Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window
Abstract
Consider the Laplacian in a straight planar strip of width
d
, with the Neumann boundary condition at a segment of length
2a
of one of the boundaries, and Dirichlet otherwise. For small enough
a
this operator has a single eigenvalue
ϵ(a)
; we show that there are positive
c
1
,
c
2
such that
−
c
1
a
4
≤ϵ(a)−
(π/d)
2
≤−
c
2
a
4
. An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length
2a
in the common boundary.