Interaction of normal modes in a plane irregular waveguide with a point of degeneracy

A unifying asymptotic approach for nonadiabatic transitions near pairs of real or complex turning points

“…The eigenvalues β and eigenfunctions φ will be found in Section 3.2 as perturbations of the eigenvalues β and eigenfunctions ϕ of (12).…”

“…We use in this Section properties of eigenvalues and eigenfunctions of the unperturbed spectral problem (12) derived in Appendix A.…”

“…. Equation ( 17) is the original spectral problem (12). Thus, we choose the principal approximation as β(0)…”

“…Upon inserting these series into (28) and equating terms with equal powers of √ , we get an infinite sequence of equations 30) is the original spectral problem (12) taken at the degeneracy point x = 0. Thus we have β(0)…”

“…On the other hand, there is a lot of physical works treating different physical phenomena caused by the presence of degeneracy points. These physical works apart from the problems for Schroedinger equation comprise, for example, the problems of radiowaves propagation [12], [18], [19], [57], acoustical waveguides [51,52], elasic waveguides [59], [56], waves in the Timoshenko beam [58], electrons in crystal [24], liquid crystals [4], graphene [61], [70], creeping waves on the smooth boundary of the convex body for particular boundary conditions [3] to mention just a few. Despite profound similarities of all the problems with degeneracy points, each physical problem from above was solved with account of its specific assumptions and without applying any of the theory developed in the abstract form in the earlier works.…”