Spectral Gap of Segments of Periodic Waveguides

The lowest spectral gap of segments of a periodic waveguide in R2 is proportional to the square of the inverse length. The aim of this letter is a brief presentation of some results concerning spectral gaps in periodic waveguides. They are a representative example of the type of results derived in the forthcoming paper [5], see also the Closing Remark. Let γ : R → R be a C-function parameterised by arc-length and denote by Γ = γ(R) the curve which is its range. Assume that the curve is periodic in the following sense: there is a p > 0 such that γ(s + p) = γ(s) + (1, 0) for all s ∈ R. Define a periodic strip of width ρ > 0 by Ω := {(x, y) | dist ( (x, y),Γ ) < ρ}. Denote the normal vector (−γ̇2, γ̇1) to γ by ν and the curvature of γ by κ. Define the mapping F : Λ:= R × (−ρ, ρ) → Ω by F(s, u) = γ(s) + u ν(s) and assume that γ and F satisfy the following conditions (1) ρ‖κ‖∞ < 1 and F is an embedding . Denote by ΛL the segment (−pL/2, pL/2)×(−ρ, ρ) and by ΩL its image F(ΛL) ⊂ Ω. Let −∆Ω be the Dirichlet Laplace operator in L(Ω) and −∆Ω,L the Laplacian in L(ΩL) with Dirichlet b.c. on ∂ΩL ∩ ∂Ω and periodic b.c. on ∂ΩL \ ∂Ω. Of course, −∆Ω,L has purely discrete spectrum. The main result of this note estimates the distance between the lowest E1,L (non-degenerate) and the second E2,L eigenvalue of −∆Ω,L. Theorem 1. There is a constant C > 0 such that for all L ∈ N satisfying pL ≥ 4ρ/ √ 3: 1 C L2 ≤ E2,L − E1,L ≤ C L2 . If the curve Γ is reflection symmetric with respect to the y-coordinate axis, the same estimate holds if we replace the periodic part of the b.c. by Neumann b.c. An analogous result was derived by Kirsch and Simon in [4] for Neumann Laplacians with periodic potential, restricted to cubes. This paper was the motivation of the present letter. Note that due to the bound (6) and the different behaviour of ground states near the boundary, Dirichlet b.c. are harder to treat than Neumann ones. The remainder of this letter explains the strategy of proof of Theorem 1 leaving out the technical details. To analyse the waveguide Laplacian it is convenient to introduce a straightening transformation, see for instance [6]. The mapping F induces the unitary operator U : L(Λ)→ L(Ω) given by Uφ = |G|−1/4φ◦F , where |G| = detG, G = diag(h, 1) 2000 Mathematics Subject Classification. Primary 81Q10, 35P15. Secondary 35J20, 35J25, 47F05.