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
in which for all , and
It is obvious that if we prove the existence of the eigenvalue (3), we obtain the following solution of the problem (1) ; , , where they are found automatically joined by a required form.
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
in which , if .
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.
To use the Fourier transformation of the distribution (generalized) function of slow growth, we must be aware of the following well-known Parseval equality:
and Plancherel’s theorem: if and only if , where
for all u and .
From now on, if it is not specifically indicated, the notation is the norm in the space .
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
for each sufficiently small and .
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 . By virtue of (8), there is a function of the Fourier transformation which coincides with . Considering (7), the real and even function could be obtained.
Lemma 1Let . The problem (6) has a nonzero solution if and only if the nonzero solution has the form
Proof Applying the Fourier transformation for (6) yields
Hence, in particular, the integral
converges absolutely. From now on, . It follows from latter relations
where means that the Fourier transformation has been determined under t. Hence, by virtue of Parseval’s equality, it follows that
Since outside Ω, then ; is the solution of the problem (11). If where in Ω we obtain for , by virtue of the latter equality . The necessity is proved.
Let us prove the sufficiency. Let be the nonzero solution of the problem (11). Consider the new problem
in which for all and outside Ω. Since , applying the Fourier transformation for (12), we obtain
From Parseval’s equality, the solution of the problem (12) exists and it is unique. In particular, when , we have
Considering this inequality and (12), we obtain , i.e., is the solution of the problem (6). Thus, the lemma is proved. □
In the case when , we consider as an integral operator, where
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 )
where Sup is determined for all the function , for which .
From the known results for self-adjoint and quite continuous operators (see ), it follows that continuously depends on μ, where
Lemma 2Let Ω be bounded when . Then
(1) at ,
(2) at .
Proof Since , , and , we have
Hence, the first statement follows from (9).
Let us prove the second statement. By virtue of (13), with outside Ω and
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:
δ will be chosen in a way such that for all and , . Since Ω is bounded, we may always obtain the latter.
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 for all . By virtue of Lemmas 1 and 2, where for , if is the eigenfunction corresponding to the eigenvalue , then
When , we have the nonzero solution of the equation (11). It follows from Lemma 1 that is the eigenvalue of the problem (6).
For the general case, we put if and ; when and . The nonzero solutions of the equation
are chosen in such a way that .
The integral operators defined by the right-hand sides of (11) and (18) are defined in , respectively. Since Ω is bounded, then both and uniformly converge by norm to and respectively. If , then
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
We can see that when ,
Taking into account that (10) is satisfied and denoting index in which is the minimum, the problem (5) can be rewritten in the following form:
when for all ; i.e., , outside , where at .
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 be the index at which is maximum. Then the problem (3) may take the form
Now, we formulate the following theorem.
Theorem 3The problem (3) does not have an eigenvalueμfor which.
Proof Multiplying the equality (22) by and integrating it in , we have
If , , then by virtue of the condition , the latter is not impossible. □
By virtue of Theorems 2 and 3, we have
Theorem 4Let be bounded. Then the problem (3) has an eigenvalueμwhich satisfies the condition.
Remark If the condition that the bounded set is not valid, then the problem may not have eigenvalues.
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