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:
partial differential equations; eigenvalue problems; Fourier transformation method1 Basic preliminaries
A dielectric waveguide is a composite of its own index of refraction for each layer.
If
where
In order to obtain
in which
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
Consider the problem
in which
It is obvious that if we prove the existence of the eigenvalue (3), we obtain the
following solution of the problem (1)
2 Formulation of the problem
We consider the eigenvalue problem (3) where
in which
The problem (5) is selfadjoint. 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
To use the Fourier transformation
and Plancherel’s theorem:
for all u and
From now on, if it is not specifically indicated, the notation
3 The existence of negative eigenvalues for the general case
Let us consider the problem:
in which
We suppose that
for each sufficiently small
Theorem 1The 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
Lemma 1Let
Proof Applying the Fourier transformation for (6) yields
Hence, in particular, the integral
converges absolutely. From now on,
where
Since
Let us prove the sufficiency. Let
in which
From Parseval’s equality, the solution of the problem (12) exists and it is unique.
In particular, when
Considering this inequality and (12), we obtain
In the case when
We remember that the operator
It is well known that (see [7])
where Sup is determined for all the function
From the known results for selfadjoint and quite continuous operators (see [7]), it follows that
Lemma 2Let Ω be bounded when
(1)
(2)
Proof Since
Hence, the first statement follows from (9).
Let us prove the second statement. By virtue of (13), with
which is applied to the last integral in Parseval’s inequality, we obtain
The following equations are correct:
In a similar way, we obtain
Thus, we have proved the following:
The following estimate is obvious:
where
δ will be chosen in a way such that
Considering (16) and (17), we obtain
Hence, by virtue of (10), the lemma is proved. □
Proof of Theorem 1 At the first stage, we suppose that
When
For the general case, we put
are chosen in such a way that
The integral operators defined by the righthand sides of (11) and (18) are defined
in
Considering the choice
From (18) and (19) it follows that
4 Application to the problem of a dielectric waveguide
In the case of
where
the condition (7) takes the form
It is clear that in the case of n arbitrary, these requirements are not satisfied. However, it takes place in the case
Let
Consider the spherical coordinates
The lefthand side of (20) takes the form
where
It follows that
where
Hence,
We can see that when
Taking into account that (10) is satisfied and denoting index
when
The theorem may be applied to the problem (21). As a consequence of this theorem, we get the following:
Theorem 2If Ω is bounded, the problem (3) has an eigenvalueμfor which
Let
when
and
Now, we formulate the following theorem.
Theorem 3The problem (3) does not have an eigenvalueμfor which.
Proof Multiplying the equality (22) by
If
By virtue of Theorems 2 and 3, we have
Theorem 4Let
Remark If the condition that the bounded set
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The idea of this paper was introduced by the first author. The second author shared the first author in calculations.
Acknowledgements
We wish to thank the referees for their valuable comments which improved the original manuscript.
References

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