# The existence of eigenvalue problems for the waveguide theory

- A Maher
^{1}Email author and - EM Karachevskii
^{2}

**2012**:133

https://doi.org/10.1186/1687-2770-2012-133

© Maher and Karachevskii; licensee Springer 2012

**Received: **5 April 2012

**Accepted: **1 November 2012

**Published: **13 November 2012

## Abstract

In this paper, the existence of the eigenvalue problem for the waveguide theory is investigated. We used the Fourier transformation method for the solution of this problem. Also, we applied this problem to a dielectric waveguide. In this study, four theorems and two lemmas are obtained.

**MSC:** 35A22, 35P10.

## Keywords

## 1 Basic preliminaries

*μ*is a spectral parameter, then the waveguide process can be written in the following form:

in which $g(x)={k}_{j}$ for all $x\in {\mathrm{\Omega}}_{j}$ and $g(x)={k}_{r}$ for all $x\in {\mathrm{\Omega}}_{r}$. If the boundary ${\mathrm{\Gamma}}_{j,r}$ is sufficiently smooth, the condition of this junction may be put down in a natural form. Indeed, the contraction of ${\mathrm{\Gamma}}_{j,r}$ is noninfinitely smooth in ${\mathrm{\Omega}}_{j}$ and ${\mathrm{\Omega}}_{r}$, the functions which deteriorate their smoothness where the conditions themselves could be impossible to write. That is how the solution of this problem was progressing.

If the boundaries of domains are bad and there are several of them, it is not clear what the condition of the junction looks like. In this situation (connection), we need another approach to the solution of the set problem.

Since results of the junction must preserve the property of solution (being a generalized solution), we propose a new circuit system to solve the set problem. In general case, it is not solved.

The existence of eigenvalue is proved in [1] for the special case $n=2$, $N=2$, ${\mathrm{\Omega}}_{1}$ - the circle. For more details, see [2–5] and [6].

It is obvious that if we prove the existence of the eigenvalue (3), we obtain the following solution of the problem (1) ${u}_{j}(x)=u(x)$; $x\in {\mathrm{\Omega}}_{j}$, $j=1,\dots ,N$, where they are found automatically joined by a required form.

## 2 Formulation of the problem

in which $c(x)=0$, if $x\in {\mathrm{\Omega}}_{j}$.

The problem (5) is self-adjoint. This can be easily seen if we use the Fourier transformation. However, it does not influence the eigenvalue existence. Some examples of the problem (5) are known (with concrete ${k}_{j}$, *N* and ${\mathrm{\Omega}}_{j}$) both with and without eigenvalues.

for all *u* and $v\in {L}^{2}({R}^{n})$.

From now on, if it is not specifically indicated, the notation $\parallel \cdot \parallel $ is the norm in the space ${L}^{2}({R}^{n})$.

## 3 The existence of negative eigenvalues for the general case

*x*and satisfying the following conditions for all $z\in {R}^{n}$:

for each sufficiently small $\delta >0$ and $\mu \to {0}^{-}$.

**Theorem 1** *The problem* (6) *has at least one negative eigenvalue if* Ω *is bounded*.

It is necessary to introduce several lemmas before proving this theorem.

In each case, we consider $\mu <0$. By virtue of (8), there is a function $h(x,\mu )\in {L}^{2}({R}^{n})$ of the Fourier transformation which coincides with ${[P(iz)-\mu ]}^{-1}$. Considering (7), the real and even function $h(x,\mu )$ could be obtained.

**Lemma 1**

*Let*$\mu <0$.

*The problem*(6)

*has a nonzero solution if and only if the nonzero solution*$v(x)$

*has the form*

*Proof*Applying the Fourier transformation for (6) yields

*t*. Hence, by virtue of Parseval’s equality, it follows that

Since $c(t)=0$ outside Ω, then $u(x)$; $x\in \mathrm{\Omega}$ is the solution of the problem (11). If $u(t)=0$ where in Ω we obtain $[c(t)u(t)]=0$ for ${R}^{n}$, by virtue of the latter equality $u(x)=0$. The necessity is proved.

Considering this inequality and (12), we obtain $c(x)f(x)=c(x)u(x)$, *i.e.*, $u(x)$ is the solution of the problem (6). Thus, the lemma is proved. □

where Sup is determined for all the function $f\in {L}_{b}^{2}(\mathrm{\Omega})$, for which $\parallel f\parallel \le 1$.

*μ*, where

**Lemma 2**

*Let*Ω

*be bounded when*$x\in \mathrm{\Omega}$.

*Then*

- (1)
${\lambda}_{k}(\mu )\to 0$

*at*$\mu \to -\mathrm{\infty}$, - (2)
${\lambda}_{1}(\mu )\to +\mathrm{\infty}$

*at*$\mu \to {0}^{-}$.

*Proof*Since ${\lambda}_{j}(\mu ){f}_{j}(x,\mu )=A(\mu )f(x,\mu )$, ${\parallel {f}_{j}(x,\mu )\parallel}_{\mathrm{\Omega}}=1$, and $|c(x)|\le a<+\mathrm{\infty}$, we have

Hence, the first statement follows from (9).

*δ* will be chosen in a way such that $|{e}^{-ix(z+\xi )}-1|\le \frac{1}{2}$ for all $x\in \mathrm{\Omega}$ and $|z|\le \delta $, $|\xi |\le \delta $. Since Ω is bounded, we may always obtain the latter.

Hence, by virtue of (10), the lemma is proved. □

*Proof of Theorem 1*At the first stage, we suppose that $c(x)\ge \nu >0$ for all $x\in \mathrm{\Omega}$. By virtue of Lemmas 1 and 2, where ${\lambda}_{1}({\mu}_{0}(\nu ))=1$ for ${\mu}_{0}(\nu )<0$, if ${f}_{1}(x)$ is the eigenfunction corresponding to the eigenvalue ${\lambda}_{1}({\mu}_{0}(\nu ))$, then

When $\mu ={\mu}_{0}(\nu )$, we have the nonzero solution of the equation (11). It follows from Lemma 1 that ${\mu}_{0}(\nu )$ is the eigenvalue of the problem (6).

are chosen in such a way that ${\parallel \varphi (\nu ,x)\parallel}_{\mathrm{\Omega}}$.

Considering the choice $\varphi (\nu ,x)$ and the property $\parallel h(x,\mu )\parallel \to 0$, if $\mu \to -\mathrm{\infty}$, we can easily prove the boundedness of ${\mu}_{0}(\nu )$. Noting that when ${\mu}_{0}$ and ${\nu}_{0}\to 0$ for which ${\mu}_{0}({\nu}_{j})\to {\mu}_{0}$, the operator $B({\mu}_{0})$ is completely continuous. In this case, as we know, the set $B({\mu}_{0})$, $\varphi (\nu ,x)$ contains the subsequence $B({\mu}_{0})$, $\varphi ({\nu}_{j},x)$ which converges by norm where $\mu \to {\mu}_{0}$.

From (18) and (19) it follows that $\{\varphi ({\nu}_{j},x)\}$ converges to $\varphi (x)$ by norm where ${\parallel u(x)\parallel}_{\mathrm{\Omega}}=1$. Then $\{B(\nu ,{\mu}_{0}({\nu}_{{j}_{1}}))\varphi ({\nu}_{{j}_{1}},x)\}$ converges to $B({\mu}_{0})\varphi (x)$ by norm and satisfies the equality $u(x)=B({\mu}_{0})\varphi (x)$, *i.e.*, when $\mu ={\mu}_{0}$, the equation (11) has a nonzero solution. Hence, the theorem is proved. □

## 4 Application to the problem of a dielectric waveguide

It is clear that in the case of *n* arbitrary, these requirements are not satisfied. However, it takes place in the case $n\le 3$ important for the application. It can easily be proved when we use the spherical coordinates. Moreover, for the case when $n\le 3$, (9) also takes place. Let us make sure that (10) is satisfied when $n\le 4$.

when $g(x)-{k}_{{j}_{m}}=c(x)=0$ for all $x\in {\mathrm{\Omega}}_{{j}_{m}}$; *i.e.*, $c(x)=0$, outside $({R}^{n}-{\mathrm{\Omega}}_{{j}_{m}})=\mathrm{\Omega}$, where $c(x)>0$ at $x\in \mathrm{\Omega}$.

The theorem may be applied to the problem (21). As a consequence of this theorem, we get the following:

**Theorem 2** *If* Ω *is bounded*, *the problem* (3) *has an eigenvalue* *μ* *for which* $\mu +{k}_{{j}_{m}}<0$.

Now, we formulate the following theorem.

**Theorem 3** *The problem* (3) *does not have an eigenvalue* *μ* *for which*
.

*Proof*Multiplying the equality (22) by $u(x)$ and integrating it in ${R}^{n}$, we have

If $g(x)-{k}_{{j}_{M}}\le 0$, $\mu +{k}_{{j}_{M}}\le 0$, then by virtue of the condition $u(x)=0$, the latter is not impossible. □

By virtue of Theorems 2 and 3, we have

**Theorem 4** *Let* $({R}^{n}-{\mathrm{\Omega}}_{{j}_{m}})$ *be bounded*. *Then the problem* (3) *has an eigenvalue* *μ* *which satisfies the condition*
.

**Remark** If the condition that the bounded set $({R}^{n}-{\mathrm{\Omega}}_{{j}_{m}})$ is not valid, then the problem may not have eigenvalues.

## 5 Conclusions

This paper deals with the existence of eigenvalue problems for the waveguide theory. These problems are very important in the study of the mathematical analysis and mathematical physics. In this paper, we introduced four theorems and two lemmas.

## Declarations

### Acknowledgements

We wish to thank the referees for their valuable comments which improved the original manuscript.

## Authors’ Affiliations

## References

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## Copyright

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