# Spectral enclosures for the damped elastic wave equation

Biagio Cassano  Dipartimento di Matematica e Fisica, Università degli Studi
della Campania “Luigi Vanvitelli”, Viale Lincoln 5, 81100 Caserta, Italy;
<EMAIL_ADDRESS>Lucrezia Cossetti  Fakultät für Mathematik,
Institut für Analysis, Karlsruher Institut für Technologie (KIT), Englerstraße
2, 76131 Karlsruhe, Germany<EMAIL_ADDRESS>Luca Fanelli
Ikerbasque $\&$ Departamento de Matematicas, Universidad del País Vasco/Euskal
Herriko Unibertsitatea (UPV/EHU), Aptdo. 644, 48080, Bilbao, Spain;
<EMAIL_ADDRESS>
(17 August 2021)

###### Abstract

In this paper we investigate spectral properties of the damped elastic wave
equation. Deducing a correspondence between the eigenvalue problem of this
model and the one of Lamé operators with non self-adjoint perturbations, we
provide quantitative bounds on the location of the point spectrum in terms of
suitable norms of the damping coefficient.

##  1 Introduction

This paper is concerned with the _damped elastic wave equation_

$u_{tt}+a(x)u_{t}-\Delta^{\ast}u=0,\qquad(x,t)\in\mathbb{R}^{d}\times(0,\infty),$
(1)

Here $a\colon\mathbb{R}^{d}\to\mathbb{C}^{d\times d}$ denotes the damping
coefficient assumed to be a (possibly) non hermitian matrix. We shall make the
standard assumption of a bounded damping, _i.e._ $a\in
L^{\infty}(\mathbb{R}^{d})^{d}.$ The symbol $-\Delta^{\ast}$ is used to denote
the Lamé operator of elasticity which is a matrix-valued differential operator
acting, w.r.t. the spacial variable $x\in\mathbb{R}^{d}$ on smooth vector
fields as

$-\Delta^{\ast}u=-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u,\qquad
u\in
C^{\infty}_{0}(\mathbb{R}^{d})^{d}:=C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{C}^{d}).$
(2)

The material-dependent Lamé parameters $\lambda,\mu\in\mathbb{R}$ are assumed
to satisfy the ellipticity condition

$\mu>0,\quad\lambda+\mu\geq 0.$ (3)

It is customarily to write the second-order evolution system (1) as a doubled
first-order system introducing the vector field $U=(u,u_{t})^{T}.$ Then (1)
can be rewritten as $U_{t}=A^{\ast}_{a}U,$ where $A_{a}^{\ast}$ is the
$2d\times 2d$ matrix-valued _damped elastic wave operator_ defined as

$A_{a}^{\ast}:=\begin{pmatrix}0&1\\\
\Delta^{\ast}&-a\end{pmatrix},\qquad\mathcal{D}(A_{a}^{\ast}):=H^{2}(\mathbb{R}^{d})^{d}\times\dot{H}^{1}(\mathbb{R}^{d})^{d}.$
(4)

The damped elastic wave equation (1) and the corresponding damped operator (4)
have attracted considerable attention in the last decades. In the constant
coefficient case, namely $a(x)=\alpha,$ $\alpha>0,$ Bocanegra-Rodríguez et al.
[10] considered the longtime dynamics of this semilinear model in the presence
of nonlinear structural forcing terms and external forces: they proved well-
posedness _à la_ Hadamard and established the existence of finite dimensional
global attractors together with the upper semicontinuity thereof. Energy decay
results in relation with stability properties of solutions to this elastic
model have been also deeply investigated. In [6] Bchatnia and Daoulatli
obtained a general energy decay estimate in a three dimensional bounded domain
in the presence of localized nonlinear damping and an external force. By
adding viscoelastic dissipation of memory type Bchatnia and Guesmia [8]
established a more general energy decay. Different viscoelastic dissipations
have been considered in [31, 32]. Strong stability of Lamé systems with
fractional order boundary damping were studied by Benaissa and Gaouar in [9].

For the undamped elastic wave equation, more commonly known as _Navier
equation_ , a more varied bibliography is available. In [3] Barceló et al.
proved uniform resolvent estimates (Limiting Absorption Principle) for this
model. With this stationary tool at hands they also proved a priori averaged
estimates for the corresponding Cauchy problem. The resolvent estimates in [3]
were generalized in [16] and then improved in [26], where a sharp result
(analogous to the one available for the Laplacian [25]) was proved.
Surprisingly, differently from the Laplacian, in [26] the authors also showed
the failure of uniform Sobolev and Carleman inequalities for the Lamé
operator. In [23] it was proved that if spacial lower-order perturbations are
replaced by temporal ones, _i.e._ if one considers the damped equation, then
those estimates become available again. In [2] the authors generalized the
results in [3] proving Agmon-Hörmander type estimates of the Navier equation
when this is perturbed by small 0-th order matrix-valued potential. From these
results Strichartz estimates for the evolution equation followed (in the same
manner as for classical wave equation, see [11, 12]). These Strichartz
estimates were then generalized in [24, 23]. In particular in [23] the
endpoint case is deduced.

The Navier equation got also attention of the inverse problem’s community. In
particular, inverse scattering was studied in [4, 5], whereas inverse boundary
problems were considered in [33, 18, 7, 22, 5]. Boundary determination of Lamé
parameters has been studied in [13, 30, 35].

In this paper we are interested in spectral properties of the damped elastic
wave equation (1), or equivalently of the elastic wave operator (4). More
precisely, we aim at deducing quantitative bounds on the location of the point
spectrum of $A_{a}^{\ast}$ in terms of suitable norms of the damping
coefficient. In order to do that we establish a correspondence (see Lemma 2.1)
between the eigenvalue problem associated to (4) and the one corresponding to
suitable Lamé operators with non self-adjoint perturbations, that is operators
of the form

$-\Delta^{\ast}+V,$ (5)

where $V$ denotes the operator of multiplication by a (possibly) non hermitian
matrix-valued function $V\colon\mathbb{R}^{d}\to\mathbb{C}^{d\times d}$.

The study of the spectrum of (5) has already a bibliography. It is well known
that the free Lamé operator $-\Delta^{\ast}$ is self-adjoint on
$H^{2}(\mathbb{R}^{d})^{d}$ and
$\sigma(-\Delta^{\ast})=\sigma_{\textup{ac}}(-\Delta^{\ast})=[0,\infty).$ It
is a natural question [15, 16, 14, 26] to ask whether and how the spectrum
changes under 0th-order perturbations, _i.e._ considering the operator (5). In
[15], adapting to the elasticity setting the method of multipliers developed
for non self-adjoint Schrödinger operators in [20] (see also [21] for similar
problems on the plane), the author showed that the point spectrum of the
perturbed Lamé operator (5) remains empty (as in the free case) under suitable
variational small perturbations (inverse-square Hardy potential with small
coupling constant is covered). Later, in [14] we showed that full spectral
stability, _i.e._
$\sigma(-\Delta^{\ast}+V)=\sigma(-\Delta^{\ast})=[0,\infty),$ can be proved in
three dimensions $d=3$ under perturbations which satisfy a smallness condition
of Hardy-type (see [14, Thm. 1.4]). Focusing on the point spectrum only, if no
stability can be proved a priori, an interesting question is related to
provide quantitative bounds on the location in the complex plane of this part
of the spectrum which, in the perturbed setting, is possibly no longer empty.
In this direction, some preliminary result valid for the _discrete_ spectrum
can be found in [16] (see also [26]). Later, these results have been extended
in [14] to cover _embedded_ eigenvalues as well. More precisely in [14] the
following result was proved.

######  Theorem 1.1 (Thm. 1.1, [14]).

Let $d\geq 2,$ $0<\gamma\leq 1/2$ if $d=2$ and $0\leq\gamma\leq 1/2$ if $d\geq
3$ and $V\in L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d};\mathbb{C}^{d\times d}).$
Then there exists a universal constant $c_{\gamma,d,\lambda,\mu}>0$
independent on $V$ such that

$\sigma_{\textup{p}}(-\Delta^{\ast}+V)\subset\left\\{z\in\mathbb{C}\colon|z|^{\gamma}\leq
c_{\gamma,d,\lambda,\mu}\|V\|_{L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d})}^{\gamma+\frac{d}{2}}\right\\}.$
(6)

In the self-adjoint case, namely for real-valued perturbations, the result
above holds for a larger class of indices $\gamma.$ More precisely, the
following result holds true.

######  Theorem 1.2 (Thm. 3.1, [16]).

Let $d\geq 2,$ $\gamma>0$ if $d=2$ and $\gamma\geq 0$ if $d\geq 3$ and $V\in
L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d};\mathbb{R}).$ Then there exists a
universal constant $c_{\gamma,d,\lambda,\mu}>0$ independent of $V$ such that
any negative eigenvalue $z$ (if any) of the self-adjoint perturbed Lamé
operator $-\Delta^{\ast}+VI_{\mathbb{R}^{d}}$ satisfies

$|z|^{\gamma}\leq
c_{\gamma,d,\lambda,\mu}\|V_{-}\|_{L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d})}^{\gamma+\frac{d}{2}},$
(7)

where $V_{-}$ is the negative part of $V,$ _i.e._
$V_{-}(x):=\max\\{-V(x),0\\}.$

Making use of Theorem 1.1 and Theorem 1.2 and the correspondence between the
eigenvalue problem associated to the damped elastic wave operator and the one
of the perturbed Lamé operator (5) (see Lemma 2.1 below) we shall prove the
following two results valid in the self-adjoint and the non self-adjoint
setting.

######  Theorem 1.3.

Let $d\geq 2$ and assume $\gamma$ satisfies the hypotheses of Theorem 1.2 and
$a\in L^{\infty}(\mathbb{R}^{d};\mathbb{R}).$ Then there exists a universal
constant $c_{\gamma,d,\lambda,\mu}>0$ independent of the damping $a$ such that
for any positive (respectively negative) eigenvalue $z$ of the damped elastic
wave operator $A_{a}^{\ast}$ and $a_{-}\in
L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d})$ (respectively $a_{+}\in
L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d})$) satisfies

$(\pm z)^{\gamma-\frac{d}{2}}\leq
c_{\gamma,d,\lambda,\mu}\|a_{\mp}\|_{L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d})}^{\gamma+\frac{d}{2}},$
(8)

Setting $\gamma=d/2$ in (8), the previous theorem provides sufficient
condition on the size of the damping coefficient to guarantee absence of
positive (respectively negative) eigenvalues.

######  Corollary 1.1.

If $d\geq 2$ and

$c_{\frac{d}{2},d,\lambda,\mu}\|a_{\mp}\|_{L^{d}(\mathbb{R}^{d})}^{d}<1,$

then $A_{a}^{\ast}$ has no positive (respectively negative) eigenvalues.

In the non self-adjoint setting we shall prove the following result.

######  Theorem 1.4.

Let $d\geq 2$ and assume $\gamma$ satisfies the hypotheses of Theorem 1.1 and
$a\in L^{\infty}(\mathbb{R}^{d};\mathbb{C}^{d\times d})$ is a (possibly) non
hermitian matrix. Then there exists a universal constant
$c_{\gamma,d,\lambda,\mu}>0$ independent of the damping $a$ such that

$\sigma_{\textup{p}}(A_{a}^{\ast})\subset\Big{\\{}z\in\mathbb{C}\colon|z|^{\gamma-\frac{d}{2}}\leq
c_{\gamma,d,\lambda,\mu}\|a\|_{L^{\gamma+\frac{d}{2}}}^{\gamma+\frac{d}{2}}\Big{\\}}.$
(9)

######  Remark 1.1.

Notice that in the non self-adjoint case, due to the more restrictive class of
indices for which Theorem 1.1 is valid compared to Theorem 1.2, no analogous
of Corollary 1.1 holds true ($\gamma=d/2$ is not admissible).

The main motivation behind our project relies on the following simple
observation: the ellipticity condition (3) allows taking $\lambda+\mu=0$ in
the definition of the Lamé operator (2). This choice turns the Lamé operator
(2) into a vector Laplacian and consequently the damped elastic wave equation
(1) into a system of classical damped wave equations. For the (scalar) damped
wave equation, results in the spirit of Theorem 1.3 and Theorem 1.4 have been
recently proved in [27]. Thus, Theorem 1.3 and Theorem 1.4 can be seen as a
generalization of the results in [27, Thm. 1, Thm. 5 and Thm. 6] in the sense
that they recover***the constants involved slightly differ due to the presence
of the coefficient $\mu$ of the vector Laplacian and due to the vectorial form
of the wave equation once $\lambda+\mu=0$ in (1). them when $\lambda+\mu=0.$

Theorem 1.3 and Theorem 1.4 are not stated for $d=1,$ as a matter of fact the
one dimensional case is rather special and it is treated separately. In $d=1$
the Lamé operator $-\Delta^{\ast}$ turns into a scalar differential operator,
more precisely it is simply a multiple of the Laplacian

$-\Delta^{\ast}:=-\mu\frac{d^{2}}{dx^{2}}-(\lambda+\mu)\frac{d^{2}}{dx^{2}}=-(\lambda+2\mu)\frac{d^{2}}{dx^{2}}.$

As a straightforward consequence of the celebrated result of Abramov, Aslanian
and Davies for 1D-Schrödinger operators (see [1, Thm. 4]), in [16] the
following result for the one dimensional non self-adjoint Lamé operator was
proved.

######  Theorem 1.5 (Thm. 1.1, [16]).

Let $d=1$ and $V\in L^{1}(\mathbb{R};\mathbb{C}).$ Then

$\sigma_{\textup{p}}(-\Delta^{\ast}+V)\subset\Big{\\{}z\in\mathbb{C}\colon|z|^{1/2}\leq\tfrac{1}{2\sqrt{\lambda+2\mu}}\|V\|_{L^{1}(\mathbb{R})}\Big{\\}}.$

######  Remark 1.2.

We stress that Theorem 1.1 in [16] was stated only for eigenvalues outside the
essential spectrum, namely for $z\in\mathbb{C}\setminus[0,\infty).$
Nevertheless, it is easy to show that embedded eigenvalues can be covered as
well (see [17, Cor. 2.16]).

In the self-adjoint case, as an immediate consequence of the Lieb-Thirring
inequalities ([28, 29]) valid for the Schrödinger operators, one has the
following result.

######  Theorem 1.6.

Let $d=1$ and $V_{-}\in L^{1}(\mathbb{R};\mathbb{R}).$ Then

$\sigma_{\textup{p}}(-\Delta^{\ast}+V)\subset\Big{\\{}z\in\mathbb{C}\colon|z|^{1/2}\leq\tfrac{1}{2\sqrt{\lambda+2\mu}}\|V_{-}\|_{L^{1}(\mathbb{R})}\Big{\\}}.$
(10)

Theorem 1.5 and Theorem 1.6 together with Lemma 2.1 below allow to deduce
properties on the point spectrum of the one dimensional damped elastic wave
operator $A_{a}^{\ast}.$ Differently from the higher dimensional setting, in
$d=1$ Theorem 1.5 does not entail any quantitative bound on the location in
the complex plane of the eigenvalues of $A_{a}^{\ast},$ on the other hand it
provides an explicit smallness condition on the size of the $L^{1}$-norm of
the damping such that $A_{a}^{\ast}$ does not have eigenvalues. More precisely
we have the following result.

######  Theorem 1.7.

Let $d=1$ and $a\in L^{1}(\mathbb{R};\mathbb{C}).$ If
$\|a\|_{L^{1}(\mathbb{R})}<2\sqrt{\lambda+2\mu},$ then
$\sigma_{\textup{p}}(A_{a}^{\ast})=\varnothing.$ Moreover, the constant
$2\sqrt{\lambda+2\mu}$ is optimal.

In the self-adjoint situation it holds true a slightly different result
compared to the ones introduced so far.

######  Theorem 1.8.

Let $d=1$ and assume that $a$ is real-valued and satisfies

$\int_{\mathbb{R}}|x||a(x)|\,dx<\infty\qquad\text{and}\qquad\lim_{R\to\infty}\|a\|_{L^{\infty}(\mathbb{R}\setminus
B_{R}(0))}=0.$ (11)

Let $z$ be a real eigenvalue of $A_{a}^{\ast}.$ If $z>0$ and
$\int_{\mathbb{R}}a<-4\sqrt{\lambda+2\mu}$ (or $z<0$ and
$\int_{\mathbb{R}}a>4\sqrt{\lambda+2\mu}$), then

$|z|\geq(\lambda+2\mu)\Bigg{(}\int_{\mathbb{R}}|x||a(x)|\,dx\Bigg{)}^{-1}.$

Moreover the following quantitative bound on the location of eigenvalues
holds.

######  Theorem 1.9.

Let $d=1$ and assume that $a$ is real-valued and satisfies (11). Moreover,
assume

$|z|<(\lambda+2\mu)\Bigg{(}\int_{\mathbb{R}}|x||a(x)|\,dx\Bigg{)}^{-1}.$

If $z>0$ and $\int_{\mathbb{R}}a<0$ (respectively, $z<0$ and
$\int_{\mathbb{R}}a>0$), then there exists exactly one $\alpha>0$ satisfying

$2\Bigg{(}\int_{\mathbb{R}}a_{-}(x)\,dx\Bigg{)}^{-1}\leq\alpha\leq-4\Bigg{(}\int_{\mathbb{R}}a(x)\,dx\Bigg{)}^{-1}\quad\Bigg{(}\text{respectively,
}2\Bigg{(}\int_{\mathbb{R}}a_{+}(x)\,dx\Bigg{)}^{-1}\leq\alpha\leq
4\Bigg{(}\int_{\mathbb{R}}a(x)\,dx\Bigg{)}^{-1}\Bigg{)}$

such that $z/\alpha$ is an eigenvalue of $A_{a}^{\ast}.$

The rest of the paper is divided as follows. In the next Section we provide
the proof of the preliminary Lemma 2.1 establishing the correspondence between
the eigenvalue problem associated to the damped elastic wave operator and the
perturbed Lamé operator. Afterwards, in Section 2.1 we show the validity of
Theorem 1.3 and Theorem 1.4 which hold in higher dimension $d\geq 2.$ The one
dimensional case, that is Theorem 1.7-Theorem 1.9, is treated separately in
Section 2.2.

##  2 Proofs

As a starting point we show how the eigenvalue problem associated to the
damped elastic wave operator $A_{a}^{\ast}$ defined in (4) is related to the
one of a perturbed Lamé operator of the form (5).

######  Lemma 2.1.

Let $d\geq 1$ and assume $a\in L^{\infty}(\mathbb{R}^{d};\mathbb{C}^{d\times
d}).$ For every $z\in\mathbb{C},$

$z\in\sigma_{\textup{p}}(A_{a}^{\ast})\quad\Longleftrightarrow\quad-z^{2}\in\sigma_{\textup{p}}(-\Delta^{\ast}+za).$

###### Proof.

Assume $z\in\sigma_{\textup{p}}(A_{a}^{\ast}),$ then there exists a non-
trivial $\Psi=(\psi_{1},\psi_{2})^{T}\in\mathcal{D}(A_{a}^{\ast})$ such that
$A_{a}^{\ast}\Psi=z\Psi.$ In other words, $\psi_{1}\in
H^{2}(\mathbb{R}^{d})^{d},$ $\psi_{2}\in\dot{H}^{1}(\mathbb{R}^{d})^{d}$ and
$\psi_{2}=z\psi_{1},$ $\Delta^{\ast}\psi_{1}-a\psi_{2}=z\psi_{2}.$ Plugging
the first equation in the second one gives
$-\Delta^{\ast}\psi_{1}+za\psi_{1}=-z^{2}\psi_{1}.$ Since $\psi_{1}\neq 0,$
then $-z^{2}\in\sigma_{\textup{p}}(-\Delta^{\ast}+za).$ Conversely, assume
$-z^{2}\in\sigma_{\textup{p}}(-\Delta^{\ast}+za),$ then there exists a non-
trivial $\psi\in H^{2}(\mathbb{R}^{d})^{d}$ such that
$(-\Delta^{\ast}+za)\psi=-z^{2}\psi.$ Defining $\Psi:=(\psi,z\psi)^{T},$ then
$\Psi\in\mathcal{D}(A_{a}^{\ast})$ and
$(A_{a}^{\ast}\Psi)^{T}=(z\psi,\Delta^{\ast}\psi-
za\psi)=z(\psi,z\psi)=z\Psi^{T}.$ Therefore,
$z\in\sigma_{\textup{p}}(A_{a}^{\ast}).$ ∎

######  Remark 2.1.

From the validity of Lemma 2.1, one has that
$0\notin\sigma_{\textup{p}}(A_{a}^{\ast})$ as the spectrum of the unperturbed
Lamé operator $-\Delta^{\ast}+0a=-\Delta^{\ast}$ is purely continuous.

###  2.1 Higher dimensions $d\geq 2:$ Proof of Theorem 1.3 and Theorem 1.4

With Lemma 2.1 at hands we now show that Theorem 1.3 and Theorem 1.4 are
consequence of Theorem 1.2 and Theorem 1.1, respectively.

###### Proof of Theorem 1.3.

From Lemma 2.1 we know that $z\in\sigma_{\textup{p}}(A_{a}^{\ast})$ if and
only if $-z^{2}\in\sigma(-\Delta^{\ast}+za).$ From Theorem 1.2 there exists
$c_{\gamma,d,\lambda,\mu}>0$ such that

$|z|^{2\gamma}\leq
c_{\gamma,d,\lambda,\mu}\|(za)_{-}\|_{L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d})}^{L^{\gamma+\frac{d}{2}}},$
(12)

where $(za)_{-}$ is the negative part of $za,$ _i.e._ $(za)_{-}=za_{+}$ if
$z\in(-\infty,0)$ and $(za)_{-}=za_{-}$ if $z\in(0,\infty).$ Using this fact
in (12) and dividing both sides of (12) by $|z|^{\gamma+d/2}$ ($z\neq 0,$ see
Remark 2.1) we obtain (8). ∎

Now we consider the non self-adjoint situation.

###### Proof of Theorem 1.4.

The proof of Theorem 1.4 is analogous to the one of Theorem 1.3. Let
$z\in\sigma_{\textup{p}}(A_{a}^{\ast}),$ then by Lemma 2.1
$-z^{2}\in\sigma_{\textup{p}}(-\Delta^{\ast}+za).$ Using the eigenvalue bound
(6) then one has

$|z|^{2\gamma}\leq
c_{\gamma,d,\lambda,\mu}|z|^{\gamma+\frac{d}{2}}\|a\|_{L^{\gamma+\frac{d}{2}}(\mathbb{R}^{d})}^{\gamma+\frac{d}{2}},$

which gives (9) and concludes the proof. ∎

###  2.2 1D: Proof of Theorem 1.7, Theorem 1.8 and Theorem 1.9

We start considering the self-adjoint situation. Let $z\in\mathbb{R}$ and let
$\\{\lambda_{n}^{\ast}(za)\\}_{n=1}^{N}$ denote the sequence of eigenvalues of
$-\Delta^{\ast}+za,$ then the following preliminary lemma on the sum of the
square root of the eigenvalues holds.

######  Lemma 2.2.

Let $d=1.$ Then

$\sum_{n=1}^{N}|\lambda_{n}^{\ast}(za)|^{1/2}\geq-\frac{z}{4\sqrt{\lambda+2\mu}}\int_{\mathbb{R}}a(x)\,dx.$
(13)

Moreover if $\int_{\mathbb{R}}|x||a(x)|\,dx<\infty,$ then the following bound
on the number $N$ of eigenvalues $\lambda_{n}^{\ast}(za)$

$N\leq 1+\frac{|z|}{\lambda+2\mu}\int_{\mathbb{R}}|x||a(x)|\,dx$ (14)

holds.

###### Proof.

If $\lambda_{n}^{\ast}(za)$ is an eigenvalue of $-\Delta^{\ast}+za,$ then
there exists $\psi\in H^{2}(\mathbb{R})$ such that
$-(\lambda+2\mu)\Delta\psi+za\psi=\lambda_{n}^{\ast}(za)\psi$ or,
equivalently,

$\Big{(}-\Delta+\frac{za}{\lambda+2\mu}\Big{)}\psi=\frac{\lambda_{n}^{\ast}(za)}{\lambda+2\mu}\psi.$
(15)

Denoting by $\lambda_{n}(V)$ the eigenvalues of the Schrödinger operator
$-\Delta+V,$ then we conclude that $\lambda_{n}^{\ast}(za)$ is an eigenvalue
of $-\Delta^{\ast}+za$ if and only if there exists $n\in\mathbb{N}$ such that
$\lambda_{n}^{\ast}(za)$ is a multiple of an eigenvalue
$\lambda_{n}(za/(\lambda+2\mu))$ of the Schrödinger operator
$-\Delta+za/(\lambda+2\mu),$ more precisely
$\lambda_{n}(za/(\lambda+2\mu))=\lambda_{n}^{\ast}(za)/(\lambda+2\mu).$ In
particular the number of eigenvalues coincides. The Buslaev-Faddeev-Zakharov
trace formula (_cf._ [19]) for 1D-Schrödinger operator $-\Delta+V$ states that

$\sum_{n=1}^{N}|\lambda_{n}(V)|^{1/2}\geq-\frac{1}{4}\int_{\mathbb{R}}V(x)\,dx,$

this and the correspondence above give immediately (13).

The Bargmann bound [34, Pb. 22] provides a control from above of the number of
eigenvalues of the 1D-Schrödinger operator $-\Delta+V$ under the condition
$\int_{\mathbb{R}}|x||V(x)\,dx<\infty.$ More precisely,

$N\leq 1+\int_{\mathbb{R}}|x||V(x)|\,dx.$ (16)

Similarly as above (that is using the correspondence between eigenvalues of
the Lamé operator $-\Delta^{\ast}+za$ and of the Schrödinger operator
$-\Delta+za/(\lambda+2\mu)$) from (16) one easily gets (14). This concludes
the proof. ∎

###### Proof of Theorem 1.8.

Let $z$ be a real eigenvalue of $A_{a}^{\ast},$ in order to prove Theorem 1.8
we will show that if
$|z|<(\lambda+2\mu)\Big{(}\int_{\mathbb{R}}|x||a(x)|\,dx\Big{)}^{-1}$ then
$\int_{\mathbb{R}}a\geq-4\sqrt{\lambda+2\mu}$ for $z>0$ and
$\int_{\mathbb{R}}a\leq 4\sqrt{\lambda+2\mu}$ for $z<0.$ First of all notice
that (13) is non-trivial only if $z\int_{\mathbb{R}}a(x)\,dx<0.$ This last
condition, in particular is known to be a sufficient condition which
guarantees that $\inf\sigma(-\Delta^{\ast}+za)<0.$ From the decay assumption
(11), then it follows that $-\Delta^{\ast}+za$ posses at least one negative
eigenvalue. From the upper bound (14) it follows that if
$|z|<(\lambda+2\mu)\Big{(}\int_{\mathbb{R}}|x||a(x)|\,dx\Big{)}^{-1}$ then
$-\Delta^{\ast}+za$ has exactly one negative eigenvalue
$\lambda_{1}^{\ast}(za).$ Thus, from (13) and the correspondence in Lemma 2.1
one has

$|z|=|\lambda_{1}(za)|^{1/2}\geq-\frac{z}{4\sqrt{\lambda+2\mu}}\int_{\mathbb{R}}a(x)\,dx.$
(17)

This implies $\int_{\mathbb{R}}a(x)\,dx\geq-4\sqrt{\lambda+2\mu}$ for $z>0$
and $\int_{\mathbb{R}}a(x)\,dx\leq 4\sqrt{\lambda+2\mu}$ for $z<0.$ ∎

###### Proof of Theorem 1.9.

From the hypotheses, as above, one has that $-\Delta^{\ast}+za$ posses exactly
one negative eigenvalue. The Lieb-Thirring type bound (10) in Theorem 1.6 and
the estimate in (17) give

$-\frac{z}{4\sqrt{\lambda+2\mu}}\int_{\mathbb{R}}a(x)\,dx\leq|\lambda_{1}(za)|^{1/2}\leq\frac{z}{2\sqrt{\lambda+2\mu}}\int_{\mathbb{R}}a_{-}(x)\,dx,\\\
\qquad\Big{(}\text{respectively
}-\frac{z}{4\sqrt{\lambda+2\mu}}\int_{\mathbb{R}}a(x)\,dx\leq|\lambda_{1}(za)|^{1/2}\leq\frac{|z|}{2\sqrt{\lambda+2\mu}}\int_{\mathbb{R}}a_{+}(x)\,dx\Big{)}.$

Using the correspondence in Lemma 2.1 the result follows. ∎

###### Proof of Theorem 1.7.

If $z\in\mathbb{C}$ is an eigenvalue of $A_{a}^{\ast},$ then from Lemma 2.1
$-z^{2}\in\sigma_{\textup{p}}(-\Delta^{\ast}+za).$ Thus, from Theorem 1.5 we
have

$|z|\leq\frac{1}{2\sqrt{\lambda+2\mu}}|z|\|a\|_{L^{1}(\mathbb{R})}.$

Dividing by $|z|,$ which cannot be zero (see Remark 2.1), one has
$1\leq\tfrac{1}{2\sqrt{\lambda+2\mu}}\|a\|_{L^{1}(\mathbb{R})}.$ If the
$L^{1}$-norm of $a$ is small, namely if
$\|a\|_{L^{1}(\mathbb{R})}<2\sqrt{\lambda+2\mu},$ then we get a contradiction.
Thus, $\sigma_{\textup{p}}(A_{a}^{\ast})=\varnothing.$ The optimality of the
result can be proved as in [27, Thm. 4]. ∎

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