True and fictitious eigenvalues of the set of Muller boundary integral equations
The authors have investigated the spectrum of a nonlinear eigenvalue problem for a system of boundary integral equations theoretically and numerically. In general, eigenvalues of this problem split into two sets. The first set is the set of the true eigenvalues corresponding to the original eigenvalue problem of the Helmholtz operator. The second set is the set of fictitious eigenvalues that correspond to the so-called "turned inside-out" problem. Therefore, it is important to determine the conditions under which the original eigenvalue problem for the Helmholtz operator on a plane and the corresponding problem for the integral operator are spectrally equivalent. The original eigenvalue problem for the Helmholtz operator is reduced to a nonlinear eigenvalue problem for the system of Muller boundary integral equations. The solutions of the original problem and the "turned inside-out" problem are found by the method of separation of variables. The solution of the system of Muller boundary integral equations is based on the Galerkin method. The paper formulates and proves the equivalence theorem for the original eigenvalue problem for the Helmholtz operator and the system of Muller boundary integral equations. It shows that the spectrum of the nonlinear eigenvalue problem of the system of Muller boundary integral equations contains fictitious eigenvalues in addition to the true eigenvalues. There is a defined area on the complex plane that consists only of fictitious eigenvalues; and it is shown that these fictitious eigenvalues are explicitly separated from true eigenvalues. The equivalence theorem provides a theoretical justification for the applicability of the Muller boundary integral equation method. The developed computer programs can be directly used in the modeling microdisk lasers.