# Spectral analysis of a viscoelastic tube conveying fluid with generalised
boundary conditions

Xiao Xuan Feng1,∗ ,  Mahyar Mahinzaeim2 and  Gen Qi Xu3

###### Abstract.

We study the spectral problem associated with the equation governing the small
transverse motions of a viscoelastic tube of finite length conveying an ideal
fluid. The boundary conditions considered are of general form, accounting for
a combination of elasticity and viscous damping acting on both the slopes and
the displacements of the ends of the tube. These include many standard
boundary conditions as special cases such as the clamped, free, hinged, and
guided conditions. We derive explicit asymptotic formulae for the eigenvalues
for the case of generalised boundary conditions and specialise these results
to the clamped case and the case in which damping acts on the slopes but not
on the displacements. In particular, the dependence of the eigenvalues on the
parameters of the problem is investigated and it is found that all eigenvalues
are located in certain sectorial sets in the complex plane.

MSC2010: Primary: 34L20; Secondary: 34B09, 37C10, 47E99
Keywords: viscoelastic tube conveying fluid, generalised boundary conditions,
asymptotics of eigenvalues, spectral problem
1 School of Mathematics and Information Science, Hebei Normal University of
Science and Technology, China.
2 Research Center for Complex Systems, Aalen University, Germany.
3 Department of Mathematics, Tianjin University, China.
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
∗ Corresponding author.

##  1\. Introduction

Consider a thin homogeneous horizontal tube of unit length, which is made of a
material modelled by the Kelvin–Voigt model for linear viscoelasticity. If we
assume that the tube satisfies the Euler–Bernoulli hypothesis for the
displacements and that it is subjected to no external tension, the equation
governing the small transverse motions of such a tube conveying an ideal
incompressible fluid is given by the mixed parabolic-hyperbolic partial
differential equation (see [23, Section 3.3])

$\frac{\partial^{4}\bm{w}\left(s,t\right)}{\partial
s^{4}}+\alpha\frac{\partial^{5}\bm{w}\left(s,t\right)}{\partial s^{4}\partial
t}+\eta\frac{\partial^{2}\bm{w}\left(s,t\right)}{\partial
s^{2}}+2\beta\eta^{1/2}\frac{\partial^{2}\bm{w}\left(s,t\right)}{\partial
s\partial t}+\delta\frac{\partial\bm{w}\left(s,t\right)}{\partial
t}+\frac{\partial^{2}\bm{w}\left(s,t\right)}{\partial t^{2}}=0.$ (1.1)

Here $\bm{w}\left(s,t\right)$ represents the transverse displacement of the
tube for $s\in\left[0,1\right]$, $t\in\mathbf{R}_{+}$. The parameter
$\alpha>0$ is the viscoelastic damping coefficient, $\eta\geq 0$ represents
the velocity of the fluid, and $\beta\in\left(0,1\right)$ is a parameter
depending only on the tube and fluid densities. The remaining parameter
$\delta\geq 0$ corresponds to the viscous damping due to friction from air.

The partial differential equation is supplemented with given initial
displacements and velocities at time $t=0$,

$\bm{w}\left(s,0\right)=g\left(s\right),\qquad\left.\frac{\partial\bm{w}\left(s,t\right)}{\partial
t}\right|_{t=0}=h\left(s\right),$ (1.2)

and generalised boundary conditions at $s=0$ and $s=1$, that is,

$\displaystyle\left.\dfrac{\partial^{2}\bm{w}\left(s,t\right)}{\partial
s^{2}}\right|_{s=0}$
$\displaystyle=\left.k_{01}\frac{\partial\bm{w}\left(s,t\right)}{\partial
s}\right|_{s=0}+\left.k_{02}\frac{\partial^{2}\bm{w}\left(s,t\right)}{\partial
s\partial t}\right|_{s=0},$ (1.3)
$\displaystyle-\left.\frac{\partial^{3}\bm{w}\left(s,t\right)}{\partial
s^{3}}\right|_{s=0}$
$\displaystyle=\left.k_{03}\bm{w}\left(0,t\right)\right.+\left.k_{04}\frac{\partial\bm{w}\left(s,t\right)}{\partial
t}\right|_{s=0},$ (1.4)
$\displaystyle-\left.\dfrac{\partial^{2}\bm{w}\left(s,t\right)}{\partial
s^{2}}\right|_{s=1}$
$\displaystyle=\left.k_{11}\frac{\partial\bm{w}\left(s,t\right)}{\partial
s}\right|_{s=1}+\left.k_{12}\frac{\partial^{2}\bm{w}\left(s,t\right)}{\partial
s\partial t}\right|_{s=1},$ (1.5)
$\displaystyle\left.\frac{\partial^{3}\bm{w}\left(s,t\right)}{\partial
s^{3}}\right|_{s=1}$
$\displaystyle=\left.k_{13}\bm{w}\left(1,t\right)\right.+\left.k_{14}\frac{\partial\bm{w}\left(s,t\right)}{\partial
t}\right|_{s=1},$ (1.6)

where the boundary parameters $k_{0j},k_{1j}\geq 0$, $j=1,2,3,4$. The system
comprising the partial differential equation (1.1) together with the initial
and boundary conditions (1.2)–(1.6) forms what we call the initial/boundary-
value problem.

It is not difficult to see that from the generalised boundary conditions a
variety of standard boundary conditions are obtained as special cases. For
example, the boundary conditions at $s=0$ would correspond to a clamped end
when $k_{01}=k_{03}=0$ and $k_{02},k_{04}\rightarrow\infty$, a free end when
$k_{01}=k_{02}=k_{03}=k_{04}=0$, a hinged end when $k_{01}=k_{02}=k_{03}=0$
and $k_{04}\rightarrow\infty$, or a guided end when $k_{01}=k_{03}=k_{04}=0$
and $k_{02}\rightarrow\infty$. For finite nonzero values of the $k_{0j}$, the
physical interpretation of the boundary conditions is that there are
viscoelastic supports at $s=0$, so that the bending moment and shear force are
proportional, respectively, to the slope and the displacement, as well as
their time rates of change. The boundary conditions at $s=1$ can be
interpreted similarly. (We note that, strictly speaking, while for $\alpha=0$
the boundary conditions would capture the “true” physics of the problem, for
$\alpha>0$, as assumed here, they do not; see [5] or [31] for explanation. But
this should not impede the significance of the technical results to be
presented in the paper.)

Let $\bm{w}\left(s,t\right)=e^{\lambda t}w\left(\lambda,s\right)$ so that we
can separate the variables and end up with the following fourth-order ordinary
differential equation:

$(1+\alpha\lambda)\,w^{(4)}\left(\lambda,s\right)+\eta
w^{(2)}\left(\lambda,s\right)+2\lambda\beta\eta^{1/2}w^{\prime}\left(\lambda,s\right)+(\delta\lambda+\lambda^{2})\,w\left(\lambda,s\right)=0,\\\
$ (1.7)

subject to the boundary conditions

$\displaystyle w^{(2)}\left(\lambda,0\right)-\left(k_{01}+\lambda
k_{02}\right)w^{\prime}\left(\lambda,0\right)$ $\displaystyle=0,$ (1.8)
$\displaystyle w^{(3)}\left(\lambda,0\right)+\left(k_{03}+\lambda
k_{04}\right)w\left(\lambda,0\right)$ $\displaystyle=0,$ (1.9) $\displaystyle
w^{(2)}\left(\lambda,1\right)+\left(k_{11}+\lambda
k_{12}\right)w^{\prime}\left(\lambda,1\right)$ $\displaystyle=0,$ (1.10)
$\displaystyle w^{(3)}\left(\lambda,1\right)-\left(k_{13}+\lambda
k_{14}\right)w\left(\lambda,1\right)$ $\displaystyle=0.$ (1.11)

Notice that the boundary conditions are all linearly dependent on the
eigenvalue parameter $\lambda$ when the quadruple
$\left\\{k_{02},k_{04},k_{12},k_{14}\right\\}$ is nonzero. The purpose of the
present paper is to give a detailed asymptotic analysis of the boundary-
eigenvalue problem (1.7)–(1.11) for general finite (but nonnegative) values of
the boundary parameters. Using these results we obtain explicit estimates and
asymptotic expressions for the eigenvalues. More specifically, we are able to
show for some special cases of the boundary conditions that the eigenvalues
asymptotically (for sufficiently large $\left|\lambda\right|$) lie in certain
sectorial sets in the complex plane. A rather interesting finding is that,
even for the limiting case $k_{04}=k_{14}=0$, the parameters $k_{02}$ and
$k_{12}$ have no effect on the asymptotic behaviour of the eigenvalues – a
property useful for the computation of instability regions.

A large number of papers have appeared over the last few decades dealing with
the spectral problems associated with tubes conveying fluid or related systems
(such as thin panels in supersonic flow, axially moving beams, etc.), the
emphasis usually being placed on studying the eigenvalues under variations in
the quadruple $\left\\{\alpha,\beta,\eta,\delta\right\\}$. To cite but a few,
these include [11, 22, 24, 8, 20, 10, 4, 33] and a series of papers written
from the viewpoint of operators notably by Pivovarchik [28, 29, 26, 27] and
Miloslavskii [18, 17, 16, 19] (see also the closely related papers [32, 30,
14]). In those papers the boundary conditions are considered typically to be
clamped, free, or hinged and thus are $\lambda$-independent. It should be
noted that the boundary-eigenvalue problem (1.7)–(1.11) is more difficult than
those associated with $\lambda$-independent boundary conditions, not
necessarily depending on whether $\eta\neq 0$ or not. One reason is that even
in the very special case $\alpha=\delta=\eta=0$ it is impossible to recast
(1.7)–(1.11) abstractly as a spectral problem for a linear operator in the
Hilbert space $\bm{L}_{2}\left(0,1\right)$, and one must therefore pose the
problem on an appropriately extended space. The other reason is that a rather
involved mathematical development will be required to derive explicit
asymptotic formulae for the eigenvalues of the boundary-eigenvalue problem.
Specifically, one must investigate carefully the exhibited dependencies of the
eigenvalue asymptotics on the boundary parameters
$\left\\{k_{02},k_{04},k_{12},k_{14}\right\\}$ for various parameter
assumptions in $\left\\{\alpha,\beta,\eta,\delta\right\\}$. In the present
paper, we will give a relatively simple analytical framework for dealing with
these matters.

The central focus of the aforementioned works has been on the study of
stability for solutions of the initial/boundary-value problems in the
corresponding situations, and some interesting results on the problem of
exponential stability may be found in the papers [18, 17, 30, 32]. More
recently in the papers [13, 1] there has been some effort focused on energy
considerations in state space terms to study the case of exponential stability
problems for time-varying fluid velocities (see [2] for an earlier example
with time-varying fluid velocities). In [13] the author considers the model of
the tube with $\alpha=\delta=0$ and with a viscoelastic memory kernel. The
tube is clamped at one end and friction exists at the other end, thus
providing damping through the boundary conditions. In [1] the authors analyse
the exponential stability of the tube with $\alpha=0$, hinged at one end, and
with damping at the other end. An important feature of the two papers is the
inclusion of “energy-conservative” external tension in the model, which
ultimately permits the investigation of exponential stability. (A similar
model but with both ends hinged and without the inclusion of external tension
appears in Miloslavskii’s paper [19].)

In [13, 1], fundamental questions concerning the location of the spectrum of
the boundary-eigenvalue problem and asymptotics of eigenvalues are not
considered. Results in this direction, as is well known, would offer one an
alternative point of departure by using a “spectral approach” to studying the
problem of exponential stability (refer to [17, 18, 30] for explanation). In
this vein, much of the motivation for the study of the boundary-eigenvalue
problem (1.7)–(1.11) has come from the study of exponential stability for
solutions of the initial/boundary-value problem (1.1)–(1.6). The need for such
study is obvious in view of the integral part that (1.1) has come to play in
the modelling of flow-induced oscillations for the purpose of control. The
results to be obtained in the present paper can be used to fill this important
gap that exists in the literature for the particular system studied.

The contents of this paper are as follows. In Section 2 the boundary-
eigenvalue problem (1.7)–(1.11) is defined in a suitable Hilbert product space
so that it will be possible to properly discuss it from a purely operator-
theoretic point of view. Some preliminary results on the structure of the
spectrum and location of eigenvalues of the boundary-eigenvalue problem are
also given in the section. Our main results concerning eigenvalue asymptotics
are contained in Sections 3 and 4, and in Section 5, we thought it useful to
indicate various open problems associated with the well-posedness and
stability of the initial/boundary-value problem.

##  2\. Abstract problem formulation and spectrum

In this section we formally obtain the abstract form of the boundary-
eigenvalue problem (1.7)–(1.11), and we begin by setting

$v=\left(1+\alpha\lambda\right)w$ (2.1)

where we assume throughout that $\lambda\neq-1/\alpha$ (as a separate
premise). Using (2.1) in (1.7), we obtain the coupled equations

$\displaystyle\lambda w$ $\displaystyle=\frac{v-w}{\alpha},$
$\displaystyle\lambda v$ $\displaystyle=-\alpha
v^{(4)}-2\beta\eta^{1/2}v^{\prime}-\frac{\alpha\delta-1}{\alpha}v-\alpha\eta
w^{(2)}+2\beta\eta^{1/2}w^{\prime}+\frac{\alpha\delta-1}{\alpha}w.$

Let us define the new variables

$\displaystyle\vartheta_{0}$ $\displaystyle\coloneqq
v^{\prime}\left(0\right),\qquad\xi_{0}$ $\displaystyle\coloneqq
v\left(0\right),$ $\displaystyle\vartheta_{1}$ $\displaystyle\coloneqq
v^{\prime}\left(1\right),\qquad\xi_{1}$ $\displaystyle\coloneqq
v\left(1\right).$

The boundary conditions (1.8)–(1.11), after using (2.1) together with the
above definitions, and on rearranging become

$\displaystyle\lambda k_{02}$ $\displaystyle\vartheta_{0}$ $\displaystyle=$
$\displaystyle v^{(2)}\left(0\right)-k_{01}v^{\prime}\left(0\right),$ (2.2)
$\displaystyle\lambda k_{04}$ $\displaystyle\xi_{0}$ $\displaystyle=-$
$\displaystyle v^{(3)}\left(0\right)-k_{03}v\left(0\right),$ (2.3)
$\displaystyle\lambda k_{12}$ $\displaystyle\vartheta_{1}$ $\displaystyle=-$
$\displaystyle v^{(2)}\left(1\right)-k_{11}v^{\prime}\left(1\right),$ (2.4)
$\displaystyle\lambda k_{14}$ $\displaystyle\xi_{1}$ $\displaystyle=$
$\displaystyle v^{(3)}\left(1\right)-k_{13}v\left(1\right).$ (2.5)

For later convenience we set

$z\coloneqq\left(\begin{array}[]{c}\vartheta_{0}\\\\[2.5pt] \xi_{0}\\\\[2.5pt]
\vartheta_{1}\\\\[2.5pt] \xi_{1}\end{array}\right)\in\mathbf{C}^{4}.$

Unless otherwise stated, it is understood that the (nonnegative) boundary
parameters $k_{0j}$, $k_{1j}$ are all nonzero and finite.

To recast (1.7)–(1.11) as a spectral problem for a linear operator in a
suitable function space, we, as usual, introduce the Sobolev–Hilbert spaces
$\bm{W}_{2}^{k}\left(0,1\right)$ of elements $w\in\bm{L}_{2}\left(0,1\right)$
for which $w,w^{\prime},\ldots,w^{(k-1)}$ exist and are absolutely continuous
on $\left[0,1\right]$, and $w^{(k)}\in\bm{L}_{2}\left(0,1\right)$. We then
define the product space

$\mathbb{X}\coloneqq\bm{W}^{2}_{2}\left(0,1\right)\times\bm{L}_{2}\left(0,1\right)\times\mathbf{C}^{4}$
(2.6)

which is a Hilbert space under the inner product

$\left(x,\tilde{x}\right)_{\mathbb{X}}\coloneqq\left(w,\tilde{w}\right)_{2}+\left(v,\tilde{v}\right)_{0}+\alpha\left(z,\tilde{z}\right),\quad
x,\tilde{x}\in\mathbb{X}.$ (2.7)

If $x$ and $\tilde{x}$ are elements of $\mathbb{X}$, then
$x=\left({w},{v},{z}\right)^{\top}$ and
$\tilde{x}=\left(\tilde{w},\tilde{v},\tilde{z}\right)^{\top}$ (the ⊤ denotes
the vector transpose). In (2.7) we use the inner products

$\displaystyle\left(w,\tilde{w}\right)_{2}$
$\displaystyle=\int^{1}_{0}w^{(2)}\left(s\right)\overline{\tilde{w}^{(2)}\left(s\right)}\,ds+k_{03}w\left(0\right)\overline{\tilde{w}\left(0\right)}+k_{01}w^{\prime}\left(0\right)\overline{\tilde{w}^{\prime}\left(0\right)}$
$\displaystyle\qquad+k_{13}w\left(1\right)\overline{\tilde{w}\left(1\right)}+k_{11}w^{\prime}\left(1\right)\overline{\tilde{w}^{\prime}\left(1\right)},$
$\displaystyle\left(v,\tilde{v}\right)_{0}$
$\displaystyle=\int^{1}_{0}v\left(s\right)\overline{\tilde{v}\left(s\right)}\,ds,$

and the “weighted” Euclidean inner product

$\left(z,\tilde{z}\right)=k_{02}\vartheta_{0}\overline{\tilde{\vartheta}_{0}}+k_{04}\xi_{0}\overline{\tilde{\xi}_{0}}+k_{12}\vartheta_{1}\overline{\tilde{\vartheta}_{1}}+k_{14}\xi_{1}\overline{\tilde{\xi}_{1}}=\left(Mz,\tilde{z}\right)_{\mathbf{C}^{4}}$

wherein the matrix $M$ is defined as
$M\coloneqq\operatorname{diag}\left(k_{02},k_{04},k_{12},k_{14}\right)$. We
remark that with the choice of inner product as (2.7), the norm in
$\mathbb{X}$ is related to the energy norm for the initial/boundary-value
problem (1.1)–(1.6) by

$E\left[\bm{w}\left(\,\cdot\,,t\right),\bm{v}\left(\,\cdot\,,t\right)\right]\coloneqq\frac{1}{2}\left\|\left(\begin{array}[]{c}{\bm{w}}\left(\,\cdot\,,t\right)\\\\[2.5pt]
\bm{v}\left(\,\cdot\,,t\right)\\\\[2.5pt]
\bm{z}\left(t\right)\end{array}\right)\right\|^{2}_{\mathbb{X}}=\frac{1}{2}\left\|\bm{w}\left(\,\cdot\,,t\right)\right\|^{2}_{2}+\frac{1}{2}\left\|\bm{v}\left(\,\cdot\,,t\right)\right\|^{2}_{0}+\frac{\alpha}{2}\left\|\bm{z}\left(t\right)\right\|^{2},$

where $\bm{w}\left(\,\cdot\,,t\right)$ is a solution of (1.1)–(1.6),
$\bm{v}\left(\,\cdot\,,t\right)=\left(\partial\bm{w}/\partial
t\right)\left(\,\cdot\,,t\right)$, and

$\bm{z}\left(t\right)=\left(\begin{array}[]{c}\left.\left(\partial\bm{v}/\partial
s\right)\left(s,t\right)\right|_{s=0}\\\\[2.5pt]
\bm{v}\left(0,t\right)\\\\[2.5pt] \left.\left(\partial\bm{v}/\partial
s\right)\left(s,t\right)\right|_{s=1}\\\\[2.5pt]
\bm{v}\left(1,t\right)\end{array}\right).$

It is now useful to rewrite the boundary conditions (2.2)–(2.5) in a more
appropriate form, for the purposes of this paper, by defining the boundary
operators $\Gamma$ and $\Lambda$ by

$\Gamma v=\left(\begin{array}[]{c}v^{\prime}\left(0\right)\\\\[2.5pt]
v\left(0\right)\\\\[2.5pt] v^{\prime}\left(1\right)\\\\[2.5pt]
v\left(1\right)\end{array}\right),\quad\Lambda
v=M^{-1}\left(\begin{array}[]{rcl}v^{(2)}\left(0\right)&-&k_{01}v^{\prime}\left(0\right)\\\\[2.5pt]
-v^{(3)}\left(0\right)&-&k_{03}v\left(0\right)\\\\[2.5pt]
-v^{(2)}\left(1\right)&-&k_{11}v^{\prime}\left(1\right)\\\\[2.5pt]
v^{(3)}\left(1\right)&-&k_{13}v\left(1\right)\end{array}\right),\quad
v\in{\bm{W}}^{4}_{2}\left(0,1\right).$

The boundary conditions can then be expressed as

$\Gamma v=z,\quad\Lambda v=\lambda z.$

Let us then define a new inner product in $\mathbb{X}$ by

$\left<x,\tilde{x}\right>_{\mathbb{X}}\coloneqq\alpha^{2}\left(w,\tilde{w}\right)_{2}+\left(v-w,\tilde{v}-\tilde{w}\right)_{0}+\alpha\left(z-\Gamma
w,\tilde{z}-\Gamma\tilde{w}\right),\quad x,\tilde{x}\in\mathbb{X}.$ (2.8)

This choice is motivated by the energy norm, and we see immediately that the
norm $\left\|\,\cdot\,\right\|^{\prime}_{\mathbb{X}}$ associated with the
inner product $\left<\,\cdot\,,\,\cdot\,\right>_{\mathbb{X}}$ is
(topologically) equivalent to the norm $\left\|\,\cdot\,\right\|_{\mathbb{X}}$
associated with $\left(\,\cdot\,,\,\cdot\,\right)_{\mathbb{X}}$. We will use
this new inner product throughout the paper when necessary.

Define now the linear operators $A_{0}$ and $A_{1}$ in $\mathbb{X}$ by

$\displaystyle A_{0}{x}$
$\displaystyle=\left(\begin{array}[]{c}\dfrac{v-w}{\alpha}\\\\[10.00002pt]
-\alpha v^{(4)}-\dfrac{\alpha\delta-1}{\alpha}\left(v-w\right)\\\\[10.00002pt]
\Lambda v\end{array}\right),$ (2.12) $\displaystyle A_{1}{x}$
$\displaystyle=\left(\begin{array}[]{c}0\\\\[10.00002pt] -\alpha\eta
w^{(2)}-2\beta\eta^{1/2}\left(v^{\prime}-w^{\prime}\right)\\\\[10.00002pt]
0\end{array}\right),$ (2.16)

with domains

$\displaystyle\bm{D}\left(A_{0}\right)$
$\displaystyle=\left\\{x=\left(\begin{array}[]{c}w\\\\[2.5pt] v\\\\[2.5pt]
z\end{array}\right)\in\mathbb{X}~{}\middle|~{}\begin{gathered}\left(\begin{array}[]{c}w\\\\[2.5pt]
v\\\\[2.5pt]
z\end{array}\right)\in\bm{W}^{2}_{2}\left(0,1\right)\times\bm{W}^{4}_{2}\left(0,1\right)\times\mathbf{C}^{4},\\\
\Gamma v=z\end{gathered}\right\\},$ (2.25)
$\displaystyle\bm{D}\left(A_{1}\right)$
$\displaystyle=\left\\{x=\left(\begin{array}[]{c}w\\\\[2.5pt] v\\\\[2.5pt]
z\end{array}\right)\in\mathbb{X}~{}\middle|~{}\left(\begin{array}[]{c}w\\\\[2.5pt]
v\\\\[2.5pt]
z\end{array}\right)\in\bm{W}^{2}_{2}\left(0,1\right)\times\bm{W}^{1}_{2}\left(0,1\right)\times\mathbf{C}^{4}\right\\}.$
(2.32)

Clearly $\bm{D}\left(A_{0}\right)\subset\bm{D}\left(A_{1}\right)$, and it will
be shown – see Proposition 2.2 below – that $A_{1}$ is $A_{0}$-bounded but not
$A_{0}$-compact (in the sense of [12, Section IV.1]). The boundary-eigenvalue
problem (1.7)–(1.11) is then easily verified to be equivalent to the abstract
spectral problem

$\left(\lambda I-A\right)x=0,\quad
x\in\bm{D}\left({A}\right),\quad\lambda\in\mathbf{C}\backslash\left\\{-1/\alpha\right\\},$

with $A\coloneqq A_{0}+A_{1}$, which implies that
$\bm{D}\left(A\right)=\bm{D}\left(A_{0}\right)$. So the eigenvalues of
(1.7)–(1.11) coincide (including multiplicities) with those of the operator
$A$.

Let us recall here in brief some standard notions from the spectral theory of
operators in a Hilbert space. Suppose $\lambda\mapsto\left(\lambda I-A\right)$
be a mapping from $\mathbf{C}$ into the set of closed linear operators in
$\mathbb{X}$. The resolvent set of $A$, denoted by $\varrho\left(A\right)$, is
the set of $\lambda$ for which $\lambda I-A$ is boundedly invertible (that is
$\left(\lambda I-A\right)^{-1}$ is closed and bounded). We call this inverse
$\left(\lambda I-A\right)^{-1}$ the resolvent of $A$. The spectrum
$\sigma\left(A\right)$ of $A$ is the set of $\lambda$ which are not in the
resolvent set. If a number $\lambda_{0}\in\mathbf{C}$ in the spectrum of $A$
has the property that
$\operatorname{ker}\left(\lambda_{0}I-A\right)\neq\left\\{0\right\\}$ then it
is called an eigenvalue of $A$ and there exists an eigenvector $x_{0}\neq 0$
of $A$ corresponding to $\lambda_{0}$ such that
$\left(\lambda_{0}I-A\right)x_{0}=0$. The set of all eigenvalues forms the
point spectrum of $A$.

We close this section with the following two propositions.

######  Proposition 2.1.

Let $A_{0}$ be defined by (2.12), (2.25). The following assertions hold:

  1. cmrmn(i)

$0\in\varrho\left(A_{0}\right)$ and $A_{0}^{-1}$ is not compact for
$\alpha\neq 0$.

  2. cmrmn(ii)

$A_{0}$ is maximal dissipative with respect to the inner product
$\left<\,\cdot\,,\,\cdot\,\right>_{\mathbb{X}}$ as defined by (2.8).

  3. cmrmn(iii)

$A_{0}$ generates a $C_{0}$-semigroup of bounded linear operators on
$\mathbb{X}$.

###### Proof.

In order to establish assertion i, let us consider the equation

$A_{0}x=\tilde{x}$

for any $\tilde{x}\in\mathbb{X}$ and $x\in\bm{D}\left(A_{0}\right)$, which is
equivalent to the problem posed by

$\displaystyle\frac{v-w}{\alpha}$ $\displaystyle=\tilde{w},$
$\displaystyle-\alpha v^{(4)}-\frac{\alpha\delta-1}{\alpha}\left(v-w\right)$
$\displaystyle=\tilde{v},$ $\displaystyle\left(\Lambda v\right)$
$\displaystyle=\tilde{z},$

together with the “compatibility” conditions

$\vartheta_{0}=v^{\prime}\left(0\right),\quad\xi_{0}=v\left(0\right),\quad\vartheta_{1}=v^{\prime}\left(1\right),\quad\xi_{1}=v\left(1\right).$

We will find it convenient to rephrase the problem in terms of

$v^{(4)}=-\frac{\left(\alpha\delta-1\right)\tilde{w}+\tilde{v}}{\alpha}\coloneqq\tilde{f}$
(2.33)

and the boundary conditions

$\displaystyle k_{02}$ $\displaystyle\tilde{\vartheta}_{0}$ $\displaystyle=$
$\displaystyle v^{(2)}\left(0\right)-k_{01}v^{\prime}\left(0\right),$ (2.34)
$\displaystyle k_{04}$ $\displaystyle\tilde{\xi}_{0}$ $\displaystyle=-$
$\displaystyle v^{(3)}\left(0\right)-k_{03}v\left(0\right),$ (2.35)
$\displaystyle k_{12}$ $\displaystyle\tilde{\vartheta}_{1}$ $\displaystyle=-$
$\displaystyle v^{(2)}\left(1\right)-k_{11}v^{\prime}\left(1\right),$ (2.36)
$\displaystyle k_{14}$ $\displaystyle\tilde{\xi}_{1}$ $\displaystyle=$
$\displaystyle v^{(3)}\left(1\right)-k_{13}v\left(1\right).$ (2.37)

It is immediate that the general solution of (2.33) takes the form

$v\left(s\right)=v\left(0\right)+sv^{\prime}\left(0\right)+\frac{s^{2}}{2}v^{(2)}\left(0\right)+\frac{s^{3}}{3!}v^{(3)}\left(0\right)+\int_{0}^{s}\frac{\left(s-r\right)^{3}}{3!}\tilde{f}\left(r\right)dr.$
(2.38)

Substituting (2.38) in (2.34)–(2.37) yields the following system of algebraic
equations for $v\left(0\right)$, $v^{\prime}\left(0\right)$,
$v^{(2)}\left(0\right)$, $v^{(3)}\left(0\right)$:

$\begin{split}-k_{01}v^{\prime}\left(0\right)+v^{(2)}\left(0\right)&=k_{02}\tilde{\vartheta}_{0},\\\
k_{03}v\left(0\right)+v^{(3)}\left(0\right)&=-k_{04}\tilde{\xi}_{0},\\\
k_{11}v^{\prime}\left(0\right)+\bigl{(}1+k_{11}\bigr{)}\,v^{(2)}\left(0\right)+\bigl{(}1+\frac{k_{11}}{2}\bigr{)}\,v^{(3)}\left(0\right)&\\\
&=-k_{12}\tilde{\vartheta}_{1}-\int_{0}^{1}\left(1-r\right)\tilde{f}\left(r\right)dr-
k_{11}\int_{0}^{1}\frac{\left(1-r\right)^{2}}{2}\tilde{f}\left(r\right)dr,\\\
-k_{13}v\left(0\right)-k_{13}v^{\prime}\left(0\right)-\frac{k_{13}}{2}v^{(2)}\left(0\right)+\bigl{(}1-\frac{k_{13}}{3!}\bigr{)}\,v^{(3)}\left(0\right)&\\\
&=k_{14}\tilde{\xi}_{1}-\int_{0}^{1}\tilde{f}\left(r\right)dr+k_{13}\int_{0}^{1}\frac{\left(1-r\right)^{3}}{3!}\tilde{f}\left(r\right)dr.\end{split}$

It can be shown by direct calculation that the determinant of the coefficient
matrix formed by this system of equations does not vanish, that is,

$\det\,\mathopen{\bBigg@{5.5}}(\begin{array}[]{cccc}0&-k_{01}&1&0\\\\[2.5pt]
k_{03}&0&0&1\\\\[2.5pt] 0&k_{11}&1+k_{11}&1+\frac{k_{11}}{2}\\\\[2.5pt]
-k_{13}&-k_{13}&-\frac{k_{13}}{2}&1-\frac{k_{13}}{3!}\end{array}\mathclose{\bBigg@{5.5}})\neq
0.$

(In fact, as we show,
$\operatorname{Re}\left<A_{0}x,x\right>_{\mathbb{X}}<0$.) So there is a
solution
$(v\left(0\right),v^{\prime}\left(0\right),v^{(2)}\left(0\right),v^{(3)}\left(0\right)\\!{)}^{\top}$
which is uniquely determined by $(\tilde{w},\tilde{v},\tilde{z}{)}^{\top}$.
Let $v$ be a solution of (2.33)–(2.37). Since

$w=-\alpha\tilde{w}+v,\quad
z=(v\left(0\right),v^{\prime}\left(0\right),v\left(1\right),v^{\prime}\left(1\right)\\!{)}^{\top},$

it is clear that

$x=\left(\begin{array}[]{c}w\\\\[2.5pt] v\\\\[2.5pt]
z\end{array}\right)\in\bm{D}\left(A_{0}\right),\quad
A_{0}x=\left(\begin{array}[]{c}\tilde{w}\\\\[2.5pt] \tilde{v}\\\\[2.5pt]
\tilde{z}\end{array}\right).$

Therefore,

$A_{0}^{-1}\left(\begin{array}[]{c}\tilde{w}\\\\[2.5pt] \tilde{v}\\\\[2.5pt]
\tilde{z}\end{array}\right)=\left(\begin{array}[]{c}w\\\\[2.5pt] v\\\\[2.5pt]
z\end{array}\right)=\left(\begin{array}[]{c}-\alpha\tilde{w}+v\\\\[2.5pt]
v\\\\[2.5pt] z\end{array}\right),$

implying that $A_{0}^{-1}$ is boundedly invertible and $A_{0}$ is hence closed
and densely defined. Thus $0\in\varrho\left(A_{0}\right)$, and it follows that
$A_{0}^{-1}$ is not a compact operator on $\mathbb{X}$ for $\alpha\neq 0$.

In order to prove assertion ii, we compute the real part of the inner product
$\left<A_{0}x,x\right>_{\mathbb{X}}$; a tedious calculation using integration
by parts together with the fact that $\Gamma v=z$ shows that

$\displaystyle 2\operatorname{Re}\left<A_{0}x,x\right>_{\mathbb{X}}$
$\displaystyle=\left<A_{0}x,x\right>_{\mathbb{X}}+\left<x,A_{0}x\right>_{\mathbb{X}}$
$\displaystyle=-2\alpha\int_{0}^{1}\bigl{|}v^{(2)}\left(s\right)-w^{(2)}\left(s\right)\bigr{|}^{2}ds-2\delta\int_{0}^{1}\bigl{|}v\left(s\right)-w\left(s\right)\bigr{|}^{2}ds$
$\displaystyle\qquad-2\left(k_{04}+\alpha
k_{03}\right)\bigl{|}v\left(0\right)-w\left(0\right)\bigr{|}^{2}-2\left(k_{02}+\alpha
k_{01}\right)\bigl{|}v^{\prime}\left(0\right)-w^{\prime}\left(0\right)\bigr{|}^{2}$
$\displaystyle\qquad-2\left(k_{14}+\alpha
k_{13}\right)\bigl{|}v\left(1\right)-w\left(1\right)\bigr{|}^{2}-2\left(k_{12}+\alpha
k_{11}\right)\bigl{|}v^{\prime}\left(1\right)-w^{\prime}\left(1\right)\bigr{|}^{2}.$

The assumptions $\alpha>0$, $\delta\geq 0$, and $k_{0j},k_{1j}\geq 0$,
$j=1,2,3,4$, made in the Introduction guarantee that
$\operatorname{Re}\left<A_{0}x,x\right>_{\mathbb{X}}\leq 0$ for all
$x\in\bm{D}\left(A_{0}\right)$, and thus $A_{0}$ is dissipative. The fact that
it is maximal dissipative follows from the fact that it is closed.

Assertion iii is an immediate consequence of the Lumer–Phillips theorem (for
example, see [25, Section 1.4]). ∎

######  Proposition 2.2.

Let the operator $A\coloneqq A_{0}+A_{1}$ be defined as in (2.12)–(2.32). The
following assertions hold:

  1. cmrmn(i)

The number $\lambda=-1/\alpha$ is in the continuous spectrum of $A_{0}$ and
$A$.

  2. cmrmn(ii)

$A_{1}$ is $A_{0}$-bounded but not $A_{0}$-compact, and
$A_{0}^{-1}A_{1}A_{0}^{-1}$ is compact.

  3. cmrmn(iii)

If
$\lambda\in\varrho\left(A_{0}\right)\subset\mathbf{C}\backslash\left\\{-1/\alpha\right\\}$,
then $\lambda\in\varrho\left(A\right)$; in particular, $\lambda I-A$ can be
factored in the form

$\lambda I-A=\bigl{[}I-A_{1}\left(\lambda
I-A_{0}\right)^{-1}\bigr{]}\left(\lambda I-A_{0}\right).$

  4. cmrmn(iv)

The eigenvalues of $A$ are distributed symmetrically with respect to the real
axis in the complex plane. If $\eta=0$, all eigenvalues lie in the closed left
half-plane, with at most finitely many of the eigenvalues lying in the right
half-plane in case $\eta\neq 0$.

###### Proof.

Let $x\in\bm{D}\left(A_{0}\right)$ and consider

$\left(\lambda I-A_{0}\right)x=\left(\begin{array}[]{c}\lambda
w-\dfrac{v-w}{\alpha}\\\\[10.00002pt] \lambda v+\alpha
v^{(4)}+\dfrac{\alpha\delta-1}{\alpha}\left(v-w\right)\\\\[10.00002pt] \lambda
z-\Lambda v\end{array}\right).$

Since the first row of $\left(\lambda I-A_{0}\right)x$ is the same as that of
$\left(\lambda I-A\right)x$ (because the first component of $A_{1}x$ is zero),
we clearly have, for $\lambda=-1/\alpha$,

$\operatorname{row}_{1}\,\bigl{(}-\frac{1}{\alpha}I-A_{0}\bigr{)}\,x=\operatorname{row}_{1}\,\bigl{(}-\frac{1}{\alpha}I-A\bigr{)}\,x=-\frac{v}{\alpha},$

where $v$ is an element of $\bm{W}^{4}_{2}\left(0,1\right)$ but not of
$\bm{W}^{2}_{2}\left(0,1\right)$. This proves assertion i.

In order to prove assertion ii, note first that from the proof of Proposition
2.1i it follows that

$A_{1}A_{0}^{-1}\left(\begin{array}[]{c}\tilde{w}\\\\[2.5pt]
\tilde{v}\\\\[2.5pt]
\tilde{z}\end{array}\right)=\left(\begin{array}[]{c}0\\\\[10.00002pt]
\alpha^{2}\eta\tilde{w}^{(2)}-2\alpha\beta\eta^{1/2}\tilde{w}^{\prime}-\alpha\eta{v}^{(2)}\\\\[10.00002pt]
0\end{array}\right)$

and $A_{1}A_{0}^{-1}$ is a bounded linear operator on $\mathbb{X}$; but since
$\tilde{w}\in\bm{W}^{2}_{2}\left(0,1\right)$ and
${v}\in\bm{W}^{4}_{2}\left(0,1\right)$, it is clear that $A_{1}A_{0}^{-1}$ is
not compact. Applying $A_{0}^{-1}$ to what we have calculated above one
readily sees that $A_{0}^{-1}A_{1}A_{0}^{-1}$ is compact and, moreover,
$A=A_{0}+A_{1}=(I+A_{1}A_{0}^{-1})\,A_{0}$, which is a relatively bounded
perturbation of the closed operator $A_{0}$ and hence is closed. Let us verify
this claim. If we introduce the equivalent norm
$\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\,\cdot\,\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbb{X}}$
in $\mathbb{X}$ defined by

$\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbb{X}}\coloneqq\left[a\left(\int^{1}_{0}\,\bigl{|}w^{(2)}\left(s\right)\bigr{|}^{2}\,ds+\int^{1}_{0}\,\bigl{|}w^{\prime}\left(s\right)\bigr{|}^{2}\,ds+\int^{1}_{0}\,\bigl{|}w\left(s\right)\bigr{|}^{2}\,ds\right)+b\int^{1}_{0}\,\bigl{|}v\left(s\right)\bigr{|}^{2}\,ds+\left\|z\right\|^{2}_{\mathbf{C}^{4}}\right]^{1/2}$

with $a$, $b$ positive constants, then, for any
$x\in\bm{D}\left(A_{0}\right)$, we obtain the estimate

$\displaystyle\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A_{1}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}^{2}_{\mathbb{X}}$
$\displaystyle=b\int^{1}_{0}\,\bigl{|}-\alpha\eta
w^{(2)}\left(s\right)-2\beta\eta^{1/2}\,(v^{\prime}\left(s\right)-w^{\prime}\left(s\right))\bigr{|}^{2}\,ds$
$\displaystyle\leq
2b\left(\alpha\eta\right)^{2}\int^{1}_{0}\,\bigl{|}w^{(2)}\left(s\right)\bigr{|}^{2}\,ds+8b\beta^{2}\eta\int^{1}_{0}\,\bigl{|}v^{\prime}\left(s\right)-w^{\prime}\left(s\right)\bigr{|}^{2}\,ds$
$\displaystyle\leq
2b\left(\alpha\eta\right)^{2}\int^{1}_{0}\,\bigl{|}w^{(2)}\left(s\right)\bigr{|}^{2}\,ds+\frac{8b\left(\alpha\beta\right)^{2}\eta}{a}\,a\int^{1}_{0}\left|\frac{v^{\prime}\left(s\right)-w^{\prime}\left(s\right)}{\alpha}\right|^{2}ds.$

If we choose $a$ and $b$ so that

$\frac{8b\left(\alpha\beta\right)^{2}\eta}{a}\eqqcolon\theta<1,$

if follows that

$\displaystyle
2b\left(\alpha\eta\right)^{2}\int^{1}_{0}\,\bigl{|}w^{(2)}\left(s\right)\bigr{|}^{2}\,ds$
$\displaystyle\leq\frac{2b\left(\alpha\beta\right)^{2}}{a}\,\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}^{2}_{\mathbb{X}},$
$\displaystyle\frac{8b\left(\alpha\beta\right)^{2}\eta}{a}\,a\int^{1}_{0}\left|\frac{v^{\prime}\left(s\right)-w^{\prime}\left(s\right)}{\alpha}\right|^{2}ds$
$\displaystyle\leq\theta\,\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A_{0}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}^{2}_{\mathbb{X}},$

where

$\displaystyle\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A_{0}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}^{2}_{\mathbb{X}}$
$\displaystyle=a\left(\int^{1}_{0}\,\biggl{|}\frac{v^{(2)}\left(s\right)-w^{(2)}\left(s\right)}{\alpha}\biggr{|}^{2}\,ds+\int^{1}_{0}\,\biggl{|}\frac{v^{\prime}\left(s\right)-w^{\prime}\left(s\right)}{\alpha}\biggr{|}^{2}\,ds+\int^{1}_{0}\,\biggl{|}\frac{v\left(s\right)-w\left(s\right)}{\alpha}\biggr{|}^{2}\,ds\right)$
$\displaystyle\qquad+b\int^{1}_{0}\,\biggl{|}-\alpha
v^{(4)}\left(s\right)-\frac{\alpha\delta-1}{\alpha}\left(v\left(s\right)-w\left(s\right)\right)\biggr{|}^{2}\,ds+\left\|\Lambda
v\right\|^{2}_{\mathbf{C}^{4}},$

and hence, that

$\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A_{1}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbb{X}}\leq\left(\frac{2b\left(\alpha\beta\right)^{2}}{a}\,\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}^{2}_{\mathbb{X}}+\theta\,\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A_{0}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}^{2}_{\mathbb{X}}\right)^{{1}/{2}}\leq\frac{\sqrt{2b}\,\alpha\beta}{\sqrt{a}}\,\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}x\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbb{X}}+\sqrt{\theta}\,\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A_{0}x\mathclose{\hbox{\set<EMAIL_ADDRESS>
This proves that the $A_{0}$-bound is less than one and thus completes the
proof that $A$ is closed (see [12, Theorem IV.1.1]). The operator $A$
therefore is a closed, densely defined linear operator in $\mathbb{X}$ with
noncompact resolvent.

Thus, we have verified that $\lambda\mapsto\left(\lambda I-A\right)$ is a
mapping from $\mathbf{C}$ into the set of closed linear operators in
$\mathbb{X}$. Note that if $\lambda\in\varrho\left(A_{0}\right)$, then
$\lambda\in\varrho\left(A\right)$ if and only if $I-A_{1}\left(\lambda
I-A_{0}\right)^{-1}$ is boundedly invertible, because

$\left(\lambda I-A\right)=\bigl{[}I-A_{1}\left(\lambda
I-A_{0}\right)^{-1}\bigr{]}\left(\lambda I-A_{0}\right).$

Since $\left(\lambda I-A_{0}\right)^{-1}A_{1}\left(\lambda
I-A_{0}\right)^{-1}$ is compact for each fixed
$\lambda\in\varrho\left(A_{0}\right)$, $A_{1}\left(\lambda
I-A_{0}\right)^{-1}$ is a bounded linear operator on $\mathbb{X}$ whose
spectrum consists only of eigenvalues. Therefore, in
$\mathbf{C}\backslash\left\\{-1/\alpha\right\\}$, we need only study the
eigenvalues of $A$. This proves assertion iii.

We now turn to the proof of assertion iv. Let ${x}\left(\neq 0\right)$ be an
eigenvector of $A$ corresponding to an eigenvalue ${\lambda}$. Then

$\left(\lambda I-A\right){x}=0,$

and it follows that

$\overline{\left(\lambda
I-A\right){x}}=(\overline{\lambda}I-A)\,\overline{{x}}=0,$

which means that $\overline{{x}}$ is an eigenvector corresponding to
$\overline{\lambda}$. This proves that the spectrum of $A$ is symmetric with
respect to the real axis. Now, setting $\eta=0$ so that $A_{1}=0$, we take the
inner product of $\left(\lambda I-A_{0}\right){x}$ with ${x}$ to obtain

$\left<\left(\lambda
I-A_{0}\right)x,x\right>_{\mathbb{X}}=\lambda\left\|x\right\|^{\prime\,2}_{\mathbb{X}}-\left<A_{0}x,x\right>_{\mathbb{X}}=0.$

The real part of this equation is

$\operatorname{Re}{\lambda}\left\|x\right\|^{\prime\,2}_{\mathbb{X}}-\operatorname{Re}\left<A_{0}x,x\right>_{\mathbb{X}}=0,$

so, in view of Proposition 2.1, we have that $\operatorname{Re}\lambda\leq 0$.
Finally, consider the case where $\eta\neq 0$. We recall that for ${\lambda}$
such that $\operatorname{Re}\lambda>0$, the operator $A_{1}\left(\lambda
I-A_{0}\right)^{-1}$ is not compact while, since
$\lambda\in\varrho\left(A_{0}\right)$ when $\operatorname{Re}\lambda>0$, the
operator $\bigl{[}A_{1}\left(\lambda I-A_{0}\right)^{-1}\bigr{]}^{2}$ is; and
therefore, as we have stated, $I-A_{1}\left(\lambda I-A_{0}\right)^{-1}$ is
boundedly invertible if and only if $\lambda$ is not an eigenvalue of $A$. The
assertion then follows from a more general perturbation result for holomorphic
Fredholm operator functions (see, for example, [9, Corollary XI.8.4]). This
completes the proof of the proposition. ∎

######  Remark 2.1.

By the well-established perturbation theory for semigroups (see [25, Chapter
3] or [12, Chapter IX]), the operator $A$ also generates a $C_{0}$-semigroup,
which ensures the solvability or well-posedness of the initial/boundary-value
problem (1.1)–(1.6) in $\mathbb{X}$.

##  3\. Asymptotics of eigenvalues

We begin this section by considering the boundary-eigenvalue problem
(1.7)–(1.11) for finite nonzero values of the $k_{0j}$, $k_{1j}$. Let us,
first of all, remark that by virtue of Proposition 2.2 the problem does not
have a pure point spectrum. Since we have assumed that $\lambda\neq-1/\alpha$,
we rewrite (1.7) in the form

$w^{(4)}\left(\lambda,s\right)+\frac{\eta}{1+\alpha\lambda}w^{(2)}\left(\lambda,s\right)+\frac{2\lambda\beta\eta^{1/2}}{1+\alpha\lambda}w^{\prime}\left(\lambda,s\right)+\frac{\delta\lambda+\lambda^{2}}{1+\alpha\lambda}w\left(\lambda,s\right)=0.$
(3.1)

The plane $\mathbf{C}\backslash\left\\{-1/\alpha\right\\}$ contains only
eigenvalues, to which we will restrict ourselves in this section.

It is easy to see that asymptotically, as
$\left|\lambda\right|\rightarrow\infty$, the differential equation (3.1)
becomes

${w}^{(4)}\left(\lambda,s\right)+\frac{2\beta\eta^{1/2}}{\alpha}{w}^{\prime}\left(\lambda,s\right)+\left[\frac{\lambda}{\alpha}+\frac{1}{\alpha}\left({\delta}-\frac{1}{\alpha}\right)\right]{w}\left(\lambda,s\right)=0.$
(3.2)

In both (3.1) and (3.2) it should be noted that the equation

${w}^{(4)}\left(\lambda,s\right)+\frac{\lambda}{\alpha}{w}\left(\lambda,s\right)=0$
(3.3)

represents the dominant terms and we shall use information on the eigenvalue
distribution of the boundary-eigenvalue problem posed by (3.3) and the
boundary conditions (1.8)–(1.11) to approximate the eigenvalues. This
technique has its origins in the classic papers of Birkhoff [7, 6] and is
connected with the concept of regularity of boundary-eigenvalue problems
introduced in those papers. (For further details the reader should consult the
book [21] by Naimark.)

The eigenvalues of (3.3), (1.8)–(1.11) lie in the closed left half of the
complex plane and are symmetric about the real axis (that is, those with
nonzero imaginary part occur in pairs). Therefore, we need only consider the
problem in the second quadrant of the complex plane, that is, we may assume
that $\arg\lambda\in\left[{\pi}/{2},\pi\right]$. If we set
${\lambda}/{\alpha}=\rho^{4}$, then
$\arg\rho\in\left[{\pi}/{8},{\pi}/{4}\right]$. To this end we define,
following the technique in [21, Section II.4.2], the sector $\mathcal{S}$ in
the complex plane by

$\mathcal{S}\coloneqq\left\\{\rho\in\mathbf{C}~{}\middle|~{}{\pi}/{8}\leq\arg\rho\leq{\pi}/{4}\right\\}$

and assume throughout the rest of the paper that $\rho\in\mathcal{S}$. Let us
set

$\displaystyle\omega_{1}$ $\displaystyle=e^{{3\pi}i/{4}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(i-1\right),\qquad\omega_{2}$
$\displaystyle=e^{{5\pi}i/{4}}$ $\displaystyle=-$
$\displaystyle\frac{1}{\sqrt{2}}\left(1+i\right),$ $\displaystyle\omega_{3}$
$\displaystyle=e^{{\pi}i/{4}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(1+i\right),\qquad\omega_{4}$
$\displaystyle=e^{{7\pi}i/{4}}$ $\displaystyle=-$
$\displaystyle\frac{1}{\sqrt{2}}\left(i-1\right).$

Clearly $\omega_{m}^{4}+1=0$, $m=1,2,3,4$, and we have the ordering

$\operatorname{Re}\rho\omega_{1}\leq\operatorname{Re}\rho\omega_{2}\leq\operatorname{Re}\rho\omega_{3}\leq\operatorname{Re}\rho\omega_{4}.$

It is easy to see then that the inequalities
$\operatorname{Re}\rho\omega_{1}\leq-c\left|\rho\right|$ and
$\operatorname{Re}\rho\omega_{2}\leq 0$ always hold ($c$ being a positive
constant) and asymptotically, that is for $\left|\rho\right|$ large,

$\left|e^{\rho\omega_{1}}\right|=\mathcal{O}\,(e^{-c\left|\rho\right|}),\qquad\left|e^{\rho\omega_{2}}\right|\leq
1.$ (3.4)

With these preliminaries, which will be used repeatedly in the sequel, we can
now produce asymptotic expressions for the eigenvalues of the boundary-
eigenvalue problem (3.1), (1.8)–(1.11). First we prove the following lemma.

######  Lemma 3.1.

Let $\lambda=\alpha\rho^{4}$. Then, in the sector $\mathcal{S}$, the
differential equation (3.3) has four linearly independent solutions
$e^{\rho\omega_{m}s}$, $m=1,2,3,4$. For $\left|\rho\right|$ large, the
differential equation (3.1) again has four linearly independent solutions
$w_{m}\left(\rho,s\right)$, $m=1,2,3,4$, which are analytic functions of
$\rho\in\mathcal{S}$ and whose asymptotic formulae are as follows:

$\left\\{\begin{split}w_{m}\left(\rho,s\right)&=e^{\rho\omega_{m}s}\,\Bigl{[}1+\dfrac{{W}_{m}\left(s\right)}{\rho}+\mathcal{O}\,(1/\rho^{2})\Bigr{]},\\\
w^{\prime}_{m}\left(\rho,s\right)&=\rho\omega_{m}e^{\rho\omega_{m}s}\,\Bigl{[}1+\dfrac{{W}_{m}\left(s\right)}{\rho}+\mathcal{O}\,(1/\rho^{2})\Bigr{]},\\\
w^{(2)}_{m}\left(\rho,s\right)&=\left(\rho\omega_{m}\right)^{2}e^{\rho\omega_{m}s}\,\Bigl{[}1+\dfrac{{W}_{m}\left(s\right)}{\rho}+\mathcal{O}\,(1/\rho^{2})\Bigr{]},\\\
w^{(3)}_{m}\left(\rho,s\right)&=\left(\rho\omega_{m}\right)^{3}e^{\rho\omega_{m}s}\,\Bigl{[}1+\dfrac{{W}_{m}\left(s\right)}{\rho}+\mathcal{O}\,(1/\rho^{2})\Bigr{]},\end{split}\right.$
(3.5)

where
${W}_{m}\left(s\right)=\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{m}s$.

###### Proof.

The first statement is obvious. The proof of the second statement is along the
lines of that given in [21, Section II.4.6] and is a simple calculation based
on the asymptotic expansions of the fundamental solutions of the differential
equation (3.1). Take $\left\\{w_{m}\left({\rho},s\right)\right\\}^{4}_{m=1}$
to be the fundamental system of solutions to (3.1), under the change from
$\lambda$ to $\alpha\rho^{4}$ associated with $\rho\in\mathcal{S}$, having the
asymptotic expansions

$w_{m}\left(\rho,s\right)=e^{\rho\omega_{m}s}\sum^{\infty}_{k=0}\frac{W_{mk}\left(s\right)}{\rho^{k}},\quad
m=1,2,3,4.$

On substituting in (3.1) and collecting terms according to powers of $\rho$,
we calculate

$W_{m0}\left(s\right)=1,\qquad
W_{m1}\left(s\right)=\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{m}s.$

Let us write $W_{m}\left(s\right)$ for $W_{m1}\left(s\right)$ and
$W_{m2}\left(\rho,s\right)$ for
$\sum^{\infty}_{k=2}\frac{W_{mk}\left(s\right)}{\rho^{k-2}}$. Then we see that
the functions $W_{m2}\left(\rho,s\right)$ are uniformly bounded with respect
to $s\in[0,1]$ and $\rho\in\mathcal{S}$ for
$\left|\rho\right|\geq\max\left\\{{1}/{\alpha},2\right\\}$, and we have the
desired result. ∎

With this lemma in place we can now determine the asymptotic expressions for
the eigenvalues of (3.1), (1.8)–(1.11).

######  Theorem 3.1.

Asymptotically, that is for large $n\in\mathbf{N}$, eigenvalues of the
boundary-eigenvalue problem (3.1), (1.8)–(1.11) are given by
${{\lambda}_{n}}=\alpha{\rho}_{n}^{4}$, where

${\rho}_{n}=\left(n+{1}/{2}\right)\pi\omega_{2}-i\left[\frac{1}{\alpha}\left(\frac{1}{k_{04}}+\frac{1}{k_{14}}\right)\omega_{1}+\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{\left(n+{1}/{2}\right)\pi}+\mathcal{O}\,(1/n^{2}).$
(3.6)

###### Proof.

Let $w\left(\rho,s\right)$ be an eigenfunction of (3.1), (1.8)–(1.11) under
the change from $\lambda$ to $\alpha\rho^{4}$ associated with
$\rho\in\mathcal{S}$. By Lemma 3.1,

$w\left(\rho,s\right)=a_{1}w_{1}\left(\rho,s\right)+a_{2}w_{2}\left(\rho,s\right)+a_{3}w_{3}\left(\rho,s\right)+a_{4}w_{4}\left(\rho,s\right)$
(3.7)

for constants $a_{m}$, $m=1,2,3,4$, possibly dependent of $\rho$. If (3.7) is
inserted into the boundary conditions, then we arrive at the following system
of algebraic equations:

$\displaystyle\bigl{[}w_{1}^{(2)}\left(\rho,0\right)-(k_{11}+\alpha\rho^{4}k_{12})\,w^{\prime}_{1}\left(\rho,0\right)\bigr{]}\,a_{1}+\bigl{[}w_{2}^{(2)}\left(\rho,0\right)-(k_{01}+\alpha\rho^{4}k_{02})\,w^{\prime}_{2}\left(\rho,0\right)\bigr{]}\,a_{2}$
$\displaystyle\quad+\bigl{[}w_{3}^{(2)}\left(\rho,0\right)-(k_{01}+\alpha\rho^{4}k_{02})\,w^{\prime}_{3}\left(\rho,0\right)\bigr{]}\,a_{3}+\bigl{[}w_{4}^{(2)}\left(\rho,0\right)-(k_{11}+\alpha\rho^{4}k_{12})\,w^{\prime}_{4}\left(\rho,0\right)\bigr{]}\,a_{4}$
$\displaystyle=0,$
$\displaystyle\bigl{[}w_{1}^{(3)}\left(\rho,0\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w_{1}\left(\rho,0\right)\bigr{]}\,a_{1}+\bigl{[}w_{2}^{(3)}\left(\rho,0\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w_{2}\left(\rho,0\right)\bigr{]}\,a_{2}$
$\displaystyle\quad+\bigl{[}w_{3}^{(3)}\left(\rho,0\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w_{3}\left(\rho,0\right)\bigr{]}\,a_{3}+\bigl{[}w_{4}^{(3)}\left(\rho,0\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w_{4}\left(\rho,0\right)\bigr{]}\,a_{4}$
$\displaystyle=0,$
$\displaystyle\bigl{[}w_{1}^{(2)}\left(\rho,1\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w^{\prime}_{1}\left(\rho,1\right)\bigr{]}\,a_{1}+\bigl{[}w_{2}^{(2)}\left(\rho,1\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w^{\prime}_{2}\left(\rho,1\right)\bigr{]}\,a_{2}$
$\displaystyle\quad+\bigl{[}w_{3}^{(2)}\left(\rho,1\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w^{\prime}_{3}\left(\rho,1\right)\bigr{]}\,a_{3}+\bigl{[}w_{4}^{(2)}\left(\rho,1\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w^{\prime}_{4}\left(\rho,1\right)\bigr{]}\,a_{4}$
$\displaystyle=0,$
$\displaystyle\bigl{[}w_{1}^{(3)}\left(\rho,1\right)-(k_{11}+\alpha\rho^{4}k_{12})\,w_{1}\left(\rho,1\right)\bigr{]}\,a_{1}+\bigl{[}w_{2}^{(3)}\left(\rho,1\right)-(k_{11}+\alpha\rho^{4}k_{12})\,w_{2}\left(\rho,1\right)\bigr{]}\,a_{2}$
$\displaystyle\quad+\bigl{[}w_{3}^{(3)}\left(\rho,1\right)-(k_{11}+\alpha\rho^{4}k_{12})\,w_{3}\left(\rho,1\right)\bigr{]}\,a_{3}+\bigl{[}w_{4}^{(3)}\left(\rho,1\right)+(k_{13}-\alpha\rho^{4}k_{14})\,w_{4}\left(\rho,1\right)\bigr{]}\,a_{4}$
$\displaystyle=0.$

Using the boundary values of the $w_{m}\left(\rho,s\right)$ and their
derivatives (as defined in (3.5)), the boundary conditions (1.8)–(1.11) are
seen to imply, for large $\left|\rho\right|$,

$\displaystyle
w_{m}^{(2)}\left(\rho,0\right)-(k_{01}+\alpha\rho^{4}k_{02})\,w^{\prime}_{m}\left(\rho,0\right)$
$\displaystyle=\left(\rho\omega_{m}\right)^{2}\left[1\right]_{m0}-(k_{01}+\alpha\rho^{4}k_{02})\,\rho\omega_{m}\left[1\right]_{m0},$
(3.8) $\displaystyle
w_{m}^{(3)}\left(\rho,0\right)+(k_{03}+\alpha\rho^{4}k_{04})\,w_{m}\left(\rho,0\right)$
$\displaystyle=\left(\rho\omega_{m}\right)^{3}\left[1\right]_{m0}+(k_{03}+\alpha\rho^{4}k_{04})\,\left[1\right]_{m0},$
$\displaystyle
w_{m}^{(2)}\left(\rho,1\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w^{\prime}_{m}\left(\rho,1\right)$
$\displaystyle=\left(\rho\omega_{m}\right)^{2}e^{\rho\omega_{m}}\left[1\right]_{m1}+(k_{11}+\alpha\rho^{4}k_{12})\,\rho\omega_{m}e^{\rho\omega_{m}}\left[1\right]_{m1},$
$\displaystyle
w_{m}^{(3)}\left(\rho,1\right)-(k_{13}+\alpha\rho^{4}k_{14})\,w_{m}\left(\rho,1\right)$
$\displaystyle=\left(\rho\omega_{m}\right)^{3}e^{\rho\omega_{m}}\left[1\right]_{m1}-(k_{13}+\alpha\rho^{4}k_{14})\,e^{\rho\omega_{m}}\left[1\right]_{m1},$

where we have used Birkhoff’s notation $\left[1\right]_{ms}\coloneqq
1+\frac{{W}_{m}\left(s\right)}{\rho}+\mathcal{O}\,(1/\rho^{2})$ for
$m=1,2,3,4$ and $s\in\left[0,1\right]$. By combining these with the above
system of algebraic equations, we have that $w\left(\rho,s\right)$ as given by
(3.7) is nontrivial if and only if

$\displaystyle\det\,$
$\displaystyle\mathopen{\bBigg@{5.5}}(\begin{matrix}-\alpha\rho^{4}k_{02}\rho\omega_{1}\left[1\right]_{10}&-\alpha\rho^{4}k_{02}\rho\omega_{2}\left[1\right]_{20}\\\\[5.0pt]
\bigl{[}\left(\rho\omega_{1}\right)^{3}+\alpha\rho^{4}k_{04}\bigr{]}\left[1\right]_{10}&\bigl{[}\left(\rho\omega_{2}\right)^{3}+\alpha\rho^{4}k_{04}\bigr{]}\left[1\right]_{20}\\\\[5.0pt]
\alpha\rho^{4}k_{12}\rho\omega_{1}e^{\rho\omega_{1}}\left[1\right]_{11}&\alpha\rho^{4}k_{12}\rho\omega_{2}e^{\rho\omega_{2}}\left[1\right]_{21}\\\\[5.0pt]
\bigl{[}\left(\rho\omega_{1}\right)^{3}-\alpha\rho^{4}k_{14}\bigr{]}\,e^{\rho\omega_{1}}\left[1\right]_{11}&\bigl{[}\left(\rho\omega_{2}\right)^{3}-\alpha\rho^{4}k_{14}\bigr{]}\,e^{\rho\omega_{2}}\left[1\right]_{21}\end{matrix}$
$\displaystyle\hskip
108.405pt\begin{matrix}-\alpha\rho^{4}k_{02}\rho\omega_{3}\left[1\right]_{30}&-\alpha\rho^{4}k_{02}\rho\omega_{4}\left[1\right]_{40}\\\\[5.0pt]
\bigl{[}\left(\rho\omega_{3}\right)^{3}+\alpha\rho^{4}k_{04}\bigr{]}\left[1\right]_{30}&\bigl{[}\left(\rho\omega_{4}\right)^{3}+\alpha\rho^{4}k_{04}\bigr{]}\left[1\right]_{40}\\\\[5.0pt]
\alpha\rho^{4}k_{12}\rho\omega_{3}e^{\rho\omega_{3}}\left[1\right]_{31}&\alpha\rho^{4}k_{12}\rho\omega_{4}e^{\rho\omega_{4}}\left[1\right]_{41}\\\\[5.0pt]
\bigl{[}\left(\rho\omega_{3}\right)^{3}-\alpha\rho^{4}k_{14}\bigr{]}\,e^{\rho\omega_{3}}\left[1\right]_{31}&\bigl{[}\left(\rho\omega_{4}\right)^{3}-\alpha\rho^{4}k_{14}\bigr{]}\,e^{\rho\omega_{4}}\left[1\right]_{41}\end{matrix}\mathclose{\bBigg@{5.5}})=0.$

Using (3.4) we have

$\det\,\mathopen{\bBigg@{5.5}}(\begin{matrix}\omega_{1}\left[1\right]_{10}&\omega_{2}\left[1\right]_{20}&-\omega_{2}e^{\rho\omega_{2}}\left[1\right]_{30}&0\\\\[5.0pt]
\bigl{(}\frac{\omega_{1}^{3}}{\alpha\rho
k_{04}}+1\bigr{)}\left[1\right]_{10}&\bigl{(}\frac{\omega_{2}^{3}}{\alpha\rho
k_{04}}+1\bigr{)}\left[1\right]_{20}&\bigl{(}1-\frac{\omega_{2}^{3}}{\alpha\rho
k_{04}}\bigr{)}\,e^{\rho\omega_{2}}\left[1\right]_{30}&0\\\\[5.0pt]
0&\omega_{2}e^{\rho\omega_{2}}\left[1\right]_{21}&-\omega_{2}\left[1\right]_{31}&-\omega_{1}\left[1\right]_{41}\\\\[5.0pt]
0&\bigl{(}1-\frac{\omega_{2}^{3}}{\alpha\rho
k_{14}}\bigr{)}\,e^{\rho\omega_{2}}\left[1\right]_{21}&\bigl{(}\frac{\omega_{2}^{3}}{\alpha\rho
k_{14}}+1\bigr{)}\left[1\right]_{31}&\bigl{(}\frac{\omega_{1}^{3}}{\alpha\rho
k_{14}}+1\bigr{)}\left[1\right]_{41}\end{matrix}\mathclose{\bBigg@{5.5}})=0,$

since $\omega_{3}=-\omega_{2}$ and $\omega_{4}=-\omega_{1}$. Then, noting that
$\left[1\right]_{m0}=1+\mathcal{O}\,(1/\rho^{2})$ and
$\left[1\right]_{m1}=1+\frac{{W}_{m}\left(1\right)}{\rho}+\mathcal{O}\,(1/\rho^{2})$,
we see that

$\displaystyle\det\,\mathopen{\bBigg@{5.5}}(\begin{matrix}\omega_{1}&\omega_{2}\\\\[5.0pt]
\frac{\omega_{1}^{3}}{\alpha\rho k_{04}}+1&\frac{\omega_{2}^{3}}{\alpha\rho
k_{04}}+1\\\\[5.0pt]
0&\omega_{2}e^{\rho\omega_{2}}\,\bigl{(}1+\frac{W_{2}\left(1\right)}{\rho}\bigr{)}\\\\[5.0pt]
0&\bigl{(}1-\frac{\omega_{2}^{3}}{\alpha\rho
k_{14}}\bigr{)}\,e^{\rho\omega_{2}}\,\bigl{(}1+\frac{W_{2}\left(1\right)}{\rho}\bigr{)}\end{matrix}$
$\displaystyle\hskip
108.405pt\begin{matrix}-\omega_{2}e^{\rho\omega_{2}}&0\\\\[5.0pt]
\bigl{(}1-\frac{\omega_{2}^{3}}{\alpha\rho
k_{04}}\bigr{)}\,e^{\rho\omega_{2}}&0\\\\[5.0pt]
-\omega_{2}\,\bigl{(}1+\frac{W_{3}\left(1\right)}{\rho}\bigr{)}&-\omega_{1}\,\bigl{(}1+\frac{W_{4}\left(1\right)}{\rho}\bigr{)}\\\\[5.0pt]
\bigl{(}\frac{\omega_{2}^{3}}{\alpha\rho
k_{14}}+1\bigr{)}\,\bigl{(}1+\frac{W_{3}\left(1\right)}{\rho}\bigr{)}&\bigl{(}\frac{\omega_{1}^{3}}{\alpha\rho
k_{14}}+1\bigr{)}\,\bigl{(}1+\frac{W_{4}\left(1\right)}{\rho}\bigr{)}\end{matrix}\mathclose{\bBigg@{5.5}})+\mathcal{O}\,(1/\rho^{2})=0,$

and a direct calculation yields

$e^{2\rho\omega_{2}}=-1+2\left[\frac{1}{\alpha}\left(\frac{1}{k_{04}}+\frac{1}{k_{14}}\right)\omega_{1}+\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{\rho}+\mathcal{O}\,(1/\rho^{2}),$
(3.9)

where we have used that $\omega_{1}-\omega_{2}=i\sqrt{2}$,
$\omega_{1}+\omega_{2}=-\sqrt{2}$, and $\omega_{1}\omega_{2}=1$. Set

$\rho_{n}=\left(n+{1}/{2}\right)\pi\omega_{2}+\gamma_{n},\quad
n\in\mathbf{N}.$ (3.10)

Substituting $\rho_{n}$ for $\rho$ in (3.9) we obtain that

$2\omega_{2}\gamma_{n}=2\left[\frac{1}{\alpha}\left(\frac{1}{k_{04}}+\frac{1}{k_{14}}\right)\omega_{1}+\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{\left(n+{1}/{2}\right)\pi\omega_{2}}+\mathcal{O}\,(1/n^{2})$

and thus, with $\omega_{2}^{2}=i$,

$\gamma_{n}=-i\left[\frac{1}{\alpha}\left(\frac{1}{k_{04}}+\frac{1}{k_{14}}\right)\omega_{1}+\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{\left(n+{1}/{2}\right)\pi}+\mathcal{O}\,(1/n^{2}).$
(3.11)

Combining (3.11) with (3.10) yields (3.6), completing the proof. ∎

######  Remark 3.1.

Using (3.6) we can readily calculate

$\displaystyle{\rho}^{4}_{n}$
$\displaystyle=-\Bigl{[}\left(n+{1}/{2}\right)\pi\Bigr{]}^{4}+4\left[\frac{1}{\alpha}\left(\frac{1}{k_{04}}+\frac{1}{k_{14}}\right)\right]\Bigl{[}\left(n+{1}/{2}\right)\pi\Bigr{]}^{2}$
$\displaystyle\qquad+i\left\\{4\left[\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\right]\Bigl{[}\left(n+{1}/{2}\right)\pi\Bigr{]}^{2}\right\\}+\mathcal{O}\,(n).$

######  Remark 3.2.

It is important to notice that the asymptotic expressions in Theorem 3.1 are
independent of the values of $\beta$ and $\eta$. In particular this means that
the theorem is still valid if $\eta=0$ (that is, when there is no flow).
Notably, the boundary parameters $k_{02}$ and $k_{12}$ also play essentially
no role, in the asymptotic sense, as they are at most within
$\mathcal{O}\,(n)$. We shall return to this matter in Section 4.

###  3.1. Eigenvalues for clamped boundary conditions

In this subsection, we will consider the boundary-eigenvalue problem
corresponding to the case in which $k_{0j},k_{1j}\rightarrow\infty$ for
$j=2,4$, that is, the case in which the ends of the tube are clamped. The
boundary-eigenvalue problem is posed by the same differential equation (3.1),
which we rewrite here with the eigenvalue parameter $\lambda$ replaced by
$\hat{\lambda}$:

$w^{(4)}\,(\hat{\lambda},s)+\frac{\eta}{1+\alpha\hat{\lambda}}w^{(2)}\,(\hat{\lambda},s)+\frac{2\hat{\lambda}\beta\eta^{1/2}}{1+\alpha\hat{\lambda}}w^{\prime}\,(\hat{\lambda},s)+\frac{\delta\hat{\lambda}+\hat{\lambda}^{2}}{1+\alpha\hat{\lambda}}w\,(\hat{\lambda},s)=0,$
(3.12)

but with (much simpler) boundary conditions

$\displaystyle w\,(\hat{\lambda},0)$ $\displaystyle=0,$ (3.13) $\displaystyle
w^{\prime}\,(\hat{\lambda},0)$ $\displaystyle=0,$ (3.14) $\displaystyle
w\,(\hat{\lambda},1)$ $\displaystyle=0,$ (3.15) $\displaystyle
w^{\prime}\,(\hat{\lambda},1)$ $\displaystyle=0.$ (3.16)

As in the previous case of generalised boundary conditions, we wish to produce
asymptotic formulae for the eigenvalues $\hat{\lambda}_{n}$ of the boundary-
eigenvalue problem (3.12)–(3.16). We are particularly concerned with the
relationship between these and the eigenvalues $\lambda_{n}$ of the boundary-
eigenvalue problem (3.1), (1.8)–(1.11). It is expected that for $n$ large
enough the eigenvalues $\lambda_{n}$ should tend to the eigenvalues
$\hat{\lambda}_{n}$ asymptotically as $k_{0j},k_{1j}\rightarrow\infty$ for
$j=2,4$. This property would mean that the eigenvalues depend continuously on
the boundary parameters.

Let us set ${\hat{\lambda}}/{\alpha}=\hat{\rho}^{4}$ and assume
$\hat{\rho}\in\mathcal{S}$. For (3.12) with
${\hat{\lambda}}={\alpha}\hat{\rho}^{4}$ essentially the result of Lemma 3.1
remains valid. So proceeding as in the proof of Theorem 3.1 but using the
boundary conditions (3.13)–(3.16) in place of (1.8)–(1.11), the determinantal
condition here reads

$\det\,\mathopen{\bBigg@{5.5}}(\begin{array}[]{cccc}\left[1\right]_{10}&\left[1\right]_{20}&\left[1\right]_{30}&\left[1\right]_{40}\\\\[5.0pt]
\hat{\rho}\omega_{1}\left[1\right]_{10}&\hat{\rho}\omega_{2}\left[1\right]_{20}&\hat{\rho}\omega_{3}\left[1\right]_{30}&\hat{\rho}\omega_{4}\left[1\right]_{40}\\\\[5.0pt]
e^{\hat{\rho}\omega_{1}}\left[1\right]_{11}&e^{\hat{\rho}\omega_{2}}\left[1\right]_{21}&e^{\hat{\rho}\omega_{3}}\left[1\right]_{31}&e^{\hat{\rho}\omega_{4}}\left[1\right]_{41}\\\\[5.0pt]
\hat{\rho}\omega_{1}e^{\hat{\rho}\omega_{1}}\left[1\right]_{11}&\hat{\rho}\omega_{2}e^{\hat{\rho}\omega_{2}}\left[1\right]_{21}&\hat{\rho}\omega_{3}e^{\hat{\rho}\omega_{3}}\left[1\right]_{31}&\hat{\rho}\omega_{4}e^{\hat{\rho}\omega_{4}}\left[1\right]_{41}\end{array}\mathclose{\bBigg@{5.5}})=0.$

Noting that $\omega_{3}=-\omega_{2}$ and $\omega_{4}=-\omega_{1}$, the
equation is equivalent to

$\det\,\mathopen{\bBigg@{5.5}}(\begin{array}[]{cccc}\left[1\right]_{10}&\left[1\right]_{20}&e^{\hat{\rho}\omega_{2}}\left[1\right]_{30}&e^{\hat{\rho}\omega_{1}}\left[1\right]_{40}\\\\[5.0pt]
\omega_{1}\left[1\right]_{10}&\omega_{2}\left[1\right]_{20}&-\omega_{2}e^{\hat{\rho}\omega_{2}}\left[1\right]_{30}&-\omega_{1}e^{\hat{\rho}\omega_{1}}\left[1\right]_{40}\\\\[5.0pt]
e^{\hat{\rho}\omega_{1}}\left[1\right]_{11}&e^{\hat{\rho}\omega_{2}}\left[1\right]_{21}&\left[1\right]_{31}&\left[1\right]_{41}\\\\[5.0pt]
\omega_{1}e^{\hat{\rho}\omega_{1}}\left[1\right]_{11}&\omega_{2}e^{\hat{\rho}\omega_{2}}\left[1\right]_{21}&-\omega_{2}\left[1\right]_{31}&-\omega_{1}\left[1\right]_{41}\end{array}\mathclose{\bBigg@{5.5}})=0$

which becomes, in view of (3.4),

$\det\,\mathopen{\bBigg@{5.5}}(\begin{array}[]{cccc}1&1&e^{\hat{\rho}\omega_{2}}&0\\\\[5.0pt]
\omega_{1}&\omega_{2}&-\omega_{2}e^{\hat{\rho}\omega_{2}}&0\\\\[5.0pt]
0&e^{\hat{\rho}\omega_{2}}\,\bigl{(}1+\frac{W_{2}\left(1\right)}{\hat{\rho}}\bigr{)}&1+\frac{W_{3}\left(1\right)}{\hat{\rho}}&1+\frac{W_{4}\left(1\right)}{\hat{\rho}}\\\\[5.0pt]
0&\omega_{2}e^{\hat{\rho}\omega_{2}}\,\bigl{(}1+\frac{W_{2}\left(1\right)}{\hat{\rho}}\bigr{)}&-\omega_{2}\,\bigl{(}1+\frac{W_{3}\left(1\right)}{\hat{\rho}}\bigr{)}&-\omega_{1}\,\bigl{(}1+\frac{W_{4}\left(1\right)}{\hat{\rho}}\bigr{)}\end{array}\mathclose{\bBigg@{5.5}})+\mathcal{O}\,(1/\hat{\rho}^{2})=0,$

whence

$e^{2\hat{\rho}\omega_{2}}=-1+2\left[\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{\hat{\rho}}+\mathcal{O}\,(1/\hat{\rho}^{2}),$
(3.17)

where we have again used that
$\left[1\right]_{m0}=1+\mathcal{O}\,(1/\hat{\rho}^{2})$ and
$\left[1\right]_{m1}=1+\frac{{W}_{m}\left(1\right)}{\hat{\rho}}+\mathcal{O}\,(1/\hat{\rho}^{2})$,
and that $\omega_{1}-\omega_{2}=i\sqrt{2}$ and
$\omega_{1}+\omega_{2}=-\sqrt{2}$. Setting

$\hat{\rho}_{n}=\left(n+{1}/{2}\right)\pi\omega_{2}+\hat{\gamma}_{n},\quad
n\in\mathbf{N},$

we calculate

$\hat{\gamma}_{n}=-i\left[\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{\left(n+{1}/{2}\right)\pi}+\mathcal{O}\,(1/n^{2}).$

The solutions of (3.17) are thus

${\hat{\rho}}_{n}=\left(n+{1}/{2}\right)\pi\omega_{2}-i\left[\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{\left(n+{1}/{2}\right)\pi}+\mathcal{O}\,(1/n^{2})$

and are precisely the solutions (3.6) of (3.9) when
$k_{04},k_{14}\rightarrow\infty$. From this we see that the dominant terms in
the ${\hat{\rho}}_{n}$ are the same as those in the ${\rho}_{n}$, when
$k_{04}$ and $k_{14}$ are finite. We summarise in the following result.

######  Theorem 3.2.

For large $n\in\mathbf{N}$ eigenvalues of (3.1), (1.8)–(1.11) are given as in
Theorem 3.1. If we take the limit as $k_{04},k_{14}\rightarrow\infty$, these
eigenvalues tend to those of (3.12)–(3.16) in the asymptotic sense.

##  4\. Asymptotics of eigenvalues for $k_{04}=k_{14}=0$

As pointed out in Remark 3.2 following the proof of Theorem 3.1, the boundary
parameters $k_{02}$ and $k_{12}$ do not appear in the asymptotic formulae for
the eigenvalues $\lambda_{n}$ of (3.1), (1.8)–(1.11) when $k_{04}$, $k_{14}$
are nonzero. The final problem in this paper consists of the question as to
whether or not the eigenvalues depend, asymptotically, on $k_{02}$ and
$k_{12}$ for the limiting case $k_{04}=k_{14}=0$. The boundary-eigenvalue
problem which we consider, then, is posed by (3.1) and the following boundary
conditions:

$\displaystyle w^{(2)}\left(\lambda,0\right)-\left(k_{01}+\lambda
k_{02}\right)w^{\prime}\left(\lambda,0\right)$ $\displaystyle=0,$ (4.1)
$\displaystyle w^{(3)}\left(\lambda,0\right)+k_{03}w\left(\lambda,0\right)$
$\displaystyle=0,$ (4.2) $\displaystyle
w^{(2)}\left(\lambda,1\right)+\left(k_{11}+\lambda
k_{12}\right)w^{\prime}\left(\lambda,1\right)$ $\displaystyle=0,$ (4.3)
$\displaystyle w^{(3)}\left(\lambda,1\right)-k_{13}w\left(\lambda,1\right)$
$\displaystyle=0.$ (4.4)

The following theorem is an analogue of Theorem 3.1 for the problem (3.1),
(4.1)–(4.4) and confirms that the eigenvalues $\lambda_{n}$ do not depend
asymptotically on the boundary parameters $k_{02}$ and $k_{12}$.

######  Theorem 4.1.

For large $n\in\mathbf{N}$ eigenvalues of the boundary-eigenvalue problem
(3.1), (4.1)–(4.4) are given by ${{\lambda}_{n}}=\alpha{\rho}_{n}^{4}$, where

${\rho}_{n}=n\pi\omega_{2}+i\left[\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{n\pi}+\mathcal{O}\,(1/n^{2}).$
(4.5)

###### Proof.

Defining $w\left(\rho,s\right)$ as in the proof of Theorem 3.1 (in (3.7)), we
have

$\displaystyle
w_{m}^{(2)}\left(\rho,0\right)-(k_{01}+\alpha\rho^{4}k_{02})\,w^{\prime}_{m}\left(\rho,0\right)$
$\displaystyle=\left(\rho\omega_{m}\right)^{2}\left[1\right]_{m0}-(k_{01}+\alpha\rho^{4}k_{02})\,\rho\omega_{m}\left[1\right]_{m0},$
$\displaystyle w_{m}^{(3)}\left(\rho,0\right)+k_{03}w_{m}\left(\rho,0\right)$
$\displaystyle=\left(\rho\omega_{m}\right)^{3}\left[1\right]_{m0}+k_{03}\left[1\right]_{m0},$
$\displaystyle
w_{m}^{(2)}\left(\rho,1\right)+(k_{11}+\alpha\rho^{4}k_{12})\,w^{\prime}_{m}\left(\rho,1\right)$
$\displaystyle=\left(\rho\omega_{m}\right)^{2}e^{\rho\omega_{m}}\left[1\right]_{m1}+(k_{11}+\alpha\rho^{4}k_{12})\,\rho\omega_{m}e^{\rho\omega_{m}}\left[1\right]_{m1},$
$\displaystyle w_{m}^{(3)}\left(\rho,1\right)-k_{13}w_{m}\left(\rho,1\right)$
$\displaystyle=\left(\rho\omega_{m}\right)^{3}e^{\rho\omega_{m}}\left[1\right]_{m1}-k_{13}e^{\rho\omega_{m}}\left[1\right]_{m1},$

which is just (3.8) with $k_{04}=k_{14}=0$. So using exactly the same
reasoning as in the proof of Theorem 3.1, we have that $w\left(\rho,s\right)$
is nontrivial if and only if

$\det\,\mathopen{\bBigg@{5.5}}(\begin{matrix}-\alpha\rho^{4}k_{02}\rho\omega_{1}\left[1\right]_{10}&-\alpha\rho^{4}k_{02}\rho\omega_{2}\left[1\right]_{20}&-\alpha\rho^{4}k_{02}\rho\omega_{3}\left[1\right]_{30}&-\alpha\rho^{4}k_{02}\rho\omega_{4}\left[1\right]_{40}\\\\[5.0pt]
\left(\rho\omega_{1}\right)^{3}\left[1\right]_{10}&\left(\rho\omega_{2}\right)^{3}\left[1\right]_{20}&\left(\rho\omega_{3}\right)^{3}\left[1\right]_{30}&\left(\rho\omega_{4}\right)^{3}\left[1\right]_{40}\\\\[5.0pt]
\alpha\rho^{4}k_{12}\rho\omega_{1}e^{\rho\omega_{1}}\left[1\right]_{11}&\alpha\rho^{4}k_{12}\rho\omega_{2}e^{\rho\omega_{2}}\left[1\right]_{21}&\alpha\rho^{4}k_{12}\rho\omega_{3}e^{\rho\omega_{3}}\left[1\right]_{31}&\alpha\rho^{4}k_{12}\rho\omega_{4}e^{\rho\omega_{4}}\left[1\right]_{41}\\\\[5.0pt]
\left(\rho\omega_{1}\right)^{3}e^{\rho\omega_{1}}\left[1\right]_{11}&\left(\rho\omega_{2}\right)^{3}e^{\rho\omega_{2}}\left[1\right]_{21}&\left(\rho\omega_{3}\right)^{3}e^{\rho\omega_{3}}\left[1\right]_{31}&\left(\rho\omega_{4}\right)^{3}e^{\rho\omega_{4}}\left[1\right]_{41}\end{matrix}\mathclose{\bBigg@{5.5}})=0.$

Recalling that $\omega_{3}=-\omega_{2}$ and $\omega_{4}=-\omega_{1}$, and
using (3.4) it follows immediately that

$\det\,\mathopen{\bBigg@{5.5}}(\begin{array}[]{cccc}1&1&e^{\rho\omega_{2}}&0\\\\[5.0pt]
\omega_{1}^{2}&\omega_{2}^{2}&\omega_{2}^{2}e^{\rho\omega_{2}}&0\\\\[5.0pt]
0&e^{\rho\omega_{2}}\,\bigl{(}1+\frac{W_{2}\left(1\right)}{\rho}\bigr{)}&1+\frac{W_{3}\left(1\right)}{\rho}&1+\frac{W_{4}\left(1\right)}{\rho}\\\\[5.0pt]
0&\omega_{2}^{2}e^{\rho\omega_{2}}\,\bigl{(}1+\frac{W_{2}\left(1\right)}{\rho}\bigr{)}&\omega_{2}^{2}\,\bigl{(}1+\frac{W_{3}\left(1\right)}{\rho}\bigr{)}&\omega_{1}^{2}\,\bigl{(}1+\frac{W_{4}\left(1\right)}{\rho}\bigr{)}\end{array}\mathclose{\bBigg@{5.5}})+\mathcal{O}\,(1/\rho^{2})=0,$

and a direct calculation gives

$e^{2\rho\omega_{3}}=1-2\left[\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{\rho}+\mathcal{O}\,(1/\rho^{2}).$

Set

$\rho_{n}=n\pi\omega_{2}+\gamma_{n},\quad n\in\mathbf{N}.$

Using the procedure leading to Theorem 3.1, we arrive at

${\gamma}_{n}=i\left[\frac{1}{4}\sqrt{\frac{1}{\alpha}}\left(\delta-\frac{1}{\alpha}\right)\omega_{2}\right]\frac{1}{n\pi}+\mathcal{O}\,(1/n^{2});$

so the $\rho_{n}$ are as given in (4.5) and the theorem is proven. ∎

##  5\. Further remarks and open problems

In this paper we have been primarily concerned with studying the boundary-
eigenvalue problem (1.7)–(1.11) and the asymptotics of eigenvalues in the
plane $\mathbf{C}\backslash\left\\{-1/\alpha\right\\}$. What we have
effectively shown in Sections 3 and 4 is that if $n$ is large enough in the
sequence $\left\\{\lambda_{n}\right\\}$, then
$\left|\operatorname{Im}\lambda_{n}\right|/\left|\operatorname{Re}\lambda_{n}\right|\leq
1$ and hence the eigenvalues lie in the sectorial set

$\left\\{z\in\mathbf{C}\backslash\left\\{-1/\alpha\right\\}~{}\middle|~{}\left|\operatorname{Im}\left(z-z_{0}\right)\right|\leq
r\left|\operatorname{Re}\left(z-z_{0}\right)\right|,~{}\operatorname{Re}\left(z-z_{0}\right)\leq
0\right\\},$

with $r$ and $z_{0}$ certain positive constants, $r$ depending on the boundary
parameters $k_{04}$, $k_{14}$ and the viscoelastic damping coefficient
$\alpha$, for all values of the other parameters. In particular, from the
results it is readily seen that the only possible accumulation points of the
eigenvalues in the extended complex plane are $-1/\alpha$ and infinity.

Among the many somewhat related matters that we have not been able to address
in this paper, one is the “freedom” in the choice of the space $\mathbb{X}$ in
which to recast (1.7)–(1.11) as a spectral problem for a linear operator.
Typically, the space
$\bm{W}^{2}_{2}\left(0,1\right)\times\bm{L}_{2}\left(0,1\right)$ would be an
alternative natural choice. Here, however, the situation is more complicated
since the inner product in it induces only a seminorm and it is necessary to
define $\mathbb{X}$ as a suitable subspace, or a quotient space, of
$\bm{W}^{2}_{2}\left(0,1\right)\times\bm{L}_{2}\left(0,1\right)$.

Let us return to the initial/boundary-value problem (1.1)–(1.6) formulated in
the Introduction, which we used to motivate our investigation. With the choice
of state space as (2.6), it is not difficult to check that the problem is
equivalent to the abstract initial-value problem

$\dot{{x}}\left(t\right)=A{{x}}\left(t\right),\qquad A\coloneqq
A_{0}+A_{1},\qquad{{x}}\left(0\right)=x_{0},$ (5.1)

for some given initial point $x_{0}\in\bm{D}\left(A_{0}\right)$, where the
state ${{x}}\left(t\right)$ can be represented formally by the column vector
$(\bm{w}\left(\,\cdot\,,t\right),\bm{v}\left(\,\cdot\,,t\right),\left.\left(\partial\bm{v}/\partial
s\right)\left(s,t\right)\right|_{s=0},\bm{v}\left(0,t\right),\left.\left(\partial\bm{v}/\partial
s\right)\left(s,t\right)\right|_{s=1},\bm{v}\left(1,t\right)\\!{)}^{\top}$. In
Section 2, we have disposed of the question of well-posedness of (5.1) by
showing that, under our assumptions, the system operator $A$ generates a
$C_{0}$-semigroup of bounded linear operators on $\mathbb{X}$. An important
question which remains is whether that semigroup is, in fact, holomorphic. To
our knowledge this question has not been discussed for our initial/boundary-
value problem.

For the problem of exponential stability of (solutions of) (5.1) it is
necessary to investigate in detail what assumptions must be placed on the
quadruples $\left\\{\alpha,\beta,\eta,\delta\right\\}$ and
$\left\\{k_{02},k_{04},k_{12},k_{14}\right\\}$ so that there exist constants
$M,\varepsilon>0$ such that

$\left\|{{x}}\left(t\right)\right\|^{\prime}_{\mathbb{X}}\leq Me^{-\varepsilon
t}\left\|x_{0}\right\|^{\prime}_{\mathbb{X}},$

where we recall that $\left\|\,\cdot\,\right\|^{\prime}_{\mathbb{X}}$ is
equivalent to the energy norm for the initial/boundary-value problem
(1.1)–(1.6). This question has been more or less settled in the papers [18,
17, 32, 30] for the initial/boundary-value problem involving the standard
boundary conditions of clamped, free, or hinged ends, and combinations
thereof. For the case of generalised boundary conditions (in the sense as it
is understood here) the problem is still open and requires a thorough
investigation, among other things, of the Riesz basis property of the root
vectors (eigen- and associated vectors) of the system operator $A$.

There are numerous problems in mechanics and mathematical physics which give
rise to initial/boundary-value problems on networks (a good overview of
several examples has very recently been given in [3]). Their stability
properties can be studied by using recent developments in the mathematical
analysis of differential equations on geometric graphs, along the lines, for
example, of [15]. There we studied the exponential stability problem for a
stretched Euler–Bernoulli beam (that is $\alpha=\beta=\delta=0$ and $\eta<0$)
on a star graph with three identical edges. However, it is not clear whether
the methods in [15] can be so modified as to encompass star-shaped networks of
tubes conveying fluid. We plan to examine this, and other related questions,
in a subsequent paper.

Acknowledgments. This work was supported in part by the National Natural
Science Foundation of China under Grant NSFC-61773277.

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# On the modularity of elliptic curves over the cyclotomic
$\mathbb{Z}_{p}$-extension of some real quadratic fields

Xinyao Zhang

###### Abstract.

The modularity of elliptic curves always intrigues number theorists. Recently,
Thorne had proved a marvelous result that for a prime $p$, every elliptic
curve defined over a $p$-cyclotomic extension of $\mathbb{Q}$ is modular. The
method is to use some automorphy lifting theorems and study non-cusp points on
some specific elliptic curves by Iwasawa theory for elliptic curves. Since the
modularity of elliptic curves over real quadratic was proved, one may ask
whether it is possible to replace $\mathbb{Q}$ with a real quadratic field
$K$. Following Thorne’s idea, we give some assumptions first and prove the
modularity of elliptic curves over the $\mathbb{Z}_{p}$-extension of some real
quadratic fields.

††footnotetext: Key words and phrases. Modularity, Iwasawa theory for elliptic
curves.

##  1\. Introduction

Let $E_{1}$, $E_{2}$ be two elliptic curves given by the following Weierstrass
equations.

$E_{1}:y^{2}+xy+y=x^{3}+x^{2}-10x-10.$ $E_{2}:y^{2}+xy+y=x^{3}+x^{2}-5x+2.$

In order to state our main theorem, we give the following assumptions first.

######  Assumptions.

Let $i=1,2$. Let $K$ be a real quadratic field, $p$ an odd prime satisfying
1\. $K\neq\mathbb{Q}(\sqrt{5})$,
2\. The Mordell-Weil group $E_{i}(K)$ is finite,
3\. The $p$-Tate-Shafarevich group X$(E_{i}/K)_{p}$ is trivial,
4\. $p$ splits completely in $K$.

Our goal in this article is to prove the following result:

######  Theorem 1.1.

Let $K$ and $p$ satisfy the Assumptions. Let $F$ be a number field contained
in the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Let $E$ be an elliptic
curve over $F$. Then $E$ is modular.

In recent years, a number of results about modularity or automorphy lifting
theorems have been proved. As arithmetic consequences, the modularity of
elliptic curves over some specific fields can be verified. By studying non-
cuspidal points on some modular curves, the following theorems had been
proved:

######  Theorem 1.2 ([FLHS15], Theorem 1).

Let $E$ be an elliptic curve over a real quadratic field $K$. Then $E$ is
modular.

######  Theorem 1.3 ([DNS20], Theorem 4).

Let $K$ be a totally real cubic number field. Let $E$ be an elliptic curve
over $K$. Then $E$ is modular.

For more discussions, one can see [Hun13].

Using Iwasawa theory for elliptic curves, Thorne proved the following
remarkable result:

######  Theorem 1.4 ([Tho19], Theorem 1).

Let $p$ be a prime, and let $F$ be a number field which is contained in the
cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$. Let $E$ be an elliptic
curve over $F$. Then $E$ is modular.

To prove Theorem 1.1, our strategy is to extend Thorne’s method, but there are
some obstrucions to study the $K_{n}$-points of $E_{i}$, where $K_{n}$ is the
$p$-cyclotomic extension of a real quadratic field $K$ satisfying
$[K_{n}:K]=p^{n}$. In Thorne’s proof, a key point is the fact that the Selmer
group $\operatorname{Sel}(E_{i}/\mathbb{Q})_{p}$ is trivial and then one can
use some results in Iwasawa theory. This is the reason why we first assume
that the Mordell-Weil group $E_{i}(K)$ is finite and the $p$-Tate-Shafarevich
group X$(E_{i}/K)_{p}$ is trivial. Under these assumptions, we use some more
general results in Iwasawa theory for elliptic curves to prove our result.

##  2\. Some data

In this section, we will give some data of $E_{i}$, most of which are from
[LMF22].

Information | $E_{1}$ | $E_{2}$
---|---|---
Mordell-Weil group | $\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$
Conductor | $15$ | $15$
$j$-invariant | $\frac{111284641}{50625}=3^{-4}\cdot 5^{-4}\cdot 13^{3}\cdot 37^{3}$ | $\frac{13997521}{225}=3^{-2}\cdot 5^{-2}\cdot 241^{3}$
Reduction type | semistable | semistable
| split multiplicative: $5$ | split multiplicative: $5$
| non-split multiplicative: $3$ | non-split multiplicative: $3$
Tamagawa product | $8$ | $4$
Analytic order of X | $1$ | $1$
Growth of $2^{\infty}$-torsion | $\mathbb{Q}(\sqrt{5}):\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{8}\mathbb{Z}$ | $\mathbb{Q}(\sqrt{5}):\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{8}\mathbb{Z}$
| $\mathbb{Q}(i):\mathbb{Z}/{4}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $\mathbb{Q}(i,\sqrt{15}):\mathbb{Z}/{4}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$

Table 1. Some data of $E_{i}$.

From table 1, we can obtain:

######  Lemma 2.1.

The Tamagawa product $\operatorname{Tam}(E_{i}/K)$ is a power of $2$.

###### Proof.

If $E_{i}$ has nonsplit multiplicative reduction at some $v$, the Tamagawa
factor at $v$ is either $1$ or $2$ (see [Sil09], Appendix C, proof of
Corollary 15.2.1). If $E_{i}$ has split multiplicative reduction at some $v$,
by Kodaira and Néron’s theorem ([Sil09], Theorem 6.1), the Tamagawa factor at
$v$ is a power of $2$ from the decomposition of the $j$-invariant. The lemma
holds since $E_{i}$ is semistable.

∎

######  Lemma 2.2.

Under our assumptions, $E_{i}(K)[2^{\infty}]=E_{i}(K_{\infty})[2^{\infty}]$,
where $K_{\infty}$ is the cyclotomic $\mathbb{Z}_{p}$-extension of $K$.

###### Proof.

Let $\rho_{E_{i},2}$ be the Galois representation associated to the Tate
module $T_{2}E_{i}$. As the elements in $E_{i}[2]$ are $\mathbb{Q}$-rational,
the image of the absolute Galois group $G_{\mathbb{Q}}$ is congruent to the
identity element$\mod 2\mathbb{Z}_{2}$ and thus a pro-$2$ group. As $K$ and
$K_{\infty}$ are both Galois extension of $\mathbb{Q}$, the index of $G_{K}$
and $G_{K_{\infty}}$ are both a power of $2$. Then the lemma holds since $p$
is an odd prime. ∎

######  Remark 2.3.

In fact, under our assumptions, we have
$E_{i}(\mathbb{Q})[2^{\infty}]=E_{i}(K_{\infty})[2^{\infty}]$ from the growth
of $2^{\infty}$-torsion of $E_{i}$ in number fields given above.

##  3\. The proof

For a number field $F$, we denote the absolute Galois group by $G_{F}$. For an
elliptic curve $E$ over $F$ and a rational prime $l$, let
$\bar{\rho}_{E,l}:G_{F}\to\operatorname{GL_{2}}(\mathbb{F}_{l})$ be the Galois
representation associated to the $l$-torsion points of $E$. Let $\zeta_{l}$ be
a primitive $l$-th root of unity.

######  Theorem 3.1 ([Tho19], Theorem 2).

Let $E$ be an elliptic curve over a totally real number field $F$, and suppose
that (at least) one of the following is true:
1\. The representation $\bar{\rho}_{E,3}|_{G_{F(\zeta_{3})}}$ is absolutely
irreducible.
2\. $\sqrt{5}\notin F$, and $\bar{\rho}_{E,5}$ is irreducible.
Then $E$ is modular.

Following [FLHS15] and [Tho19], we consider modular curves $X(s3,b5)$ and
$X(b3,b5)$. The next corollary is an analogue to [[Tho19], Lemma 3] .

######  Corollary 3.2.

Let $F$ be a totally real field such that $\sqrt{5}\notin F$.
1\. If $E/F$ is an elliptic curve which is not modular, then $E$ determines an
$F$-rational point of one of the curves $X(s3,b5)$, $X(b3,b5)$.
2\. Recall that $K$ is a real quadratic field satisfying
$K\neq\mathbb{Q}(\sqrt{5})$. If $F/K$ is cyclic and $X(s3,b5)(F)=X(s3,b5)(K)$,
$X(b3,b5)(F)=X(b3,b5)(K)$, then all elliptic curves over $F$ are modular.

###### Proof.

The first assertion is a consequence of Theorem 3.1 and [[FLHS15], Proposition
4.1]. The second part follows from the first one, Theorem 1.2 and base change
for $\operatorname{GL_{2}}$ (see [Lan80]). ∎

######  Proposition 3.3 ([Tho19], Proposition 4).

The modular curves $X(b3,b5)$, $X(s3,b5)$ are isomorphic over $\mathbb{Q}$ to
$E_{1}$, $E_{2}$, respectively.

######  Remark 3.4.

$E_{1}$ and $E_{2}$ are elliptic curves of Cremona label $15A1$ and $15A3$,
respectively. See [Cre97].

######  Lemma 3.5.

Let $l$ be an odd prime. Under our assumptions, we have
$E_{i}(\mathbb{Q})[l^{\infty}]=E_{i}(K_{\infty})[l^{\infty}]$.

###### Proof.

Considering the Galois representation
$\bar{\rho}_{E,l}:G_{\mathbb{Q}}\to\operatorname{GL_{2}}(\mathbb{F}_{l})$, we
know that $\bar{\rho}_{E,l}$ is surjective (see [[Tho19], proof of Theorem 5]
or [[Yos16], Lemma 2.3]). If $K_{\infty}$ has an $l$-torsion point, then the
order of $\bar{\rho}_{E,l}|_{G_{K_{\infty}}}$ is a factor of $l(l-1)$. Since
$[K:\mathbb{Q}]=2$ and $K_{\infty}$ is the $\mathbb{Z}_{p}$-extension of $K$,
the index of $\bar{\rho}_{E,l}|_{G_{K_{\infty}}}$ in
$\operatorname{GL_{2}}(\mathbb{F}_{l})$ is either $p^{n}$ or $2p^{n}$ for some
$n$. It is known that the order of the finite group
$\operatorname{GL_{2}}(\mathbb{F}_{l})$ is $l(l+1)(l-1)^{2}$, so we have
$(l+1)(l-1)\mid 2p^{n}$. The only possibility is $p=2$, but it is against our
assumptions. ∎

######  Corollary 3.6.

Under our assumptions, the $p$-Selmer group $\operatorname{Sel}(E_{i}/K)_{p}$
is trivial.

###### Proof.

It is a direct consequence of Lemma 3.5. ∎

To prove the next proposition, we will use Iwasawa theory. We denote a
generator of the characteristic ideal of the Pontryagin dual of the Selmer
group $\operatorname{Sel}(E_{i}/K_{\infty})_{p}$ by $f_{E_{i}}$. Let the
$\lambda$-invariant $\lambda_{E_{i}}$ be the $\mathbb{Z}_{p}$-corank of
$\operatorname{Sel}(E_{i}/K_{\infty})_{p}$.

######  Proposition 3.7.

Under our assumptions, we have $E_{i}(K)=E_{i}(K_{\infty})$.

###### Proof.

By Lemma 2.2 and Lemma 3.5, we just need to prove that $E_{i}(K_{\infty})$ is
finite. Since $p$ splits completely in $K$, we can divide the proof into three
cases.

Case 1. $E_{i}$ has good ordinary reduction at all primes lying above $p$.

In this case, we have the formula [[Gre99], Theorem 4.1]:

$f_{E_{i}}(0)\sim\left(\prod_{v~{}\mathrm{bad}}c_{v}^{(p)}\right)\left(\prod_{v\mid
p}\left|\tilde{E_{i}}_{,v}\left(\mathbb{F}_{v}\right)_{p}\right|^{2}\right)\left|\operatorname{Sel}(E_{i}/K)_{p}\right|/\left|E_{i}(K)_{p}\right|^{2}.$

Here $a\sim b$ when $a$ and $b$ have the same $p$-adic valuation, the highest
power of $p$ dividing the Tamagawa factor $c_{v}$ at $v$ is $c_{v}^{(p)}$, and
$\mathbb{F}_{v}$ is the residue field at $v$.

As $|E_{i}(\mathbb{Q})|=8$, if $p$ divides $|E_{i}(\mathbb{F}_{p})|$ , then
$8p$ divides $|E_{i}(\mathbb{F}_{p})|$ and it contradicts the Hasse bound.
Since $p$ splits completely in $K$, we have
$p\nmid|\tilde{E_{i}}_{,v}(\mathbb{F}_{v})_{p}|$. By Lemma 2.1 and Corollary
3.6, $f_{E_{i}}(0)$ is a $p$-adic unit and $\lambda_{E_{i}}=0$. Hence,
$E_{i}(K_{\infty})$ is finite.

Case 2. $E_{i}$ has multiplicative reduction at all primes lying above $p$.

In this case, we have the formula [[Gre99], pp. 92-93]:

$f_{E_{i}}(0)\sim\left(\prod_{v\mid
p}l_{v}\right)\left(\prod_{v~{}\mathrm{bad}}c_{v}^{(p)}\right)\left|\operatorname{Sel}(E_{i}/K)_{p}\right|/\left|E_{i}(K)_{p}\right|^{2}$

If $E_{i}$ has non-split multiplicative reduction at $v$, we can take
$l_{v}=2$. If $E_{i}$ has split multiplicative reduction at $v$, we can take

$l_{v}=\frac{\log_{p}(N_{{K_{v}}/\mathbb{Q}_{p}}(q_{E_{i}}^{(v)}))}{\operatorname{ord}_{p}(N_{K_{v}/\mathbb{Q}_{p}}(q_{E_{i}}^{(v)}))}\cdot\frac{[K_{v}\cap\mathbb{Q}_{p}^{\mathrm{unr}}:\mathbb{Q}_{p}]}{2p[K_{v}\cap\mathbb{Q}_{p}^{\mathrm{cyc}}:\mathbb{Q}_{p}]}$

Here $q_{E_{i}}$ is the Tate period, and $\mathbb{Q}_{p}^{\mathrm{unr}}$ and
$\mathbb{Q}_{p}^{\mathrm{cyc}}$ are the maximal unramified extension and the
cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}_{p}$, respectively. From
[[Tho19], proof of Theorem 5], we know that
$\frac{\log_{p}(N_{{K_{v}}/\mathbb{Q}_{p}}(q_{E_{i}}^{(v)}))}{\operatorname{ord}_{p}(N_{K_{v}/\mathbb{Q}_{p}}(q_{E_{i}}^{(v)}))}$
lies in $p\mathbb{Z}_{p}^{\times}$ since $p$ splits completely in $K$. By
Lemma 2.1 and Corollary 3.6 and the formula, $E_{i}(K_{\infty})$ is finite.

Case 3. $E_{i}$ has good supersingular reduction at all primes lying above
$p$.

In this case, our assertion follows from [[IP06], Theorem 5.1] as $a_{p}=0$
under our assumptions. ∎

Theorem 1.1 follows from Corollary 3.2 and Proposition 3.7.

At last, we give some data from [LMF22].

Base field | Mordell-Weil group | Tamagawa product | Analytic order of X
---|---|---|---
$\mathbb{Q}(\sqrt{2})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:16,~{}E_{2}:4$ | $E_{i}:1$
$\mathbb{Q}(\sqrt{6})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:16,~{}E_{2}:4$ | $E_{i}:1$
$\mathbb{Q}(\sqrt{17})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:16,~{}E_{2}:4$ | $E_{i}:1$
$\mathbb{Q}(\sqrt{21})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:32,~{}E_{2}:8$ | $E_{i}:1$
$\mathbb{Q}(\sqrt{53})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:16,~{}E_{2}:4$ | $E_{i}:1$
$\mathbb{Q}(\sqrt{61})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:64,~{}E_{2}:16$ | $E_{i}:1$
$\mathbb{Q}(\sqrt{65})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:32,~{}E_{2}:8$ | $E_{i}:1$
$\mathbb{Q}(\sqrt{69})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:32,~{}E_{2}:8$ | $E_{i}:1$
$\mathbb{Q}(\sqrt{77})$ | $E_{i}:\mathbb{Z}/{2}\mathbb{Z}\times\mathbb{Z}/{4}\mathbb{Z}$ | $E_{1}:16,~{}E_{2}:4$ | $E_{i}:4$

Table 2. $E_{i}$ in some real quadratic fields.

From table 2, we can find many real quadratic fields and infinitely many odd
primes satisfying our assumptions.

######  Acknowledgements.

The author would like to express appreciation to his supervisor Professor
Takeshi Saito for his comments and suggestions. Also, the author would like to
thank Professor Jack Thorne and Dr. Sho Yoshikawa for some helpful
communications.

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Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro,
Tokyo 153-8914, Japan

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