2 v 1 2 3 Ja n 19 96 Bound states in a locally deformed waveguide : the critical case

We consider the Dirichlet Laplacian for a strip in IR2 with one straight boundary and a width a(1 + λf(x)) , where f is a smooth function of a compact support with a length 2b . We show that in the critical case, ∫ b −b f(x) dx = 0 , the operator has no bound states for small |λ| if b < ( √ 3/4)a . On the other hand, a weakly bound state exists provided ‖f ′‖ < 1.56a−1‖f‖ ; in that case there are positive c1, c2 such that the corresponding eigenvalue satisfies −c1λ ≤ ǫ(λ)− (π/a)2 ≤ −c2λ for all |λ| sufficiently small.