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 method
1 Basic preliminaries
A dielectric waveguide is a composite of its own index of refraction for each layer. If is a layer, where the index of refraction is and μ is a spectral parameter, then the waveguide process can be written in the following form:
In order to obtain and , the process in all the waveguide for the common boundary of domains and is evaluated. and must be joined in the way that the obtained known functions for and for will be the generalized solution of the equation
in which for all and for all . If the boundary is sufficiently smooth, the condition of this junction may be put down in a natural form. Indeed, the contraction of is noninfinitely smooth in and , 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.
Consider the problem
2 Formulation of the problem
We consider the eigenvalue problem (3) where and , are mutually exclusive (disjoint) measurable sets with a positive measure. If we introduce a new spectral parameter , then the problem (1) takes the form
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 , N and ) both with and without eigenvalues.
3 The existence of negative eigenvalues for the general case
Let us consider the problem:
in which is a measurable function, for all , almost everywhere in Ω, outside Ω, Ω is measurable and is a linear pseudo-differential operator with constant coefficients. Here argument quasi-polynomial , not depending on x and satisfying the following conditions for all :
We suppose that
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.
Proof Applying the Fourier transformation for (6) yields
Hence, in particular, the integral
We remember that the operator is defined only when . Since , thus the Fourier transformation for the functions , coincides. That is why . If Ω is bounded, then the kernel of the integrated operator belongs to . It follows that the operator is completely continuous. Its self-adjointness and positiveness are obvious. This enables us to write down the eigenvalues of the operator :
It is well known that (see )
From the known results for self-adjoint and quite continuous operators (see ), it follows that continuously depends on μ, where
Hence, the first statement follows from (9).
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:
Considering (16) and (17), we obtain
Hence, by virtue of (10), the lemma is proved. □
Considering the choice and the property , if , we can easily prove the boundedness of . Noting that when and for which , the operator is completely continuous. In this case, as we know, the set , contains the subsequence , which converges by norm where .
From (18) and (19) it follows that converges to by norm where . Then converges to by norm and satisfies the equality , i.e., when , the equation (11) has a nonzero solution. Hence, the theorem is proved. □
4 Application to the problem of a dielectric waveguide
In the case of
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 important for the application. It can easily be proved when we use the spherical coordinates. Moreover, for the case when , (9) also takes place. Let us make sure that (10) is satisfied when .
Consider the spherical coordinates
The left-hand side of (20) takes the form
It follows that
The theorem may be applied to the problem (21). As a consequence of this theorem, we get the following:
Now, we formulate the following theorem.
Theorem 3The problem (3) does not have an eigenvalueμfor which.
By virtue of Theorems 2 and 3, we have
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.
The authors declare that they have no competing interests.
The idea of this paper was introduced by the first author. The second author shared the first author in calculations.
We wish to thank the referees for their valuable comments which improved the original manuscript.
Karchevskii, EM: The fundamental wave problem for cylindrical dielectric waveguides. Differ. Equ.. 36, 998–999 (2000). Publisher Full Text