  A two-dimensional circular quantum billiard with unusual boundary conditions
introduced by Berry and Dennis (\emph{J Phys A} {\bf 41} (2008) 135203) is
considered in detail. It is demonstrated that most of its eigenfunctions are
strongly localized and the corresponding eigenvalues are close to eigenvalues
of the circular billiard with Neumann boundary conditions. Deviations from
strong localization are also discussed. These results agree well with numerical
calculations.
