Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold

Considering the example of a mixed boundary value problem for the Helmholtz operator we discuss two methods for finding eigenvalues below the continuous spectrum threshold: one variational and the other – asymptotic. We construct asymptotics for the eigenvalue arising near the threshold as a small obstacle appears in the cylindrical waveguide. The resulting asymptotic formula, its derivation and justification differ substantially from the case of a bounded domain.