Spectrum of the Dirichlet Laplacian in sheared waveguides

Spectral analysis in broken sheared waveguides

“…in the quadratic form sense. Proposition 3 and Remark 1 in [31] imply that σ ess (H DN ext ) = [E 1 (β), ∞). By minimax principle, and since the spectrum of…”

“…For each β ∈ (0, ∞), define the spatial curve rβ (x) := (x, 0, βx), x ∈ R, and the mapping Lβ (x, y 1 , y 2 ) := rβ (x) + y 1 e 2 + y 2 e 3 , (x, y 1 , y 2 ) ∈ R × S. Consider the straight waveguide Ωβ := Lβ (R × S). Proposition 3 of [31] shows that the Dirichlet Laplacian in Ωβ has a purely essential spectrum and it is equal to the interval [E 1 (β), ∞). Since the essential spectrum of the Dirichlet Laplacian in tubular domains is determined by the geometry of the region at infinity only, the statement of Proposition 1 in the Introduction is expected.…”

“…Again, by the considerations in [31], it is possible to construct a Weyl sequence supported in Ω ext associated with λ ≥ E 1 (β). This shows that [E 1 (β), ∞) ⊆ σ ess (−∆ D Ω β ).…”

“…Remark 1. The model considered in this work was inspired by [31]. In that work the author considered a sheared waveguide constructed by the parallel transporting of a two-dimensional cross section along a spatial curve.…”

Spectral analysis in broken sheared waveguides

“…For each 𝛽 ∈ (0, ∞), define the spatial curve r𝛽 (x) ∶= (x, 0, 𝛽x), x ∈ R, and the mapping 𝛽 (x, 𝑦 1 , 𝑦 2 ) ∶= r𝛽 (x) + 𝑦 1 e 2 + 𝑦 2 e 3 , (x, 𝑦 1 , 𝑦 2 ) ∈ R × S. Consider the straight waveguide Ω𝛽 ∶= 𝛽 (R × S). Proposition 3 of [19] shows that the Dirichlet Laplacian in Ω𝛽 has a purely essential spectrum and it is equal to the interval [E 1 (𝛽), ∞). Since the essential spectrum of the Dirichlet Laplacian in tubular domains is determined by the geometry of the region at infinity only, the statement of Proposition 1 in the Introduction is expected.…”