Asymptotics of bound states, bands and resonances for laterally coupled quantum waveguides and layers

Igor Popov

Leningrad Institute of Fine Mechanics and Optics, Leningrad Institute of Fine Mechanics and Optics

Spectral problem for the Dirichlet Laplacian in two-dimensional strips (threedimensional layers) of widths d_{+}, d_{−}, d_{+} > d_{−} coupled through small openings aω_{i}, i = 1, ...n, (a is small parameter) is considered. Method of matching of asymptotic expansions of solutions of boundary value problem is used. The asymptotics (in a) of a bound state λ_{a} close to the threshold π^{2}/d_{+}^{2} is obtained:

λ_{a} = \{ \begin{matrix}\frac{π^{2}}{d_{+}^{2}} − (\frac{π^{3}}{d_{+}^{3}} ∑_{i=1}^{n} c_{ω_{i}})^{2}a^{4} + o(a^{4}), & R^{2}, \\ \frac{π^{2}}{d_{+}^{2}} − (\frac{1}{d_{+}^{2}} exp (−\frac{2d_{+}^{3}}{3π^{2}} (∑_{i=1}^{n} b_{ω_{i}})^{−1}a^{−3}(1 + o(1))), & R^{3}. \end{matrix}

Here c_{ω_{i}} is a capacity of ω_{i} in R^{2}, b_{ω_{i}} is an average virtual mass of ω_{i} in R^{3}. The case of two identical waveguides (layers) is considered too.

The asymptotics of resonances close to N−th threshold is determined for 2D case.. 2D waveguides in a magnetic field and curved waveguides are considered.

Asymptotics of bands for the case of periodic system of coupling windows (period L) for two waveguides is constructed:

[\frac{π^{2}}{d_{+}^{2}} − \frac{3π^{3}}{2Ld_{+}^{3}} a^{2} + o(a^{2}), \frac{π^{2}}{d_{+}^{2}} − \frac{π^{3}}{2Ld_{+}^{3}} a^{2} + o(a^{2})], R^{2}.

[λ_{1}^{+} − \frac{18π(ψ_{1}^{+0})^{2}b_{ω}}{L} a^{3} + o(a^{3}), λ_{1}^{+} − \frac{6π(ψ_{1}^{+0})^{2}b_{ω}}{L} a^{3} + o(a^{3})], , R^{3}.

Here λ_{1}^{+} is the first transversal eigenvalue, ψ_{1}^{+0} is the value of the normal derivative of the corresponding transversal eigenfunction at the centre of the opening. There is a gap for sufficiently small a. For the case of layers coupled through singly (Λ_{1}) and doubly (Λ_{2}) periodic system of windows there is no gap, and the asymptotics of the lower bound for the continuous spectrum is:

λ_{min} = \frac{π^{2}}{d_{+}^{2}} − \frac{π^{6}b_{ω}^{2}}{Ld_{+}^{6}} a^{6} + o(a^{6}), for Λ_{1},

λ_{min} = \frac{π^{2}}{d_{+}^{2}} − \frac{πb_{ω}|Λ̂|}{d_{+}^{3}}a^{3} + o(a^{3}), for Λ_{2},

where |Λ̂| is a square of the Brillouin zone for Λ_{2}.