{k}^{\pm}=v_{k}^{\pm}\mathcal{E}_{k}$, $k\in\\{1,2\\}$. With account of
this notations and definition (10.42) of $\alpha_{k}^{\pm}$, we derive from
(10.20) and (10.63) that

$C+DW(\ell)\Phi_{0}(\ell,\lambda)=\begin{pmatrix}1&1&0&0\\\
\alpha_{1}^{-}w_{1}^{-}e^{-i\lambda b_{1}}&\alpha_{1}^{+}w_{1}^{+}e^{i\lambda
b_{1}}&\gamma_{1}w_{2}^{-}e^{-i\lambda b_{2}}&\gamma_{1}w_{2}^{+}e^{i\lambda
b_{2}}\\\ 0&0&1&1\\\ \gamma_{2}w_{1}^{-}e^{-i\lambda
b_{1}}&\gamma_{2}w_{1}^{+}e^{i\lambda
b_{1}}&\alpha_{2}^{-}w_{2}^{-}e^{-i\lambda
b_{2}}&\alpha_{2}^{+}w_{2}^{+}e^{i\lambda b_{2}}\\\ \end{pmatrix}.$ (10.66)

It now follows from (10.66) after straightforward calculations that

$\widetilde{\Delta}_{0}(\lambda)=(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})w_{1}^{+}w_{2}^{+}\cdot
e^{i\lambda(b_{1}+b_{2})}+(\alpha_{1}^{-}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})w_{1}^{-}w_{2}^{-}\cdot
e^{-i\lambda(b_{1}+b_{2})}\\\
-(\alpha_{1}^{+}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})w_{1}^{+}w_{2}^{-}\cdot
e^{i\lambda(b_{1}-b_{2})}-(\alpha_{1}^{-}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})w_{1}^{-}w_{2}^{+}\cdot
e^{i\lambda(-b_{1}+b_{2})}\\\ =\left(\alpha_{1}^{+}w_{1}^{+}e^{i\lambda
b_{1}}-\alpha_{1}^{-}w_{1}^{-}e^{-i\lambda
b_{1}}\right)\left(\alpha_{2}^{+}w_{2}^{+}e^{i\lambda
b_{2}}-\alpha_{2}^{-}w_{2}^{-}e^{-i\lambda b_{2}}\right)\\\
-\gamma_{1}\gamma_{2}\left(w_{1}^{+}e^{i\lambda b_{1}}-w_{1}^{-}e^{-i\lambda
b_{1}}\right)\left(w_{2}^{+}e^{i\lambda b_{2}}-w_{2}^{-}e^{-i\lambda
b_{2}}\right).$ (10.67)

It is clear from observation $w_{k}^{\pm}=v_{k}^{\pm}\mathcal{E}_{k}$ above
and definition (10.43) of $\Delta^{\operatorname{Tim}}_{0}(\cdot)$ that

$\widetilde{\Delta}_{0}(\lambda)=\mathcal{E}_{1}\mathcal{E}_{2}\Delta^{\operatorname{Tim}}_{0}(\lambda),\qquad\lambda\in\mathbb{C},$
(10.68)

which finishes the proof of part (i).

(ii) If $\gamma_{1}\gamma_{2}=0$ then with account of (10.68), formula (10.67)
simplifies to

$\Delta^{\operatorname{Tim}}_{0}(\lambda)=\left(\alpha_{1}^{+}v_{1}^{+}e^{i\lambda
b_{1}}-\alpha_{1}^{-}v_{1}^{-}e^{-i\lambda
b_{1}}\right)\left(\alpha_{2}^{+}v_{2}^{+}e^{i\lambda
b_{2}}-\alpha_{2}^{-}v_{2}^{-}e^{-i\lambda b_{2}}\right).$ (10.69)

Condition (10.50) on $\alpha_{1},\alpha_{2}$ implies that
$\alpha_{1}^{\pm},\alpha_{2}^{\pm}\neq 0$. It is clear that arithmetic
progressions $\Lambda_{1}^{0}$ and $\Lambda_{2}^{0}$,

$\Lambda_{k}^{0}:=\\{\lambda_{k,m}^{0}\\}_{m\in\mathbb{Z}},\quad\lambda_{k,m}^{0}:=\frac{2\pi
m-i\ln\tau_{k}^{0}}{2b_{k}},\qquad\tau_{k}^{0}:=\frac{\alpha_{k}^{-}v_{k}^{-}}{\alpha_{k}^{+}v_{k}^{+}}\neq
0,\qquad m\in\mathbb{Z},\quad k\in\\{1,2\\},$ (10.70)

are zeros of the first and second factor in the r.h.s. of (10.69),
respectively. It is easily seen from definition (10.42) of $v_{k}^{\pm}$ that

$\frac{v_{k}^{-}}{v_{k}^{+}}=\frac{h_{k}(0)}{h_{k}(\ell)},\qquad
k\in\\{1,2\\}.$ (10.71)

Hence $\tau_{k}^{0}=\tau_{k}$, where $\tau_{k}$ if given by (10.51).
Asymptotical formula (10.51) now follows from part (i) of the theorem. Note
that $\sigma_{1}=2b_{1}$ and $\sigma_{2}=2b_{2}$ in notations of Lemma 5.20.
Hence Lemma 5.20(iii) implies that the sequence $\Lambda$ is asymptotically
separated if and only if condition (10.52) holds, which finishes the proof of
part (ii).

(iii) It follows from (10.53) and definition (10.42) of $\alpha_{k}^{\pm}$
that

$\alpha_{1}^{\pm}\alpha_{2}^{\mp}=(\alpha_{1}\pm h_{1}(\ell))(\alpha_{2}\mp
h_{2}(\ell))=(\alpha_{1}\pm
h_{1}(\ell))\left(\frac{h_{2}(\ell)}{h_{1}(\ell)}\alpha_{1}\mp
h_{2}(\ell)\right)\\\ =\frac{h_{2}(\ell)}{h_{1}(\ell)}(\alpha_{1}\pm
h_{1}(\ell))(\alpha_{1}\mp
h_{1}(\ell))=\frac{h_{2}(\ell)}{h_{1}(\ell)}(\alpha_{1}^{2}-h_{1}^{2}(\ell))=\gamma_{1}\gamma_{2}.$
(10.72)

I.e.
$\alpha_{1}^{+}\alpha_{2}^{-}=\alpha_{1}^{-}\alpha_{2}^{+}=\gamma_{1}\gamma_{2}$.
Hence formula (10.67) simplifies to

$\Delta^{\operatorname{Tim}}_{0}(\lambda)=(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})v_{1}^{+}v_{2}^{+}\cdot
e^{i\lambda(b_{1}+b_{2})}+(\alpha_{1}^{-}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})v_{1}^{-}v_{2}^{-}\cdot
e^{-i\lambda(b_{1}+b_{2})},\qquad\lambda\in\mathbb{C}.$ (10.73)

It follows that $\alpha_{1}^{-}=\frac{\gamma_{1}\gamma_{2}}{\alpha_{2}^{+}}$,
$\alpha_{2}^{-}=\frac{\gamma_{1}\gamma_{2}}{\alpha_{1}^{+}}$. Hence

$\alpha_{1}^{-}\alpha_{2}^{-}-\gamma_{1}\gamma_{2}=\frac{(\gamma_{1}\gamma_{2})^{2}}{\alpha_{1}^{+}\alpha_{2}^{+}}-\gamma_{1}\gamma_{2}=-\frac{\gamma_{1}\gamma_{2}}{\alpha_{1}^{+}\alpha_{2}^{+}}(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2}).$
(10.74)

With account of this observation, formula (10.73) simplifies further,

$\Delta^{\operatorname{Tim}}_{0}(\lambda)=(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})\left(v_{1}^{+}v_{2}^{+}\cdot
e^{i\lambda(b_{1}+b_{2})}-\frac{\gamma_{1}\gamma_{2}}{\alpha_{1}^{+}\alpha_{2}^{+}}v_{1}^{-}v_{2}^{-}\cdot
e^{-i\lambda(b_{1}+b_{2})}\right),\qquad\lambda\in\mathbb{C}.$ (10.75)

Since $\gamma_{1}\gamma_{2}\neq 0$, it follow from (10.72) that
$\alpha_{1}^{\pm},\alpha_{2}^{\pm}\neq 0$. Hence, using (10.72) again, taking
into account definition (10.42) of $\alpha_{k}^{\pm}$ and the fact that
$h_{1}$ is positive function, we arrive at

$\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2}=\alpha_{1}^{+}\alpha_{2}^{+}-\alpha_{1}^{-}\alpha_{2}^{+}=\alpha_{2}^{+}(\alpha_{1}^{+}-\alpha_{1}^{-})=2\alpha_{2}^{+}h_{1}(\ell)\neq
0.$ (10.76)

Combining (10.75) with (10.76), we see that
$\Delta^{\operatorname{Tim}}_{0}(\cdot)$ is not identically zero and the
sequence $\Lambda_{0}$ of its zeros is the following arithmetic progression,

$\Lambda_{0}:=\\{\lambda_{m}^{0}\\}_{m\in\mathbb{Z}},\quad\lambda_{m}^{0}:=\frac{2\pi
m-i\ln\tau_{0}}{2(b_{1}+b_{2})},\qquad\tau_{0}:=\frac{\gamma_{1}\gamma_{2}\cdot
v_{1}^{-}v_{2}^{-}}{\alpha_{1}^{+}\alpha_{2}^{+}\cdot v_{1}^{+}v_{2}^{+}}\neq
0,\qquad m\in\mathbb{Z}.$ (10.77)

It follows from (10.72) and (10.71) that

$\tau_{0}=\frac{\gamma_{1}\gamma_{2}\cdot
v_{1}^{-}v_{2}^{-}}{\alpha_{1}^{+}\alpha_{2}^{+}\cdot
v_{1}^{+}v_{2}^{+}}=\frac{\alpha_{1}^{-}\alpha_{2}^{+}\cdot
h_{1}(0)h_{2}(0)}{\alpha_{1}^{+}\alpha_{2}^{+}\cdot
h_{1}(\ell)h_{2}(\ell)}=\tau,$

where $\tau$ is given by (10.54). Asymptotical formula (10.54) now follows
from (10.77) and part (i) of the theorem.

(iv) It is clear that the polynomial $\mathcal{P}(\cdot)$ is of the following
form,

$\mathcal{P}(z)=(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})v_{1}^{+}v_{2}^{+}\cdot
z^{2(n_{1}+n_{2})}+(\alpha_{1}^{-}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})v_{1}^{-}v_{2}^{-}\\\
-(\alpha_{1}^{+}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})v_{1}^{+}v_{2}^{-}\cdot
z^{2n_{1}}-(\alpha_{1}^{-}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})v_{1}^{-}v_{2}^{+}\cdot
z^{2n_{2}},\quad\ \ \lambda\in\mathbb{C}.$ (10.78)

It follows from the definition (10.42) of $\alpha_{k}^{\pm}$ and $v_{k}^{\pm}$
and condition (10.48), that the coefficient of $\mathcal{P}(z)$ at $z^{N}$ is
non-zero,
$(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})v_{1}^{+}v_{2}^{+}\neq 0$,
and the coefficient of $\mathcal{P}(z)$ at $z^{0}$ is non-zero,
$(\alpha_{1}^{-}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})v_{1}^{-}v_{2}^{-}\neq 0$.
Hence

$\deg\mathcal{P}=N=2(n_{1}+n_{2}),\quad\text{and}\quad\mathcal{P}(0)\neq 0.$

This implies that $\mathcal{P}$ has $N$ non-zero roots $z_{1},\ldots,z_{N}$
(counting multiplicity). Therefore, the sequence $\Lambda_{0}$ of zeros of the
exponential polynomial $\Delta^{\operatorname{Tim}}_{0}(\cdot)$ given by
(10.43) is the union of $N$ arithmetic progressions,

$\Lambda_{0}=\\{\widetilde{\Lambda}_{k}^{0}\\}_{k=1}^{N},\quad\widetilde{\Lambda}_{k}^{0}:=\\{\widetilde{\lambda}_{k,m}^{0}\\}_{m\in\mathbb{Z}},\quad\widetilde{\lambda}_{k,m}^{0}:=\frac{2\pi
m}{b}-\frac{i\ln z_{k}}{b},\qquad m\in\mathbb{Z},\quad
k\in\\{1,\ldots,{N}\\}.$ (10.79)

Part (i) of the theorem now finishes the proof. ∎

Now we are ready to formulate the main result on Riesz basis property
_(without parentheses)_ for the dynamic generator $\mathcal{L}$ of the
Timoshenko beam model.

######  Theorem 10.7.

Assume conditions of Theorem 10.6(i) and let the sequence $\Lambda_{0}$ of
zeros of the exponential polynomial $\Delta^{\operatorname{Tim}}_{0}(\cdot)$
given by (10.43) be asymptotically separated (see Definition 5.13(ii)). Then
the system of root vectors of the operator $\mathcal{L}$ forms a Riesz basis
_(without parentheses)_ in the Hilbert space $\mathcal{H}$. In particular,
this is the case if one of the following conditions holds:

(a) $\gamma_{1}\gamma_{2}=0$, $\alpha_{1}\neq\pm h_{1}(\ell)$,
$\alpha_{2}\neq\pm h_{2}(\ell)$ and numbers $\tau_{1}$ and $\tau_{2}$ given by
(10.51) satisfy condition (10.52):

(b) $\gamma_{1}\gamma_{2}\neq 0$ and numbers $\alpha_{1}$ and $\alpha_{2}$
satisfy condition (10.53).

(c) $b_{1}/b_{2}\in\mathbb{Q}$ and the polynomial $\mathcal{P}(\cdot)$ given
by (10.78) has $N=2(n_{1}+n_{2})$ distinct roots.

###### Proof.

According to the proof of Theorem 10.6 operator $\mathcal{L}$ is similar to
$4\times 4$ Dirac-type operator $L_{U}(Q)$ with matrix functions $B(\cdot)$
and $Q(\cdot)$ satisfying conditions (4.3)–(4.7) (after reordering entries of
the $B(\cdot)$). It also follows from (10.68) and condition of the theorem
that the modified characteristic determinant $\widetilde{\Delta}_{0}(\cdot)$
given by (6.70) has asymptotically separated sequence of zeros (counting
multiplicity). Hence BVP (5.1)–(5.2) is strictly regular according to
Definition 6.13. Therefore, Theorem 9.11 implies that the system of root
vectors of the operator $L_{U}(Q)$ forms a Riesz basis _(without parentheses)_
in the Hilbert space $\mathfrak{H}$. Similarity of the operators $L_{U}(Q)$
and $\mathcal{L}$ implies that the system of root vectors of the operator
$\mathcal{L}$ forms a Riesz basis _(without parentheses)_ in the Hilbert space
$\mathcal{H}$. Moreover, according to parts (ii), (iii) and (iv) of Theorem
10.6, each of the conditions (a), (b), (c) imply that the modified
characteristic determinant $\widetilde{\Delta}_{0}(\cdot)$ has asymptotically
separated sequence of zeros (counting multiplicity), which finishes the proof.
∎

If $\beta_{1}(\cdot)=\beta_{2}(\cdot)=:\beta(\cdot)$ we can obtain results
similar to Theorems 10.6 and 10.7. In fact, we can obtain explicit criterion
for eigenvalues of $\mathcal{L}$ to be asymptotically separated, but in terms
of solutions of certain $2\times 2$ Cauchy problems that do not have explicit
form. Indeed, in this case $B=\operatorname{diag}(-\beta,\beta,-\beta,\beta)$
and the block diagonal of $Q$ with respect to $B$ is of the following form,

$Q_{\operatorname{diag}}:=\Theta^{-1}\begin{pmatrix}p_{1}+h_{1}^{\prime}&0&h_{2}&0\\\
0&p_{1}-h_{1}^{\prime}&0&-h_{2}\\\ -h_{2}&0&p_{2}+h_{2}^{\prime}&0\\\
0&h_{2}&0&p_{2}-h_{2}^{\prime}\end{pmatrix},\qquad x\in[0,\ell],$ (10.80)

where $\Theta(\cdot)$ is given by (10.15). Let $W$ be a solution of the
following $4\times 4$ Cauchy problem,

$W^{\prime}(x)+Q_{\operatorname{diag}}W(x)=0,\quad x\in[0,\ell],\qquad
W(0)=I_{4},$ (10.81)

Then the characteristic determinant $\Delta^{\operatorname{Tim}}_{0}(\cdot)$
has the following form

$\Delta^{\operatorname{Tim}}_{0}(\lambda)=\det(C+DW(\ell)\Phi_{0}(\ell,\lambda)),\qquad\lambda\in\mathbb{C}.$
(10.82)

One can easily see that the matrix function $W$ has a similar form to
$Q_{\operatorname{diag}}$,

$W=:\begin{pmatrix}w_{11}^{-}&0&w_{12}^{-}&0\\\ 0&w_{11}^{+}&0&w_{12}^{+}\\\
w_{21}^{-}&0&w_{22}^{-}&0\\\ 0&w_{21}^{+}&0&w_{22}^{+}\end{pmatrix},\qquad
W_{\pm}:=\begin{pmatrix}w_{11}^{\pm}&w_{12}^{\pm}\\\
w_{21}^{\pm}&w_{22}^{\pm}\\\ \end{pmatrix},$ (10.83)

where matrix functions $W_{\pm}$ are solutions of the following $2\times 2$
Cauchy problems,

$\displaystyle
W_{-}^{\prime}+\frac{1}{2}\begin{pmatrix}\frac{p_{1}+h_{1}^{\prime}}{h_{1}}&\frac{h_{2}}{h_{1}}\\\
-1&\frac{p_{2}+h_{2}^{\prime}}{h_{2}}\\\ \end{pmatrix}W_{-}=0,\qquad
x\in[0,\ell],\qquad W_{-}(0)=I_{2},$ (10.84) $\displaystyle
W_{+}^{\prime}+\frac{1}{2}\begin{pmatrix}\frac{p_{1}-h_{1}^{\prime}}{h_{1}}&-\frac{h_{2}}{h_{1}}\\\
1&\frac{p_{2}-h_{2}^{\prime}}{h_{2}}\\\ \end{pmatrix}W_{+}=0,\qquad
x\in[0,\ell],\qquad W_{+}(0)=I_{2}.$ (10.85)

This in turn implies that

$DW(\ell)=:\begin{pmatrix}0&0&0&0\\\
d_{11}^{-}&d_{11}^{+}&d_{12}^{-}&d_{12}^{+}\\\ 0&0&0&0\\\
d_{21}^{-}&d_{21}^{+}&d_{22}^{-}&d_{22}^{+}\\\
\end{pmatrix},\quad\text{where}\quad\begin{cases}d_{11}^{\pm}:=\alpha_{1}^{\pm}w_{11}^{\pm}(\ell)+\gamma_{2}w_{21}^{\pm}(\ell),\\\
d_{12}^{\pm}:=\alpha_{1}^{\pm}w_{12}^{\pm}(\ell)+\gamma_{2}w_{22}^{\pm}(\ell),\\\
d_{21}^{\pm}:=\alpha_{2}^{\pm}w_{21}^{\pm}(\ell)+\gamma_{1}w_{11}^{\pm}(\ell),\\\
d_{22}^{\pm}:=\alpha_{2}^{\pm}w_{22}^{\pm}(\ell)+\gamma_{1}w_{12}^{\pm}(\ell).\end{cases}$
(10.86)

Note that since $\beta_{1}\equiv\beta_{2}$ then

$b_{1}=b_{2}=\int_{0}^{\ell}\sqrt{\frac{\rho(t)}{K(t)}}dt=:b>0.$

With account of this observation, formulas (10.86), (10.20) and (10.63) imply
that

$C+DW(\ell)\Phi_{0}(\ell,\lambda)=\begin{pmatrix}1&1&0&0\\\
d_{11}^{-}e^{-i\lambda b}&d_{11}^{+}e^{i\lambda b}&d_{12}^{-}e^{-i\lambda
b}&d_{12}^{+}e^{i\lambda b}\\\ 0&0&1&1\\\ d_{21}^{-}e^{-i\lambda
b}&d_{21}^{+}e^{i\lambda b}&d_{22}^{-}e^{-i\lambda b}&d_{22}^{+}e^{i\lambda
b}\\\ \end{pmatrix},$ (10.87)

which in turn implies that

$\Delta^{\operatorname{Tim}}_{0}(\lambda)=(d_{11}^{+}d_{22}^{+}-d_{12}^{+}d_{21}^{+})\cdot
e^{2i\lambda b}+(d_{11}^{-}d_{22}^{-}-d_{12}^{-}d_{21}^{-})\cdot e^{-2i\lambda
b}\\\
-(d_{11}^{+}d_{22}^{-}-d_{12}^{-}d_{21}^{+})-(d_{11}^{-}d_{22}^{+}-d_{12}^{-}d_{21}^{+}).$
(10.88)

This can be rewritten as follows,

$e^{-2i\lambda
b}\Delta^{\operatorname{Tim}}_{0}(\lambda)=:\mathcal{P}(e^{2i\lambda
b}),\quad\lambda\in\mathbb{C},\qquad\mathcal{P}(z)=:d_{+}z^{2}-d_{0}z+d_{-},\quad
z\in\mathbb{C}.$ (10.89)

I.e. $\mathcal{P}(z)$ is a quadratic polynomial at $z$ and hence it has two
(possibly equal) non-zero roots $z_{1},z_{2}$. Hence the sequence of zeros of
$\Delta^{\operatorname{Tim}}_{0}(\cdot)$ is the union of two arithmetic
progressions that are asymptotically separated if and only if $z_{1}\neq
z_{2}$, which is equivalent to $d_{0}^{2}\neq 4d_{+}d_{-}$ (the discriminant
of $\mathcal{P}$ is non-zero). Summarizing all of the above observations and
following the proofs of Theorems 10.6 and 10.7 we can establish the following
result.

######  Theorem 10.8.

Let parameters $\rho,I_{\rho},K,EI,p_{1},p_{2}$ of the Timoshenko beam model
(10.1)–(10.2) satisfy conditions (10.44)–(10.45) and additionally

$\frac{K(x)}{\rho(x)}=\frac{E(x)}{I_{\rho}(x)},\qquad x\in[0,\ell],\qquad
b:=\int_{0}^{\ell}\sqrt{\frac{\rho(t)}{K(t)}}dt=\int_{0}^{\ell}\sqrt{\frac{E(t)}{I_{\rho}(t)}}dt,$
(10.90)

i.e. $\beta_{1}\equiv\beta_{2}$. Further, let parameters
$\alpha_{1},\alpha_{2},\gamma_{1},\gamma_{2}\in\mathbb{C}$ from boundary
conditions (10.4)–(10.5) satisfy “regularity” condition (10.48). Then the
following statements hold:

(i) Dynamic generator $\mathcal{L}$ of the general Timoshenko beam model
(10.1)–(10.5) has a countable sequence of eigenvalues
$\Lambda:=\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ counting multiplicity. This
sequence is the union of two sequences asymptotically close to arithmetic
progressions:

$\Lambda=\Lambda_{1}\cup\Lambda_{2},\qquad\Lambda_{k}:=\\{\lambda_{k,m}\\}_{m\in\mathbb{Z}},\quad\lambda_{k,m}:=\frac{\pi
m}{b}-\frac{i\ln z_{k}}{2b}+o(1),\qquad m\in\mathbb{Z},\quad k\in\\{1,2\\},$
(10.91)

where $z_{1}$ and $z_{2}$ are the roots of the polynomial $\mathcal{P}(\cdot)$
given by (10.89). Moreover, the system of root vectors of the operator
$\mathcal{L}$ forms a Riesz basis without parentheses in the Hilbert space
$\mathcal{H}$, where each block has a size at most two.

(ii) Sequence $\Lambda=\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ of eigenvalues of
the dynamic generator $\mathcal{L}$ is asymptotically separated (see
Definition 5.13(ii)) if and only if $z_{1}\neq z_{2}$, or equivalently if

$(d_{11}^{+}d_{22}^{-}-d_{12}^{-}d_{21}^{+}+d_{11}^{-}d_{22}^{+}-d_{12}^{-}d_{21}^{+})^{2}\neq
4(d_{11}^{+}d_{22}^{+}-d_{12}^{+}d_{21}^{+})\cdot(d_{11}^{-}d_{22}^{-}-d_{12}^{-}d_{21}^{-}).$
(10.92)

In this case the system of root vectors of the operator $\mathcal{L}$ forms a
Riesz basis _(without parentheses)_ in the Hilbert space $\mathcal{H}$.

###### Proof.

Let us only comment on the Riesz basis with parentheses property. This
property follows from Theorem 10.2, if we note that condition (10.90) implies
that $\nu\equiv 0$, where $\nu(\cdot)$ is given by (10.32). In turn, Remark
9.22(iii) explains why the block sizes are at most two in our case. ∎

######  Remark 10.9.

(i) One can easily see that

$d_{+}=d_{11}^{+}d_{22}^{+}-d_{12}^{+}d_{21}^{+}=(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})(w_{11}^{+}(\ell)w_{22}^{+}(\ell)-w_{12}^{+}(\ell)w_{21}^{+}(\ell))=(\alpha_{1}^{+}\alpha_{2}^{+}-\gamma_{1}\gamma_{2})\det
W_{+}(\ell).$

A classical Liouville’s formula (see (4.48)–(4.49)) implies that

$\det
W_{+}(\ell)=\prod_{k=1}^{2}\exp\left(-\int_{0}^{\ell}\left(\frac{p_{k}(t)}{2h_{k}(t)}-\frac{h_{k}^{\prime}(t)}{2h_{k}(t)}\right)dt\right).$

Similarly

$d_{-}=(\alpha_{1}^{-}\alpha_{2}^{-}-\gamma_{1}\gamma_{2})\det
W_{-}(\ell),\qquad\det
W_{-}(\ell)=\prod_{k=1}^{2}\exp\left(-\int_{0}^{\ell}\left(\frac{p_{k}(t)}{2h_{k}(t)}+\frac{h_{k}^{\prime}(t)}{2h_{k}(t)}\right)dt\right).$

In particular,

$d_{+}d_{-}=\bigl{(}(\alpha_{1}^{-}\alpha_{2}^{-})^{2}-(\gamma_{1}\gamma_{2})^{2}\bigr{)}\exp\left(-\int_{0}^{\ell}\left(\frac{p_{1}(t)}{h_{1}(t)}+\frac{p_{2}(t)}{h_{2}(t)}\right)dt\right)$

has a particularly simple explicit form. Unfortunately,

$d_{0}=d_{11}^{+}d_{22}^{-}-d_{12}^{-}d_{21}^{+}+d_{11}^{-}d_{22}^{+}-d_{12}^{-}d_{21}^{+}$

does not have such an explicit form, which makes formula (10.91) for the
eigenvalues asymptotic to be somewhat implicit in nature. It still represents
a sharp practical formula if one uses numeric methods to solve the Cauchy
problem.

(ii) Using Theorem 6.9 we can obtain explicit sharp asymptotic formulas for
eigenvectors of the operator $\mathcal{L}$, provided that characteristic
determinant $\Delta^{\operatorname{Tim}}_{0}(\cdot)$ has asymptotically
separated zeros. Namely, if $y$ is the eigenvector of the operator $L_{U}(Q)$,
then $\mathbf{y}=\mathcal{U}^{-1}\widetilde{\mathcal{U}}y$ is the eigenvector