Perturbation of an eigenvalue in the continuous spectrum of a waveguide with an asymmetric obstacle
The author considers a two-dimensional infinitely long waveguide formed by two parallel lines with an asymmetric obstacle. The problem is mathematically related to an eigenvalue problem of the Laplace operator in the corresponding domain with Neumann boundary conditions. The spectrum of the eigenvalue problem is continuous. The eigenvalues fill the closed real half line \([0, \infty)\). Additionally, a point spectrum may exist which is associated with trapped modes. \textit{D. V. Evans, M. Levitin}, and \textit{D. Vassiliev} proved in [J. Fluid Mech. 261, 21--31 (1994; Zbl 0804.76075)] the existence of at least one trapped mode, which provokes resonance phenomena, for the case of obstacles which are symmetric about the centreline of the waveguide. Based on the method used in this article and imposing artificial Dirichlet boundary conditions on the midline of the waveguide with the symmetric obstacle, the author showed in [Sib. Math. J. 51, No. 5, 866--878 (2010); translation from Sib. Mat. Zh. 51, 1086--1101 (2010; Zbl 1205.35178)] that the problem, under certain inequality condition, has an eigenvalue in the interval \((0, {\pi}^2)\) of the continuous spectrum. These results and the finding in [\textit{A. Aslanyan, L. Parnovski} and \textit{D. Vassiliev}, Q. J. Mech. Appl. Math. 53, No. 3, 429--447 (2000; Zbl 0972.76091)] that the shift of the symmetric obstacle by a small distance along the vertical axis (that is, changeover to an asymmetric obstacle) gives a complex eigenvalue, have led to the belief that the formation of the eigenvalue in the continuous spectrum requires certain symmetry. In the present article it is shown, using an asymptotic representation of eigenvalues inside a continuous spectrum, that the eigenvalue persists in the continuous spectrum if the perturbed boundary of the obstacle satisfies certain integral conditions. This incomplete stability of the point spectrum is called forced stability.
eigenvalue problem, eigenvalue perturbation
scientific article; zbMATH DE number 5995578