Muller Boundary Integral Equations in the Microring Lasers Theory

Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes

“…The assertion in the opposite direction relative to the statement of this theorem is not true, as, as in [18], we did not substitute representations ( 10)-( 12) into ( 4) and ( 5), but added the limit values of them and their normal derivatives from both sides of the boundaries Γ 1 and Γ 2 term by term. However, the following result holds true (see Theorem 4 [21]). For each γ ∈ R and k ∈ I + problem ( 16) has only the trivial solution w = 0, w ∈ W. Here, I + denotes the strictly positive imaginary semi-axis of L 0 .…”

“…If u ∈ U is an eigenfunction of problem ( 1)-( 6) corresponding to an eigenvalue k ∈ L for a value of the parameter γ ∈ R, then, defined by ( 13)-( 15), functions u j and v j belong to the Banach spaces C j , j = 1, 2, respectively, and form a nontrivial solution w ∈ W of ( 16) with the same values of k and γ. This was proved in Theorem 3 of [21]. The assertion in the opposite direction relative to the statement of this theorem is not true, as, as in [18], we did not substitute representations ( 10)-( 12) into ( 4) and ( 5), but added the limit values of them and their normal derivatives from both sides of the boundaries Γ 1 and Γ 2 term by term.…”

“…This makes the theoretical analysis more difficult compared with [8,14], as well as [15], where the problems with one boundary were investigated, as it was done originally by Muller [9]. In [21], the authors generalized results of [15] and clarified the connection between GCFEP and the eigenvalue problem for the system of Muller BIEs in this more complicated situation.…”

“…The main idea of the present work is to provide, using results of [21], a rigorous proof of convergence of the Galerkin method proposed previously in [19,20] for the numerical modeling of lasers with piercing holes, and to derive the accuracy estimates for the approximate eigenvalues and eigenfunctions. Our consideration, similar to [8], is based on the fundamental results of the theory of holomorphic operator-valued functions.…”

“…In Theorem 4 from [21], it was established that, for any k ∈ I + , Equation ( 16) has only a trivial solution in the space W. The operator B(k) is weakly singular, therefore, any solution of Equation ( 16) from H must belong to the space W and can only be trivial for k ∈ I + . The operator-valued function A(k) : H → H is a Fredholm one, so for it we have I + ⊂ ρ(A).…”