Bound states in a locally deformed waveguide: the critical case
Abstract
We consider the Dirichlet Laplacian for a strip in $\,\R^2$ 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
′
∥<1.56
a
−1
∥f∥
; in that case there are positive
c
1
,
c
2
such that the corresponding eigenvalue satisfies
−
c
1
λ
4
≤ϵ(λ)−(π/a
)
2
≤−
c
2
λ
4
for all
|λ|
sufficiently small.