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Gap opening and split band edges in waveguides coupled by a periodic system of small windows. (English. Russian original) Zbl 1288.81046
Math. Notes 93, No. 5, 660-675 (2013); translation from Mat. Zametki 93, No. 5, 665-683 (2013).
The authors provide a mathematical rigorous description of the split band wedge in coupled waveguides. More precisely, let $$H^{\varepsilon}$$ be the Laplacian in an infinite straight strip, $$\mathbb{R} \times (-d,\pi), 0<d< \pi$$, with Dirichlet boundary conditions at the lines $$x_{2}=-d, x_{2}=\pi$$ and Neumann boundary condition at the line $$x_{2}=0$$, except at the intervals $$\omega^{\varepsilon}_{n} = (2nh- \varepsilon, 2nh+ \varepsilon) \times \left\{ 0 \right\}, n \in \mathbb{Z}, \varepsilon \geq 0$$. The Hamiltonian $$H^{\varepsilon}$$ describes a system of two waveguides coupled by the $$2h$$-periodic system of windows $$\omega^{\varepsilon}_{n}$$. The Hamiltonian $$H^{0}$$ can be decomposed as $$H^{0}=H_{+} \oplus H_{-}$$, where $$H_{\pm}$$ are Laplacians with Dirichlet boundary conditions at $$x_{2}=\pi$$, respectively at $$x_{2}=-d$$ and Neumann conditions at $$x_{2}=0$$. Let $$H_{\pm}=\int_{(-\pi,\pi]}^{\oplus} H_{\pm}(k)\, dk$$ be the Bloch-Floquet decomposition of the operators $$H_{\pm}$$, with elementary cell $$(-h,h)\times (0,\pi)$$, respectively $$(-h,h)\times (-d,0)$$. The main result of the paper gives sufficient conditions on the eigenvalues of $$H_{\pm}(k)$$ in order that $$H^{\varepsilon}$$ admits a spectral gap. Asymptotics of the endpoints of the spectral gap, for $$\varepsilon \rightarrow 0$$, are also provided.

##### MSC:
 81Q37 Quantum dots, waveguides, ratchets, etc. 47A10 Spectrum, resolvent 35J25 Boundary value problems for second-order elliptic equations 35P99 Spectral theory and eigenvalue problems for partial differential equations 82D77 Quantum waveguides, quantum wires
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