Planar waveguide with "twisted" boundary conditions: Small width

We consider a planar waveguide with "twisted" boundary conditions. By twisting we mean a special combination of Dirichlet and Neumann boundary conditions. Assuming that the width of the waveguide goes to zero, we identify the effective (limiting) operator as the width of the waveguide tends to zero, establishes the uniform resolvent convergence in various possible operator norms, and gives the estimates for the rates of convergence. We show that studying the resolvent convergence can be treated as a certain threshold effect and we present an elegant technique which justifies such point of view. (C) 2012 American Institute of Physics. [doi:10.1063/1.3681895]