Frame of Exponentials Related to Analytic Families Operators and Application to a Non-self Adjoint Problem of Radiation of a Vibrating Structure in a Light Fluid

Abstract

In the present paper, we investigate under sufficient conditions the existence of frames of exponential families, where the exponents coincide with the eigenvalues of the perturbed operator

$$\begin{aligned} T(\varepsilon ):=T_0+\varepsilon T_1 +\varepsilon ^2T_2+\cdots +\varepsilon ^k T_k+\cdots ,~\varepsilon \in \mathbb {C}. \end{aligned}$$

Here $T_0$ is a closed densely defined linear operator on a separable Hilbert space $\mathcal{H}$ with domain $\mathcal{D}(T_0)$ having isolated eigenvalues with multiplicity one and $T_1, T_2,\ldots$ are linear operators on $\mathcal{H}$ having the same domain $\mathcal{D}\supset \mathcal{D}(T_0)$ and satisfying a specific growing inequality. The obtained results are applied to a non-self adjoint problem deduced from a perturbation method for sound radiation.

1 Introduction

Frames, which are systems that provide robust, stable and usually non-unique representations of vectors, have been well studied in literature with important applications where redundancy plays a vital and useful role. Frames were first introduced by Duffin and Schaeffer [13], reintroduced by Daubechies et al. [12] and developed later by several authors such as Christensen [8,9,10,11] and Young [28]. They have many nice properties which make them very useful in the characterization of function spaces, signal processing and many other fields. In fact, a frame can be considered as a generalized basis in the sense that every element of a separable Hilbert space $\mathcal{H}$ can be written as a linear combination of the frame elements; while differing in a very important aspect. Indeed, they may be linearly dependent and therefore the uniqueness of representation characteristic of bases may be lost.

Actually, one has that a sequence of vectors is a Riesz basis if and only if it is a frame and is $\omega$-linearly independent. Nevertheless, we find that the Riesz basis property and especially Riesz basis of exponential families attracts the attention of many researchers and have many applications in mathematical physics. Among these applications, we cite the problem of radiation of a vibrating structure in a light fluid initially which was motivated by Filippi et al. [18] and widely considered in literature [5, 14,15,16,17, 20, 21]. Indeed, in [5] the authors studied the existence of a sequence of complex numbers $(\varepsilon _n)_{n\in \mathbb {N^*}}$ such that the family of exponentials associated to the eigenvalues of the operator

$$\begin{aligned} \displaystyle \frac{d^2}{dx^2}-\varepsilon _n K\displaystyle \frac{d^2}{dx^2}+\varepsilon _n^2 K^2\displaystyle \frac{d^2}{dx^2}+\cdots +(-1)^n\varepsilon _n^n K^n\displaystyle \frac{d^2}{dx^2}+\cdots , \end{aligned}$$

(1.1)

forms a Riesz basis in $L^2\big (0,T\big )$, for some $T>0$. Here K is the integral operator with kernel the Hankel function of the first kind and order 0 and $\vert \varepsilon _n\vert <\frac{1}{\Vert K\Vert }$. More precisely, they have extended Eq. (1.1) to an abstract setting and they have considered the following operator introduced by Nagy [24]:

$$\begin{aligned} T(\varepsilon )=T_0+\varepsilon T_1 +\varepsilon ^2T_2+\cdots +\varepsilon ^k T_k+\cdots , \end{aligned}$$

(1.2)

where $\varepsilon \in \mathbb {C}$ and $T_0$ is a closed densely defined linear operator on a separable Hilbert space $\mathcal{H}$ with domain $\mathcal{D}(T_0)$, while $T_1, T_2,\ldots$ are linear operators on $\mathcal{H}$ having the same domain $\mathcal{D}\supset \mathcal{D}(T_0)$ and satisfying

$$\begin{aligned} \Vert T_k\varphi \Vert \le q^{k-1}(a\Vert \varphi \Vert +b\Vert T_0\varphi \Vert )~~\hbox {for all}~\varphi \in \mathcal{D}(T_0)~\hbox {and}~k\ge 1, \end{aligned}$$

(1.3)

where a, b and q are positive constants.

On the other hand, based on the spectral analysis developed by Nagy [24] and on a stability result of Riesz bases due to Schueller [26], Charfi et al. [5] assured, for each eigenvalue $\lambda _n$ of $T_0$, the existence of a sequence of complex numbers $(\varepsilon _n)_{n\in \mathbb {N}^*}$ and a sequence of eigenvalues $(\lambda _n(\varepsilon _n))_{n\in \mathbb {N}^*}$, which can be developed as entire series of $(\varepsilon _n)_{n\in \mathbb {N^*}}$, such that the system $\{e^{i\lambda _n(\varepsilon _n)t}\}_{n\in \mathbb {N^*}}$ forms a Riesz basis in $L^2(0,T)$.

However, the Riesz basis of exponentials hence obtained is related to the eigenvalues of a sequence of operators $(T(\varepsilon _n))_{n\in \mathbb {N}^*}$ and depend on the sequence $(\varepsilon _n)_{n\in \mathbb {N}^*}$. In order to guarantee the existence of a Riesz basis of exponentials where the exponents coincide with the eigenvalues of the perturbed operator (1.2) for a fixed complex number $\varepsilon$, the authors in [14] used the following condition

$$\begin{aligned} \Vert T_k\varphi \Vert \le q^{k-1}(a\Vert \varphi \Vert +b\Vert T_0\varphi \Vert ^{\beta }\Vert \varphi \Vert ^{1-\beta })~~\hbox {for all}~\varphi \in \mathcal{D}(T_0)~\hbox {and}~k\ge 1, \end{aligned}$$

(1.4)

where $\beta \in ]0,1]$ and a, b and $q>0$. They have proved that if the system $\{e^{i\lambda _nt}\}_{n\in \mathbb {N^*}}$ forms a Riesz basis in $L^2(0,T)$, then for $\vert \varepsilon \vert$ enough small, there exists a sequence of eigenvalues $\lambda _n(\varepsilon )$ of the perturbed operator $T(\varepsilon )$ having the form

$$\begin{aligned} \lambda _n(\varepsilon )=\lambda _n+\varepsilon \lambda _{n,1}+\varepsilon ^2\lambda _{n,2}+\cdots \end{aligned}$$

(1.5)

such that the family $\{e^{i\lambda _n(\varepsilon )t}\}_{n\in \mathbb {N^*}}$ forms a Riesz basis in $L^2(0,T)$. As an application, they have considered the following integro-differential operator

$$\begin{aligned} \displaystyle \frac{d^4}{dx^4}-\varepsilon K\displaystyle \frac{d^2}{dx^2}+\varepsilon ^2 K^2\displaystyle \frac{d^2}{dx^2}+\cdots +(-1)^n\varepsilon ^n K^n\displaystyle \frac{d^2}{dx^2}+\cdots , \end{aligned}$$

where $\vert \varepsilon \vert <\frac{1}{\Vert K\Vert }$.

It is interesting to remember here that Duffin and Schaeffer [13] introduced the concept of frames in the context of systems of complex exponentials. Indeed, frames of exponentials are very special types of frames and their study is ultimately related to sampling theories tracing back to Paley–Wiener [25]. Moreover, it was shown in [4, 19, 27] that the frame properties for systems of exponentials $\{e^{i\lambda _{n}t}\}_{n\in \mathbb {Z}}$ are closely related to density issues concerning the sequence $(\lambda _{n})_{n\in \mathbb {Z}}$. So, since frames are a much more flexible tool than orthonormal and Riesz bases, we try to extend the results established in [5] and [14] to the notion of frames. Indeed, it is very difficult to determine the density of the system of eigenvalues $(\lambda _{n}(\varepsilon ))_{n\in \mathbb {N^*}}$ of $T(\varepsilon )$.

To grapple with such difficulties, we continue the analysis of the perturbed operator (1.2) under the new specific growing inequality for all $\varphi \in \mathcal{D}(T_0)$:

$$\begin{aligned} \displaystyle {\Vert T_k\varphi \Vert \le q^{k-1}\sum _{i=1}^{N}b_{i}\Vert T_0\varphi \Vert ^{\beta _{i}}\Vert \varphi \Vert ^{(1-\beta _{i})}}~~\hbox {for all}~\varphi \in \mathcal{D}(T_0)~\hbox {and}~k\ge 1, \end{aligned}$$

(1.6)

where $q, b_{1}, b_{2}, \ldots ,b_{N}$ are positive constants and $\{\beta _{1}, \beta _{2}, \ldots ,\beta _{N}\}\subset ]0,1]$, with $\beta _{i}\ne \beta _{j}$, for all $(i,j) \in \{1, \ldots , N\}^{2}.$

Clearly, Eq. (1.6) generalizes inequalities given in Eqs. (1.3) and (1.4). On the other hand, we would like to mention that, for $k=1$, Eq. (1.6) was recently introduced by Abdelmoumen et al. in [1] as the concept of generalized subordination that combines the two notions of relatively boundedness with order $\beta$ and $\beta$-subordination used in [2, 6].

So, based on this new concept Eq. (1.6), we investigate under sufficient conditions ensuring the existence of a sequence of complex numbers $(\varepsilon _n)_{n\in \mathbb {N^*}}$ such that the family of exponentials $\{e^{i\lambda _{n}(\varepsilon _n)t}\}_{n\in \mathbb {N^*}}$ is a frame for $L^2(0,T)$, where the exponents coincide with the eigenvalues of the perturbed operator $T(\varepsilon _n)$ which can be developed as entire series of $\varepsilon _n$. Furthermore, in order to improve this result, we study the existence of a fixed complex number $\varepsilon$ for at least the first $M\ge 1$ vectors. More precisely, we show that for $|\varepsilon |$ enough small the systems $\{e^{i\lambda _n(\varepsilon ) t}\}_1^M \cup \{e^{i\lambda _n(\varepsilon _n)t}\}_{M+1}^{\infty }$ and $\{e^{i\lambda _n(\varepsilon ) t}\}_1^M \cup \{e^{i\lambda _nt}\}_{M+1}^{\infty }$ are frames for $L^2(0,T)$.

Clearly, the frames of exponential families that we obtain depend more or less on $\varepsilon _n$. Actually, the exponents coincide with the eigenvalues of the perturbed operators $(T(\varepsilon _n))_{n\in \mathbb {N}^*}$ or even with the first M eigenvalues associated to the perturbed operator $T(\varepsilon )$. It is along this line of thoughts that we try to give some supplements to these results. More accurately, we provide sufficient conditions ensuring the existence of a frame of exponentials for a fixed complex number $\varepsilon$. Indeed, based on the spectral analysis developed in [7], we prove that for $|\varepsilon |$ enough small, there exist a sequence of eigenvalues $(\lambda _n(\varepsilon ))_{n\in \mathbb {N^*}}$ related to the perturbed operator (1.2) having the form (1.5) such that the system $\{e^{i\lambda _n(\varepsilon ) t}\}_{n\in \mathbb {N^*}}$ forms a frame in $L^2(0,T)$. After that, we generalize these results using a H-lipschitz function.

This paper is organized as follows: In the next section, we state our main results concerning the existence of frames of exponentials related to the perturbed operator (1.2). Section 3 is devoted to apply the obtained results to a non-self adjoint problem of radiation of a vibrating structure.

2 Main Results

In this part, we are mainly concerned with the frames properties of the family of nonharmonic exponentials related to the perturbed operator $T(\varepsilon )$ given by (1.2). Hence, we start with putting forward the following definition:

Definition 2.1

[8] A family $\{\varphi _n\}_{n\in I}$ is said to be a frame for a separable Hilbert space $\mathcal{H}$ if there exist positive constants ${\varvec{A}},{\varvec{B}}>0$ such that

$$\begin{aligned} {\varvec{A}}\Vert \varphi \Vert ^2\le \displaystyle \sum _{n\in I}|\langle \varphi ,\varphi _n\rangle |^2\le {\varvec{B}}\Vert \varphi \Vert ^2,~~\hbox {for all}~~ \varphi \in \mathcal{H}, \end{aligned}$$

where I a countable index set. The numbers ${\varvec{A}}$ and ${\varvec{B}}$ are called lower and upper frame bounds.$\diamondsuit$

In this paper, we consider a linear operator $T_0$ acting on a separable Hilbert space $\mathcal{H}$ verifying the following hypotheses:

(H1):

$T_0$ is closed with domain $\mathcal{D}(T_0)$ dense in $\mathcal{H}$.

(H2):

The eigenvalues $(\lambda _n)_n$ of $T_0$ are isolated, with multiplicity one and

$$\begin{aligned} \forall n\ge 1,~~|Im \lambda _{n}|\le h\quad \hbox { where }h:=\displaystyle \sup _{n}|Im\lambda _{n}|. \end{aligned}$$

(H3):

The family of exponentials $\{e^{i\lambda _nt}\}_1^{\infty }$ forms a frame in $L^2(0,T)$, for some $T>0$ with frame bounds A and B.

Let $T_1, T_2, T_3,\ldots$ be linear operators on $\mathcal{H}$ having the same domain $\mathcal{D}$ and satisfying the hypothesis:

(H4) $\mathcal{D}\supset \mathcal{D}(T_0)$ and there exist $q, b_{1}, b_{2}, \ldots ,b_{N}$ positive constants and $\{\beta _{1}, \beta _{2}, \ldots ,\beta _{N}\}\subset ]0,1]$, with $\beta _{i}\ne \beta _{j}$, for all $(i,j) \in \{1, \ldots , N\}^{2}$, such that for all $k\ge 1$

$$\begin{aligned} \Vert T_k\varphi \Vert \le q^{k-1}\displaystyle \sum _{i=1}^{N}b_i\Vert T_0\varphi \Vert ^{\beta _i}\Vert \varphi \Vert ^{1-\beta _i}~~\hbox {for all}~~\varphi \in \mathcal{D}(T_0). \end{aligned}$$

Let $\varepsilon$ be a non zero complex number and consider the eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{l} T_0\varphi +\varepsilon T_1\varphi +\varepsilon ^2T_2\varphi +\cdots +\varepsilon ^kT_k\varphi +\cdots =\lambda \varphi \\ \varphi \in \mathcal{D}(T_0) \end{array}\right. \end{aligned}$$

Now, we state the following results developed in [7].

Theorem 2.1

[7,  Theorem 3.1] Assume that assumptions (H1) and (H4) hold. Then,

(i) For $\vert \varepsilon \vert <q^{-1}$, the series $\sum _{k\ge 0}\varepsilon ^kT_k \varphi$ converges for all $\varphi \in \mathcal {D}(T_0).$ If $T(\varepsilon )\varphi$ denotes its limit, then $T(\varepsilon )$ is a linear operator with domain $\mathcal {D}(T_0)$.

(ii) For $\vert \varepsilon \vert <(q+\sum _{i=1}^N\beta _i b_i)^{-1},$ the operator $T(\varepsilon )$ is closed.$\diamondsuit$

In the sequel, we designate by $\lambda _n$ the eigenvalue number n of the operator $T_0$. Since $(T_{0}-zI)^{-1}$ is an analytic function of z, $\Vert (T_{0}-z)^{-1}\Vert$ is a continuous function of z. Hence, we denote by:

$$\begin{aligned} M_{n}:= & {} \displaystyle \max _{z\in \mathcal{C}_{n}}\Vert (T_{0}-z)^{-1}\Vert ,\\ N_{n}:= & {} \displaystyle \max _{z\in \mathcal{C}_{n}}\Vert T_{0}(T_{0}-z)^{-1}\Vert =\displaystyle \max _{z\in \mathcal{C}_{n}}\Vert I+z(T_{0}-z)^{-1}\Vert , \end{aligned}$$

$\mathcal{C}_{n}=\mathcal{C}(\lambda _{n},r_{n})$ the circle with center $\lambda _{n}$ and with radii $r_{n}=\frac{d_{n}}{2}$ and $d_{n}=d(\lambda _{n},\sigma (T_{0}){\setminus }\{\lambda _{n}\})$ is the distance between $\lambda _{n}$ and $\sigma (T_{0}){\setminus }\{\lambda _{n}\}$.

Theorem 2.2

[7,  Theorem 3.4] Assume that the hypotheses (H1), (H2) and (H4) are verified. Let $\varphi _n$ (respectively, $\varphi _n^{*})$ be an eigenvector of $T_0$ (respectively, $T_0^{*}$: the adjoint of $T_0)$ associated to the eigenvalue $\lambda _n$ (respectively, $\overline{\lambda _n})$ such that $\Vert \varphi _n\Vert =\Vert \varphi _n^{*}\Vert$ and $\langle \varphi _n,\varphi _n^{*}\rangle =1.$ Then:

(i) For $\vert \varepsilon \vert <\frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+r_nM_n\sum _{i=1}^N\alpha _{n,i}},$$T(\varepsilon )$ will have a unique point of the spectrum in the neighborhood of $\lambda _n$, and this point $\lambda _n (\varepsilon )$ will be also with multiplicity one.

(ii) For $\vert \varepsilon \vert <\frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i}},$$\lambda _n(\varepsilon )$ and the corresponding eigenvector $\varphi _n(\varepsilon )$ of $T(\varepsilon )$ can be developed into an entire series of $\varepsilon {:}$

$$\begin{aligned} \lambda _n(\varepsilon )= & {} \lambda _n +\varepsilon \lambda _{n,1}+ \varepsilon ^2 \lambda _{n,2}+\cdots ,\\ \varphi _n(\varepsilon )= & {} \varphi _n +\varepsilon \varphi _{n,1}+\varepsilon ^2 \varphi _{n,2}+\cdots , \end{aligned}$$

and we have

$$\begin{aligned} \vert \lambda _{n,j}\vert\le & {} \omega _n^2 r_n^2 M_n\sum _{i=1}^N\alpha _{n,i} \left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2 r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) ^{j-1}\hbox {~for all~} j\ge 1,\\ \Vert \varphi _{n,j}\Vert\le & {} \omega _n r_n M_n\left( q+\sum _{i=1}^N\alpha _{n,i} +\omega _n^2 r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) ^{j}\hbox {~for all~} j\ge 1, \end{aligned}$$

where $\omega _n=\Vert \varphi _n\Vert$ and $\alpha _{n,i}:=b_iN_n^{\beta _i}M_n^{1-\beta _i}.$$\diamondsuit$

Our result is formulated in the following theorem.

Theorem 2.3

Assume that hypotheses (H1)–(H4) are satisfied. Then, there exists a sequence of complex numbers $(\varepsilon _n)_{n\in \mathbb {N}^*}$ and a sequence of eigenvectors $\{\lambda _n(\varepsilon _n)\}_{n\in \mathbb {N}^*}$ having the form

$$\begin{aligned} \lambda _n(\varepsilon _n)=\lambda _n+\varepsilon _n\lambda _{n,1}+\varepsilon _n^2\lambda _{n,2}+\cdots , \end{aligned}$$

such that for $\vert \varepsilon _n\vert <\frac{\sqrt{A}}{\pi \sqrt{\frac{2}{3}T}\omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}nte^{t(r_n+h)}+ \sqrt{A}(q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i})}$ the system $\{e^{i\lambda _n(\varepsilon _n)t}\}_{n\in \mathbb {N^*}}$ forms a frame in $L^{2}(0,T)$.$\diamondsuit$

To prove this result, we need the following result due to [9].

Lemma 2.1

[9,  Theorem 1] Let $\{\varphi _n\}_{n\in I}$ be a frame for $\mathcal{H}$ with bounds ${\varvec{A}}$ and ${\varvec{B}}$ and let $\{\psi _n\}_{n\in I}$ be a family of vectors in $\mathcal{H}$ such that

$$\begin{aligned} M:=\displaystyle \sum _{n\in I}\Vert \varphi _n-\psi _n\Vert ^2<{\varvec{A}}. \end{aligned}$$

Then, the family $\{\psi _n\}_{n\in I}$ is a frame for $\mathcal{H}$ with bounds ${{\varvec{A}}}\left( 1-\sqrt{\frac{M}{{{\varvec{A}}}}}\right) ^2$ and ${{\varvec{B}}}\left( 1+\sqrt{\frac{M}{{{\varvec{B}}}}}\right) ^2$. $\diamondsuit$

Proof of Theorem 2.3

Let $n\in \mathbb {N}^*$ and $\lambda _n$ the eigenvalue number n of $T_0$. We have,

$$\begin{aligned} \begin{array}{ll} \displaystyle |e^{i\lambda _{n}(\varepsilon )t}-e^{i\lambda _{n}t}|&{}=\displaystyle |e^{i\lambda _{n}t}(e^{i(\lambda _{n}(\varepsilon )-\lambda _{n})t}-1)|\\ &{}= \displaystyle |e^{i\lambda _{n}t}||e^{i(\lambda _{n}(\varepsilon )-\lambda _{n})t}-1|\\ &{}\le \displaystyle 2 e^{| Im\lambda _{n}|t}e^{- Im\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\frac{t}{2}\right) }\left| \sin \left( ~(\lambda _{n}(\varepsilon )-\lambda _{n})~\frac{t}{2}\right) \right| . \end{array} \end{aligned}$$

Using hypothesis (H2), we obtain

$$\begin{aligned} \displaystyle |e^{i\lambda _{n}(\varepsilon )t}-e^{i\lambda _{n}t}|\le & {} \displaystyle 2 e^{ht}e^{\left| Im\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\frac{t}{2}\right) \right| }\left[ \sin ^2\left( Re\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\displaystyle \frac{t}{2}\right) \right) \right. \nonumber \\&\left. +\sinh ^2\left( Im\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\displaystyle \frac{t}{2}\right) \right) \right] ^{\frac{1}{2}}\nonumber \\&\nonumber \\\le & {} \displaystyle 2 e^{ht}e^{\left| Im\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\frac{t}{2}\right) \right| }\left[ ~\left| \sin \left( Re\left( (\lambda _{n}(\varepsilon ) -\lambda _{n})\displaystyle \frac{t}{2}\right) \right) \right| \right. \nonumber \\&\left. +\left| \sinh \left( Im \left( (\lambda _{n}(\varepsilon )-\lambda _{n})\displaystyle \frac{t}{2}\right) \right) \right| ~\right] \nonumber \\&\nonumber \\\le & {} \displaystyle 2 e^{ht}e^{\left| Im\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\frac{t}{2}\right) \right| }\left[ ~\left| Re\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\displaystyle \frac{t}{2}\right) \right| \right. \nonumber \\&\left. +\sinh \left| Im\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\displaystyle \frac{t}{2}\right) \right| ~\right] \nonumber \\&\nonumber \\\le & {} \displaystyle 2 e^{ht}e^{\left| Im\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\frac{t}{2}\right) \right| } \left[ ~\left| Re\left( (\lambda _{n}(\varepsilon )-\lambda _{n})\displaystyle \frac{t}{2}\right) \right| \right. \nonumber \\&\left. +\left| Im \left( (\lambda _{n}(\varepsilon ) -\lambda _{n})\displaystyle \frac{t}{2}\right) \right| \displaystyle e^{\left| Im \left( (\lambda _{n}(\varepsilon )-\lambda _{n})\frac{t}{2}\right) \right| } \right] \nonumber \\\le & {} \displaystyle 2 e^{ht}e^{\left| Im\left( (\lambda _{n}(\varepsilon )-\lambda _{n})t\right) \right| }\left| (\lambda _{n}(\varepsilon )-\lambda _{n})t\right| . \end{aligned}$$

(2.1)

However, in view of Theorem 2.2, for $|\varepsilon |<\frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}}$, the perturbed operator $T(\varepsilon )$ will have a unique point of the spectrum in the neighborhood of $\lambda _n$, and this point $\lambda _{n}(\varepsilon )$ will be also with multiplicity one. Further, for $|\varepsilon |<\frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+\omega _{n}^{2} r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}}$, $\lambda _{n}(\varepsilon )$ can be developed into an entire series of $\varepsilon$:

$$\begin{aligned} \lambda _{n}(\varepsilon )=\lambda _{n}+\varepsilon \lambda _{n,1}+\varepsilon ^{2}\lambda _{n,2}+\cdots , \end{aligned}$$

(2.2)

where $\lambda _{n,j}$ satisfies

$$\begin{aligned} \vert \lambda _{n,j}\vert \le \omega _{n}^2r_{n}^2M_{n}\sum _{i=1}^N\alpha _{n,i}\left( q+\sum _{i=1}^N\alpha _{n,i} +\omega _{n}^{2}r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}\right) ^{j-1},~~ \hbox {for all} ~j\ge 1. \end{aligned}$$

(2.3)

For each eigenvalue $\lambda _n$, we fix an $\varepsilon _n$ such that

$$\begin{aligned} \vert \varepsilon _n\vert <\frac{\sqrt{A}}{\pi \sqrt{\frac{2}{3}T}\omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}nte^{t(r_{n}+h)}+ \sqrt{A}(q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i})}. \end{aligned}$$

Obviously, we have

$$\begin{aligned} \vert \varepsilon _n\vert <\displaystyle \frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+\omega _{n}^{2} r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}}, \end{aligned}$$

then the eigenvalue $\lambda _n(\varepsilon _n)$ is inside $\mathcal{C}_n$ and we get

$$\begin{aligned} e^{| Im((\lambda _{n}(\varepsilon _n)-\lambda _{n})t)|}\le e^{r_{n}t}. \end{aligned}$$

(2.4)

Moreover, Eqs. (2.2) and (2.3) are satisfied. So, by combining Eqs. (2.1), (2.2), (2.3) and (2.4) we obtain

$$\begin{aligned} \displaystyle \Vert e^{i\lambda _{n}(\varepsilon _n)t}-e^{i\lambda _{n}t}\Vert ^2\le & {} 4\displaystyle \int _{0}^Tt^2e^{2(r_{n}+h)t}\left| \lambda _{n}(\varepsilon _n)-\lambda _{n}\right| ^2dt\nonumber \\\le & {} 4\displaystyle \int _{0}^Tt^2e^{2t(r_{n}+h)}\left( \displaystyle {\sum _{j=1}^{\infty }}\vert \varepsilon _n\vert ^j\vert \lambda _{n,j}\vert \right) ^2dt\nonumber \\\le & {} 4\displaystyle \int _{0}^T t^2e^{2t(r_{n}+h)}\nonumber \\&\left( \displaystyle {\sum _{j=1}^{\infty }}\vert \varepsilon _n\vert ^j\omega _n^{2}r_n^{2}M_n\displaystyle \sum _{i=1}^N\alpha _{n,i} \left( q+\displaystyle \sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) ^{j-1}\right) ^2dt\nonumber \\\le & {} 4\displaystyle \int _{0}^T t^2e^{2t(r_{n}+h)}\left( \omega _n^{2}r_n^{2}M_n\displaystyle \sum _{i=1}^N\alpha _{n,i}\right) ^2|\varepsilon _n|^2\nonumber \\&\quad \left[ \displaystyle {\sum _{j=0}^{\infty }}\left( \vert \varepsilon _n\vert \left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) \right) ^j\right] ^2dt\nonumber \\\le & {} \displaystyle \int _{0}^T\frac{4t^2e^{2t(r_{n}+h)}\left( \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}\right) ^2\vert \varepsilon _n\vert ^2}{\left( 1-\vert \varepsilon _n\vert \left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) \right) ^2}~dt\nonumber \\< & {} \displaystyle \int _{0}^{T}\frac{6A}{\pi ^{2} n^2T}~dt\nonumber \\\le & {} \displaystyle \frac{6A}{\pi ^{2}n^{2}}. \end{aligned}$$

(2.5)

Therefore, we have

$$\begin{aligned} \begin{array}{ll} \displaystyle \sum _{n=1}^{\infty }\Vert e^{i\lambda _n(\varepsilon )t} -e^{i\lambda _n t}\Vert ^2&<\displaystyle \sum _{n=1}^{\infty }\displaystyle \frac{6A}{\pi ^{2}n^{2}}= A. \end{array} \end{aligned}$$

Consequently, Lemma 2.1 implies that the family $\{e^{i\lambda _n(\varepsilon _n)t}\}_1^\infty$ forms a frame in $L^{2}(0,T)$. $\square$

Now, we will show the existence of exponential frames where the first M exponents coincide with the eigenvalues of the perturbed operator $T(\varepsilon )$ for a fixed complex number $\varepsilon$.

Theorem 2.4

Suppose that hypotheses (H1)–(H4) hold. Then, there exists a sequence of complex numbers $(\varepsilon _n)_{n\in \mathbb {N}^*}$ and two sequences of eigenvalues $\{\lambda _n(\varepsilon )\}_{n\in \mathbb {N}^*}$ and $\{\lambda _n(\varepsilon _n)\}_{n\in \mathbb {N}^*}$ having the form

$$\begin{aligned} \lambda _n(\varepsilon )= & {} \lambda _n+\varepsilon \lambda _{n,1}+\varepsilon ^2\lambda _{n,2}+\cdots \\ \lambda _n(\varepsilon _n)= & {} \lambda _n+\varepsilon _n\lambda _{n,1}+\varepsilon _n^2\lambda _{n,2}+\cdots \end{aligned}$$

such that for $\vert \varepsilon \vert <\min _{n\in [1, M]}\frac{\sqrt{A}}{\pi \sqrt{\frac{2}{3}T}\omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}nte^{t(r_{n}+h)}+\sqrt{A}(q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n \sum _{i=1}^N\alpha _{n,i})}$, the systems

(i) $\{e^{i\lambda _n(\varepsilon ) t}\}_1^M \cup \{e^{i\lambda _n(\varepsilon _n)t}\}_{M+1}^{\infty }$

(ii) $\{e^{i\lambda _n(\varepsilon ) t}\}_1^M \cup \{e^{i\lambda _nt}\}_{M+1}^{\infty }$

form frames in $L^{2}(0,T)$.$\diamondsuit$

Proof

(i) Let $n\in [1,M],~M\ge 1$. We reveal that

$$\begin{aligned} \begin{array}{ll} \vert \varepsilon \vert &{}<\displaystyle \min _{n\in [1, M]}\frac{\sqrt{A}}{\pi \sqrt{\frac{2}{3}T}\omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}nte^{t(r_{n}+h)}+\sqrt{A}(q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n \sum _{i=1}^N\alpha _{n,i})}\\ &{}\le \displaystyle \frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+\omega _{n}^{2} r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}}. \end{array} \end{aligned}$$

Hence, Theorem 2.2 entails the existence of a unique point of the spectrum of the perturbed operator $T(\varepsilon )$ in the neighborhood of $\lambda _n$, and this point $\lambda _{n}(\varepsilon )$ will also be with multiplicity one. Also, Eqs. (2.2) and (2.3) are verified. Consequently, Eq. (2.5) yields

$$\begin{aligned} \displaystyle \sum _{n=1}^M\displaystyle \Vert e^{i\lambda _{n}(\varepsilon )t}-e^{i\lambda _{n}t}\Vert ^2\le & {} \displaystyle \sum _{n=1}^M4\displaystyle \int _{0}^T t^2e^{2t(r_{n}+h)}\left( \omega _n^{2}r_n^{2}M_n\displaystyle \sum _{i=1}^N\alpha _{n,i}\right) ^2|\varepsilon |^2\nonumber \\&\left[ \displaystyle {\sum _{j=0}^{\infty }}\left( \vert \varepsilon \vert \left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) \right) ^j\right] ^2dt\nonumber \\\le & {} \displaystyle \sum _{n=1}^M\displaystyle \int _{0}^T\frac{4t^2e^{2t(r_{n}+h)}\left( \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}\right) ^2\vert \varepsilon \vert ^2dt}{\left( 1-\vert \varepsilon \vert \left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) \right) ^2}.\nonumber \\ \end{aligned}$$

(2.6)

Then, Eq. (2.6) implies

$$\begin{aligned} \displaystyle \sum _{n=1}^M\displaystyle \Vert e^{i\lambda _{n}(\varepsilon )t}-e^{i\lambda _{n}t}\Vert ^2< & {} \displaystyle \sum _{n=1}^M\displaystyle \int _{0}^{T}\frac{6A}{\pi ^{2} n^2T}~dt\nonumber \\\le & {} \displaystyle \sum _{n=1}^M \displaystyle \frac{6A}{\pi ^{2}n^{2}}. \end{aligned}$$

(2.7)

Now, let $n\ge M+1$. For each eigenvalue $\lambda _n$ of $T_0$, we fix an $\varepsilon _n\in \mathbb {C}$ such that

$$\begin{aligned} \vert \varepsilon _n\vert <\displaystyle \frac{\sqrt{A}}{\pi \sqrt{\frac{2}{3}T}\omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}nte^{t(r_{n}+h)}+\sqrt{A}(q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n \sum _{i=1}^N\alpha _{n,i})}. \end{aligned}$$

Evidently, we have

$$\begin{aligned} \vert \varepsilon _n\vert <\displaystyle \frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+\omega _{n}^{2} r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}}. \end{aligned}$$

Similarly as mentioned above, we prove that

$$\begin{aligned} \displaystyle \sum _{n=M+1}^{\infty }\displaystyle \Vert e^{i\lambda _{n}(\varepsilon _n)t}-e^{i\lambda _{n}t}\Vert ^2<\displaystyle \sum _{n=M+1}^{\infty } \displaystyle \frac{6A}{\pi ^{2}n^{2}}. \end{aligned}$$

(2.8)

Setting $f_n\in \{e^{i\lambda _n(\varepsilon )t}\}_1^M \cup \{e^{i\lambda _n(\varepsilon _n)t}\}_{M+1}^{\infty }$. Using Eqs (2.7) and (2.8), we can see that

$$\begin{aligned} \begin{array}{ll}\displaystyle {\sum _{n=1}^{\infty }}\Vert f_n -e^{i\lambda _n t}\Vert ^2 &{}=\displaystyle {\sum _{n=1}^M}\Vert e^{i\lambda _n(\varepsilon )t} -e^{i\lambda _n t}\Vert ^2+ \displaystyle {\sum _{n=M+1}^{\infty }}\Vert e^{i\lambda _n(\varepsilon _n)t} -e^{i\lambda _n t}\Vert ^2\\ &{}<\displaystyle \sum _{n=1}^M \displaystyle \frac{6A}{\pi ^{2}n^{2}}+\displaystyle \sum _{n=M+1}^{\infty } \displaystyle \frac{6A}{\pi ^{2}n^{2}}\\ &{}=A. \end{array} \end{aligned}$$

Hence, according to Lemma 2.1, the system $(f_n)_{n\in \mathbb {N}^{*}}$ forms a frame in $L^{2}(0,T)$. Thus, the family $\{e^{i\lambda _n(\varepsilon )t}\}_1^M \cup \{e^{i\lambda _n(\varepsilon _n)t}\}_{M+1}^{\infty }$ forms a frame in $L^{2}(0,T)$. This achieves the proof of the first item.

(ii) The proof of the second item is similar to (i). $\square$

It is interesting to mention here that the frames of exponentials constructed in Theorems 2.3 and 2.4 rely totally on $\varepsilon _n$ or partially on $\varepsilon$. Hence, we intend to study the existence of a frame of exponentials related to $T(\varepsilon )$.

Theorem 2.5

Assume that hypotheses (H1)–(H4) are verified. Suppose further that for all $n\in \mathbb {N}^*$ there exists a sequence $(r_n)_{n\ge 1}$ in $\mathbb {R}_+^*$ verifying

• (i) $\displaystyle \{z\in \mathbb {C}~\hbox {such that}~|z-\lambda _n|\le r_n\}\cap \sigma (T_0)=\{\lambda _n\}$

• (ii) $K:=\displaystyle \sup _{n\ge 1}\left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^{2}r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) <\infty$

• (iii) $L:=\displaystyle \sum _{n=1}^{\infty }\left( \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}\right) ^{2}\displaystyle \int _0^{T}t^2e^{2t(r_n+h)}dt<\infty$.

Then, for $\vert \varepsilon \vert <\frac{\sqrt{A}}{2\sqrt{L}+ \sqrt{A}K}$ there exists a sequence of eigenvalues $(\lambda _n(\varepsilon ))_{n\in \mathbb {N}^*}$ of $T(\varepsilon )$ that can be developed as entire series of $\varepsilon$ such that the system $\{e^{i\lambda _n(\varepsilon ) t}\}_1^{\infty }$ forms a frame in $L^{2}(0,T)$.$\diamondsuit$

Proof

Let $n\in \mathbb {N}^*$ and $\lambda _n$ the eigenvalue number n of $T_0$. Clearly,

$$\begin{aligned} \begin{array}{ll} \vert \varepsilon \vert &{}<\displaystyle \frac{\sqrt{A}}{2\sqrt{L}+\sqrt{A}K}\\ &{}\le \displaystyle \frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+\omega _{n}^{2} r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}}. \end{array} \end{aligned}$$

So, it follows from Theorem 2.2 that the perturbed operator $T(\varepsilon )$ will have a unique point of the spectrum in the neighborhood of $\lambda _n$, denoted by $\lambda _{n}(\varepsilon )$. Moreover, $\lambda _{n}(\varepsilon )$ will be simple and can be developed as

$$\begin{aligned} \lambda _{n}(\varepsilon )=\lambda _{n}+\varepsilon \lambda _{n,1}+\varepsilon ^{2}\lambda _{n,2}+\cdots , \end{aligned}$$

with

$$\begin{aligned} \vert \lambda _{n,j}\vert \le \omega _{n}^2r_{n}^2M_{n}\sum _{i=1}^N\alpha _{n,i}\left( q+\sum _{i=1}^N\alpha _{n,i}+ \omega _{n}^{2}r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}\right) ^{j-1},~~ \hbox {for all} ~j\ge 1. \end{aligned}$$

Therefore, in view of Eq. (2.5) we get

$$\begin{aligned} \begin{array}{ll} \displaystyle \Vert e^{i\lambda _{n}(\varepsilon )t}-e^{i\lambda _{n}t}\Vert ^2 &{}\le 4\displaystyle \int _{0}^T t^2e^{2t(r_{n}+h)}\left( \omega _n^{2}r_n^{2}M_n\displaystyle \sum _{i=1}^N\alpha _{n,i}\right) ^2\vert \varepsilon \vert ^2\\ &{}\quad \left[ \displaystyle {\sum _{j=0}^{\infty }}\left( \vert \varepsilon \vert \left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) \right) ^j\right] ^2dt. \end{array} \end{aligned}$$

As

$$\begin{aligned} \vert \varepsilon \vert \left( q+\displaystyle \sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\displaystyle \sum _{i=1}^N\alpha _{n,i}\right) <1, \end{aligned}$$

we infer that

$$\begin{aligned} \Vert e^{i\lambda _{n}(\varepsilon )t}-e^{i\lambda _{n}t}\Vert ^2\le & {} \displaystyle \int _{0}^T\frac{4t^2e^{2t(r_{n}+h)}\left( \omega _n^{2}r_n^{2}M_n\sum \nolimits _{i=1}^N\alpha _{n,i}\right) ^2\vert \varepsilon \vert ^2}{\left( 1-\vert \varepsilon \vert (q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i})\right) ^2}~dt\nonumber \\< & {} \displaystyle \frac{A\left( \omega _n^{2}r_n^{2}M_n\displaystyle \sum \nolimits _{i=1}^N\alpha _{n,i}\right) ^{2}\displaystyle \int _{0}^{T}t^2e^{2t(r_n+h)}dt}{\displaystyle \displaystyle {\sum _{n=1}^\infty } \left( \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}\right) ^{2}\displaystyle \int _0^{T}t^2e^{2t(r_n+h)}dt}. \end{aligned}$$

(2.9)

Hence, Eq. (2.9) entails the estimate

$$\begin{aligned} \displaystyle \sum _{n=1}^{\infty }\Vert e^{i\lambda _n(\varepsilon )t} -e^{i\lambda _n t}\Vert ^2 < A. \end{aligned}$$

Then, it follows from Lemma 2.1 that the family $\{e^{i\lambda _n(\varepsilon )t}\}_1^\infty$ forms a frame in $L^{2}(0,T)$. $\square$

Now, we are going to generalize these results by supposing, instead of (H3), that the exponential family $\{e^{if(\lambda _n)t}\}_1^{\infty }$ forms a frame in $L^2(0,T)$, where f is a H-lipschitz function, i.e.,

$$\begin{aligned} \exists ~ H>0~\hbox {such that~}\forall x,y\in [c,+\infty [,~c>0,~ |f(x)-f(y)|\le H|x-y|. \end{aligned}$$

To reach this objective, let us consider the following hypothesis:

$(H3')$ The family $\{e^{if(\lambda _n )t}\}_1^{\infty }$ forms a frame in $L^{2}(0,T)$ with lower frame bound A, where $T>0$ and f is a H-lipschitz function.

Theorem 2.6

Suppose that hypotheses (H1), (H2), $(H3')$ and (H4) hold. Then,

• (i) For $\vert \varepsilon _n\vert <\frac{\sqrt{6A}}{\pi \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}n\sqrt{T}tH+\sqrt{6A}(q+\sum _{i=1}^N\alpha _{n,i}+ \omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i})}$, the system $\{e^{if(\lambda _n(\varepsilon _n) )t}\}_1^{\infty }$ forms a frame in $L^{2}(0,T)$.

• (ii) For $\vert \varepsilon \vert <\min _{n\in [1, M]}\frac{\sqrt{6A}}{\pi \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}n\sqrt{T}tH+\sqrt{6A}(q+\sum _{i=1}^N\alpha _{n,i}+ \omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i})}$, the families $\{e^{if(\lambda _n(\varepsilon )) t}\}_1^M \cup \{e^{if(\lambda _n(\varepsilon _n))t}\}_{M+1}^{\infty }$ and $\{e^{if(\lambda _n(\varepsilon )) t}\}_1^M \cup \{e^{if(\lambda _n)t}\}_{M+1}^{\infty }$ are frames in $L^{2}(0,T)$.

• (iii) If we assume further that for all $n\in \mathbb {N}^*$, there exists a sequence $(r_n)_{n\ge 1}$ in $\mathbb {R}_+^*$ such that:

  $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \{z\in \mathbb {C}~\hbox {such that}~|z-\lambda _n|\le r_n\}\cap \sigma (T_0)=\{\lambda _n\}\\ S:=\displaystyle \sup _{n\ge 1}\left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^{2}r_nM_n\sum _{i=1}^N\alpha _{n,i}\right)<\infty \\ Q:=\displaystyle \sum _{n=1}^{\infty }\left( \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}\right) ^{2}<\infty , \end{array}\right. \end{aligned}$$

hence for $\vert \varepsilon \vert <\frac{\sqrt{A}}{H\sqrt{\frac{QT^3}{3}} +S\sqrt{A}}$ the system $\{e^{if(\lambda _n(\varepsilon ) )t}\}_1^{\infty }$ forms a frame in $L^{2}(0,T)$.$\diamondsuit$

Proof

Let $n\in \mathbb {N}^*$, $\lambda _n$ the eigenvalue number n of $T_0$. Clearly, we have

$$\begin{aligned} \displaystyle |e^{if(\lambda _{n}(\varepsilon ))t}-e^{if(\lambda _{n})t}|=\displaystyle |e^{if(\lambda _{n})t}(e^{i(f(\lambda _{n}(\varepsilon ))-f(\lambda _{n}))t}-1)|. \end{aligned}$$

As f is a H-lipschitz function, we obtain

$$\begin{aligned} \begin{array}{ll} \displaystyle \vert e^{if(\lambda _{n}(\varepsilon ))t}-e^{if(\lambda _{n})t}\vert &{}\le t\left| f(\lambda _n(\varepsilon ))-f(\lambda _n) \right| \\ &{}\le tH \left| \lambda _n(\varepsilon )-\lambda _n\right| . \end{array} \end{aligned}$$

The rest of the proof of (i) (respectively, (ii) and (iii)) is similar to that of Theorem 2.3 (respectively, Theorems 2.4 and 2.5). $\square$

Remark 2.1

It should be noted here that our results extend the outcomes developed in [5] and [14] since Riesz bases are frames with the property of $\omega$-linearly independent.$\diamondsuit$

3 Application to a Problem of Radiation of a Vibrating Structure in a Light Fluid

In the domain $-a<x<a$ of the plan $y=0$, we consider an elastic membrane excited by a harmonic force $\exp (-iwt)$ with an amplitude f(x) which is independent of the third space variable. This is embedded along the two straights $x=-a$ and $x=a$, in the two half-planes perfectly rigid $(x<-a,\ y=0)$ and $(x>a,\ y=0)$. The two half-spaces $y<0$ and $y>0$ are filled with gas. The fluid motion is described by a Helmholtz equation in $\mathbb {R}^2$ whereas the motion’s equation of the membrane is reduced to the equation of the vibrant cord. Let u denote the displacement of the membrane, the problem is described by the boundary value problem:

$\forall -a<x<a,$

$$\begin{aligned}&\left( \displaystyle \frac{d^2}{dx^2}+\frac{\rho _1\omega ^2}{T_1}\right) u(x)+i\rho _0\displaystyle \int _{-a}^a H_0(k_{0}|x-x'|)\nonumber \\&\quad \left( \displaystyle \frac{\omega ^2}{T_1}+\frac{1}{\rho _1}\left( \displaystyle \frac{d^2}{dx^2}+\left( -\frac{d^2}{dx^2}\right) ^{\frac{1}{2}}\right) \right) u(x')dx'=\displaystyle \frac{f(x)}{T_1}, \end{aligned}$$

(3.1)

and satisfying $u(-a)=u(a)=0.$

Here where $H_0(z)=J_0(z)+iY_0(z)$ is the Hankel function of the first kind and order 0, while the physical characteristics of the system are $\rho _0$ the fluid density, $c_0$ the fluid sound speed, $\rho _1$ and $T_1$ are respectively the surface density and the tightness of the membrane. $k_0=\displaystyle {\frac{\omega }{c_0}}$ is the wave number of the fluid. The flexion wave speed in the membrane is characterized by $c_1$: $=(\frac{\rho _1}{T_1})^{-\frac{1}{2}}$. We denote by p the acoustic pressure in the fluid and satisfies

$$\begin{aligned} p(x,y)= & {} -i \frac{\rho _0}{2}\int _{-a}^{a} H_0\left( k_0 \sqrt{(x-x')^2+y^2}\right) \nonumber \\&\left( \displaystyle \omega ^2+\frac{T_1}{\rho _1}\left( \displaystyle \frac{d^2}{dx^2}+ \left( -\frac{d^2}{dx^2}\right) ^{\frac{1}{2}}\right) \right) u(x')\text {d}x',\,\, \hbox {for}\, y>0\\ p(x,y)= & {} i \frac{\rho _0}{2}\int _{-a}^{a} H_0\left( k_0 \sqrt{(x-x')^2+y^2}\right) \nonumber \\&\left( \displaystyle \omega ^2+\frac{T_1}{\rho _1}\left( \displaystyle \frac{d^2}{dx^2} +\left( -\frac{d^2}{dx^2}\right) ^{\frac{1}{2}}\right) \right) u(x')\text {d}x',\,\, \hbox {for}\, y<0. \end{aligned}$$

The problem (3.1) satisfies the following conditions:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {\left( \frac{d^2}{dx^2}+\omega ^2\frac{\rho _1}{T_1}\right) u(x) =\frac{1}{T_1}\left( f(x)-P(x)\right) } \,\, \forall x \in (-a,a),\\ u(-a)=u(a)=0, \end{array}\right. \end{aligned}$$

where $P(x)=\displaystyle \lim _{y\rightarrow 0^{+}} p(x,y)-\lim _{y\rightarrow 0^{-}}p(x,y).$

In the sequel, we shall need the following operators:

$$\begin{aligned} \left\{ \begin{array}{ll} T_0:{\mathcal {D}}(T_0)\subset L^2(-a,a) \longrightarrow L^2(-a,a), \\ \qquad \qquad \qquad \qquad \!\psi \longrightarrow T_0\psi (x)=- \displaystyle {\frac{d^2 \psi }{dx^2}}(x),\\ {\mathcal { D}}(T_0)=H_0^1(-a,a) {\cap } H^2(-a,a) \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{rl} K:L^2(-a,a) &{}\longrightarrow L^2(-a,a) \\ \psi &{} \longrightarrow K\psi (x)=\displaystyle {\frac{i}{2}} \displaystyle \int _{-a}^{a} H_0(k_0\vert x-x'\vert )\psi (x')dx'. \end{array} \right. \end{aligned}$$

Now, we state a useful result from [3].

Lemma 3.1

[3,  Lemma 4.1] The following assertions hold:

• (i) $T_0$ is a self-adjoint operator with dense domain.

• (ii) The injection from $\mathcal{D}(T_0)$ into $L^2(-a,a)$ is compact.

• (iii) The resolvent set of $T_0$ is not empty. In fact, $0\in \rho (T_0)$.

• (iv) The spectrum of $T_0$ is constituted only of point spectrums which are positive, denumerable and of which the multiplicity is one and which have no finite limit points and satisfies

  $$\begin{aligned} 0<\lambda _1\le \lambda _2\le \cdots \le \lambda _n\rightarrow +\infty . \end{aligned}$$

  Further,

  $$\begin{aligned} \lambda _n=\left( \displaystyle \frac{n\pi }{2a}\right) ^2. \end{aligned}$$

$\diamondsuit$

Remark 3.1

In view of Lemma 3.1 the operator $T_0$ is self-adjoint with compact resolvent. Then, it follows from [23,  p. 20] that its spectral decomposition is given by

$$\begin{aligned} T_0\varphi =\displaystyle \sum _{n=1}^\infty \lambda _n\langle \varphi ,\varphi _n\rangle \varphi _n \end{aligned}$$

where $\lambda _n=\left( \frac{n\pi }{2a}\right) ^2$ is the eigenvalue number n of $T_0$ associated to the eigenvector $\varphi _n(x)=\mu e^{i\sqrt{\lambda _n}x}+\eta e^{-i\sqrt{\lambda _n}x}$.$\diamondsuit$

In the sequel, we consider the following operator:

$$\begin{aligned} \left\{ \begin{array}{l} B= T_0^{\frac{1}{2}}:{\mathcal {D}}(B)\subset L^2(-a,a)\longrightarrow L^2(-a,a)\\ \displaystyle {\qquad \qquad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ~~\varphi \longrightarrow B\varphi (x)=\left( -\frac{d^2}{dx^2}\right) ^{\frac{1}{2}}}\\ {\mathcal {D}}(B)=\left\{ \varphi \in L^2(-a,a)\hbox {~such that~} \displaystyle \sum _{n=1}^{\infty }\vert \lambda _n\vert \vert \langle \varphi ,\varphi _n\rangle \vert ^2<\infty \right\} \end{array} \right. \end{aligned}$$

and the following eigenvalue problem:

Find the values $\lambda (\varepsilon )\in \mathbb {C}$ for which there is a solution $u\in H_0^1(-a,a)\cap H^2(-a,a),$$u\ne 0$ for the equation

$$\begin{aligned} T_0u+\varepsilon K(T_0-B)u=\lambda (\varepsilon )(I+\varepsilon K)u \end{aligned}$$

(3.2)

where $\displaystyle {\lambda =\frac{\omega ^2 \rho _1}{T_1}}$ and $\displaystyle {\varepsilon =\frac{2\rho _0}{\rho _1}}$.

Note that $\lambda$ and $\varphi$ each depends on the value of $\varepsilon$. So, we designate by $\lambda :=\lambda (\varepsilon )$ and $\varphi :=\varphi (\varepsilon )$.

We remark that for $|\varepsilon |<\frac{1}{\Vert K\Vert }$, the operator $I+\varepsilon K$ is invertible. Then, the problem (3.2) becomes:

Find the values $\lambda (\varepsilon )\in \mathbb {C}$ for which there is a solution $u\in H_0^1(-a,a)\cap H^2(-a,a),$$u\ne 0$ for the equation

$$\begin{aligned} (I+\varepsilon K)^{-1}\displaystyle T_0+\varepsilon (I+\varepsilon K)^{-1}K(T_0-B)u=\lambda (\varepsilon )u. \end{aligned}$$

(3.3)

The problem (3.3) is equivalent to:

Find the values $\lambda (\varepsilon )\in \mathbb {C}$ for which there is a solution $u\in H_0^1(-a,a)\cap H^2(-a,a),$$u\ne 0$ for the equation

$$\begin{aligned} \left( \displaystyle T_{0}+\varepsilon T_{1}+\varepsilon ^2T_{2}+\cdots +\varepsilon ^nT_{n}+\cdots \right) u= \lambda (\varepsilon )u, \end{aligned}$$

(3.4)

where $T_n:=(-1)^nK^nB$ with $\mathcal{D}(T_n)=H^1(-a,a)$ for all $n\ge 1$.

In order to prove our theorem, we shall prove some auxiliary results.

Proposition 3.1

There exist positive constants $b_1=q=\Vert K\Vert ,~\beta _1=\frac{1}{2}$ such that

$$\begin{aligned} \Vert T_k\varphi \Vert \le q^{k-1}b_1\Vert T_0\varphi \Vert ^{\frac{1}{2}}\Vert \varphi \Vert ^{\frac{1}{2}} \end{aligned}$$

for all $\varphi \in \mathcal{D}(T_0)$ and for all $k\ge 1.$$\diamondsuit$

Proof

Let $\varphi \in \mathcal{D}(T_0)$. We have $\mathcal {D}(T_0)\subset \mathcal {D}(B)$ and for every $\varphi \in \mathcal {D}(T_0)$

$$\begin{aligned} \Vert B\varphi \Vert ^2&=\displaystyle \sum _{n=1}^\infty \lambda _n \vert \langle \varphi ,\varphi _n\rangle \vert ^2\\&=\displaystyle \sum _{n=1}^\infty \left( \frac{n\pi }{2a}\right) ^2 \vert \langle \varphi ,\varphi _n\rangle \vert ^2\\&=\displaystyle \sum _{n=1}^\infty \left( \frac{n\pi }{2a}\right) ^{4 \frac{1}{2}} \vert \langle \varphi ,\varphi _n\rangle \vert ^{2\frac{1}{2}} \vert \langle \varphi ,\varphi _n\rangle \vert ^{2(1-\frac{1}{2})}. \end{aligned}$$

Using Hölder’s inequality, we get

$$\begin{aligned} \Vert B\varphi \Vert ^2&\le \displaystyle \left( \sum _{n=1}^\infty \left( \frac{n\pi }{2a}\right) ^4 \vert \langle \varphi ,\varphi _n\rangle \vert ^{2}\right) ^{\frac{1}{2}} \left( \sum _{n=1}^\infty \vert \langle \varphi ,\varphi _n\rangle \vert ^{2}\right) ^{1-\frac{1}{2}} \\&\le \Vert T_0\varphi \Vert \Vert \varphi \Vert . \end{aligned}$$

Hence,

$$\begin{aligned} \Vert B\varphi \Vert \le \Vert T_0\varphi \Vert ^{\frac{1}{2}}\Vert \varphi \Vert ^{\frac{1}{2}}\quad \hbox {~for all~}\varphi \in \mathcal {D}(T_0). \end{aligned}$$

However, we have

$$\begin{aligned} \Vert T_k\varphi \Vert&=\Vert K^kB\varphi \Vert \\&\le \Vert K^k\Vert \Vert B\varphi \Vert \\&\le \Vert K\Vert ^k \Vert T_0\varphi \Vert ^{\frac{1}{2}}\Vert \varphi \Vert ^{\frac{1}{2}}, \end{aligned}$$

for all $\varphi \in \mathcal {D}(T_0)$ and $k\ge 1.$ Hence, it suffices to take $N=1$, $b_1=q=\Vert K\Vert$ and $\beta _1=\frac{1}{2}$. $\square$

Proposition 3.2

For $\vert \varepsilon \vert <\frac{1}{\Vert K\Vert },$ the series $\sum _{k\ge 0} \varepsilon ^kT_k \varphi$ converges for all $\varphi \in \mathcal{D}(T_0).$ If we designate its sum by $T(\varepsilon )\varphi ,$ we define a linear operator $T(\varepsilon )$ with domain $\mathcal{D}(T_0).$ For $\vert \varepsilon \vert < \frac{2}{3\Vert K\Vert },$ the operator $T(\varepsilon )$ is closed.$\diamondsuit$

The main result of this section asserts:

Proposition 3.3

The family $\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }$ forms a frame in $L^2(0,a_1), a_1<4a$.$\diamondsuit$

Proof

In view of [5,  Theorem  4.1], the family $\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }$ forms a Riesz basis in $L^2(0,4a)$. Then, there exist positive constants C and D such that

$$\begin{aligned} C\Vert f\Vert ^2\le \displaystyle \sum _{n=1}^{\infty }\vert \langle f,e^{i\frac{n\pi }{2a}t}\rangle \vert ^2\le D\Vert f\Vert ^2,~~\forall f\in L^2(0,4a). \end{aligned}$$

(3.5)

Given $f\in L^2(0,a_1)$, extend it to (0, 4a) by setting $f(x)=0$ for $a_1<x< 4a$. Then, we can apply Eq. (3.5), but because of the zero extension, the norm and inner product are from $L^2(0,a_1)$. So, $\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }$ forms a frame in $L^2(0,a_1)$. $\square$

Remark 3.2

In [5,  Theorem  4.1], it was shown that $\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }$ forms a Riesz basis in $L^2(0,4a)$. However, if we reduce the interval (0, 4a), the Riesz basis properties fail to exist. Indeed, the situation is more complicated since the set $\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }$ is over complete on an interval less than 4a.$\diamondsuit$

Now, we are ready to state the objective of this section.

Theorem 3.1

For $\vert \varepsilon _n\vert$ enough small and $\vert \varepsilon \vert$ enough small there exist two sequences of eigenvalues $\{\lambda _n(\varepsilon _n)\}_{n\in \mathbb {N^*}}$ and $\{\lambda _n(\varepsilon )\}_{n\in \mathbb {N^*}}$ of $T(\varepsilon )$ having the form

$$\begin{aligned} \lambda _n(\varepsilon _n)= & {} \lambda _n+\varepsilon _n\lambda _{n,1}+\varepsilon _n^2\lambda _{n,2}+\cdots \\ \lambda _n(\varepsilon )= & {} \lambda _n+\varepsilon \lambda _{n,1}+\varepsilon ^2\lambda _{n,2}+\cdots \end{aligned}$$

such that the systems

(i) $\{e^{i\sqrt{|\lambda _n(\varepsilon _n)|}t}\}_{1}^{\infty }$

(ii) $\{e^{i\sqrt{|\lambda _n(\varepsilon )|}t}\}_1^M \cup \{e^{i\sqrt{|\lambda _n(\varepsilon _n)|}t}\}_{M+1}^{\infty }~~~~$

(iii) $\{e^{i\sqrt{|\lambda _n(\varepsilon )|}t}\}_1^M \cup \{e^{i\sqrt{\lambda _n}t}\}_{M+1}^{\infty }$

are frames in $L^2(0,a_1).$$\diamondsuit$

Proof

The results follow from Theorem 2.6, Lemma 3.1 and Propositions 3.1 and 3.3. $\square$

Theorem 3.2

There exists a sequence of eigenvalues $(\lambda _n(\varepsilon ))_{n\in \mathbb {N}^*}$ of $T(\varepsilon )$ having the form

$$\begin{aligned} \lambda _n(\varepsilon )=\lambda _n+\varepsilon \lambda _{n,1}+\varepsilon ^2\lambda _{n,2}+\cdots \end{aligned}$$

such that for $|\varepsilon |$ enough small the family $\{e^{i\sqrt{|\lambda _n(\varepsilon )|}t}\}_1^\infty$ forms a frame in $L^2(0,a_1)$.$\diamondsuit$

Proof

Let $n\in \mathbb {N}^*$, $\lambda _n$ the nth eigenvalue of $T_0$ and $r_n=\displaystyle \frac{\lambda _{n+1}-\lambda _n}{2}$.

We have $\{z\in \mathbb {C},~|z-\lambda _n|\le r_n\}\cap \sigma (T_0)=\{\lambda _n\}$. Let $\mathcal{{C}}_n=\mathcal{{C}}(\lambda _n,r_n)$ the closed circle with center $\lambda _n$ and radius $r_n$ and $z\in \mathcal{{C}}_n$.

Since $T_0$ is self-adjoint, then it follows from [22] that

$\Vert R_z\Vert =\Vert (T_0-zI)^{-1}\Vert =\displaystyle \frac{1}{d(z,\sigma (T_0))}.$

So,

$$\begin{aligned} M_n=\displaystyle \max _{z\in \mathcal{{C}}_n}\Vert R_z\Vert =\displaystyle \frac{1}{r_n}. \end{aligned}$$

(3.6)

Further, as $z\in \mathcal{{C}}_n$ then $z=\lambda _n+r_ne^{it}$ with $t\in [0,2\pi [$. Hence, $|z|\le \lambda _n+r_n$.

Consequently,

$$\begin{aligned} \Vert T_0 R_z\Vert= & {} \Vert I+zR_z\Vert \\\le & {} 1+\vert z\vert \Vert R_z\Vert \\\le & {} 1+(\lambda _n+r_n)\Vert R_z\Vert , \end{aligned}$$

entails the estimate

$$\begin{aligned} N_n= & {} \displaystyle \max _{z\in \mathcal{{C}}_n}\Vert T_0 R_z\Vert \nonumber \\\le & {} 1+(\lambda _n+r_n)\displaystyle \max _{z\in \mathcal{{C}}_n}\Vert R_z\Vert \nonumber \\\le & {} \left( 2+\displaystyle \frac{\lambda _n}{r_n}\right) . \end{aligned}$$

(3.7)

Since

$$\begin{aligned} \alpha _n =\alpha _{n,1}=b_1N_n^{\frac{1}{2}} M_n^{\frac{1}{2}}, \end{aligned}$$

Eqs. (3.6) and (3.7) imply that

$$\begin{aligned} \alpha _n \le b_1\left( 2+\displaystyle \frac{\lambda _n}{r_n} \right) ^{\frac{1}{2}}\left( \displaystyle \frac{1}{r_n} \right) ^{\frac{1}{2}}. \end{aligned}$$

(3.8)

As

$$\begin{aligned} r_n=\displaystyle \frac{\left( \frac{\pi }{2a}\right) ^2(n+1)^2-\left( \frac{\pi }{2a}\right) ^2 n^2}{2}=\displaystyle \frac{\left( \frac{\pi }{2a} \right) ^2 (2n+1)}{2}>\left( \frac{\pi }{2a}\right) ^2n, \end{aligned}$$

we obtain

$$\begin{aligned} \displaystyle \frac{1}{r_n}<\displaystyle \frac{4a^2}{n\pi ^2}. \end{aligned}$$

So,

$$\begin{aligned} \displaystyle b_1\left( 2+\frac{\lambda _n}{r_n}\right) ^{\frac{1}{2}}\left( \displaystyle \frac{1}{r_n}\right) ^{\frac{1}{2}}< b_1\left( 2+n\right) ^{\frac{1}{2}}\left( \displaystyle \frac{4a^2}{n\pi ^2}\right) ^{\frac{1}{2}}. \end{aligned}$$

(3.9)

Then, Eqs. (3.8) and (3.9) yield

$$\begin{aligned} q+\alpha _n+\omega _n^{2}r_n M_n\alpha _n\le & {} \Vert K\Vert +2\alpha _n\\< & {} \Vert K\Vert +b_1\displaystyle \frac{4a}{\pi }\sqrt{\displaystyle \frac{2+n}{n}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \displaystyle \sup _{n\ge 1}(q+\alpha _n+\omega _n^{2}r_n M_n\alpha _n)<\Vert K\Vert +\sqrt{3}b_1\displaystyle \frac{4a}{\pi }<\infty . \end{aligned}$$

Now, if we set $\widetilde{\varphi _n}(z)=\displaystyle \frac{1}{(n+1)^2}\varphi _n(z)$, we get that $(\widetilde{\varphi _n})_n$ is a system of eigenvectors of $T_0$ associated to the eigenvalues $(\lambda _n)_n$ and $\widetilde{\omega _n}=\Vert \widetilde{\varphi _n}\Vert =\displaystyle \frac{1}{(n+1)^2}$.

Moreover,

$$\begin{aligned} \begin{array}{ll} \displaystyle \sum _{n=1}^{\infty }\left( \widetilde{ \omega _n}^{2}r_n^2 M_n\alpha _n\right) ^2&{}\le \displaystyle \sum _{n=1}^{\infty }\left( \widetilde{\omega _n}^{2}r_n\displaystyle \max _{n\ge 1 }\alpha _n\right) ^2\\ &{}< \displaystyle \sum _{n=1}^{\infty }\left( \sqrt{3}b_1\displaystyle \frac{2a}{\pi }\widetilde{\omega _n}^{2}r_n\right) ^2\\ &{}<\displaystyle \sum _{n=1}^{\infty }\left( \sqrt{3}b_1\displaystyle \frac{2a}{\pi }\widetilde{\omega _n}^{2}\lambda _{n+1}\right) ^2\\ &{}<\displaystyle \sum _{n=1}^{\infty }\left( \sqrt{3}b_1\displaystyle \frac{\pi }{2a}\displaystyle \frac{(n+1)^2}{(n+1)^4}\right) ^2\\ &{}<\displaystyle \sum _{n=1}^{\infty }\left( \displaystyle \frac{\sqrt{3}b_1\pi }{2a(n+1)^2}\right) ^2<\infty . \end{array} \end{aligned}$$

Consequently, according to Theorem 2.6, Lemma 3.1 and Propositions 3.1 and 3.3 the family $\{e^{i\sqrt{|\lambda _n(\varepsilon )|}t}\}_1^\infty$ forms a frame in $L^2(0,a_1)$. $\square$