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The spectrum of the Dirichlet problem is studied in the case of a periodic infinite planar domain having the form of an accommodation ladder: two parallel strips-uprights of thickness h > 0 are linked by treads of the same thickness. It is shown that, for h small, a gap is always opened between the second and third segments of the essential spectrum of the problem operator. The gap between the first and second segments is also discussed: its presence and characteristics depend on the distance between the uprights. It is shown that variation of the thickness of finitely many treads leads to the arising of any prescribed number of discrete spectrum points below the essential spectrum as well as inside the open gap. §1. Problem setting Let Γ be the union of the lines {x = (x1, x2) ∈ R : x1 = ±l/2, x2 ∈ R} and the infinite collection of segments γp = {x : |x1| < l/2, x2 = p} joining these lines; here p ∈ Z = {0,±1,±2, . . . } and l is a fixed positive parameter. We introduce the domain (1.1) Ω = { x : dist(x,Γ) < h/2 } , which has the form of an accommodation ladder (see Figure 1) with thin, of thickness h 1, uprights and treads. Consider the Dirichlet spectral problem for the Laplace operator (1.2) −Δxu(x) = λu(x), x ∈ Ω, u(x) = 0, x ∈ ∂Ω; the variation form [1] of this problem looks like this: (1.3) (∇xu,∇xv)Ωh = λ(u, v)Ωh , v ∈ H̊(Ω). Here Δx = ∇x · ∇x is the Laplacian, ∇x = (∂1, ∂2) is the gradient operator, the dot denotes the scalar product on R, and ∂k = ∂/∂xk, k = 1, 2. Also, λ h is the spectral parameter, ( , )Ωh is the scalar product in the Lebesgue space L2(Ω ), and H̊(Ω) is the space of functions of the Sobolev class H(Ω) that vanish of the boundary ∂Ω. In accordance with [2, §10.1], the variational problem (1.3) gives rise to an unbounded, positive definite, selfadjoint operator A in the Hilbert space L2(Ω ), with domain D(A) ⊂ H̊(Ω). Since the inclusion H̊(Ω) ⊂ L2(Ω) is noncompact (because Ω is unbounded), the spectrum σ(A) of A has a nonempty essential component σe(A ) (see Theorem 10.1.5 in [2]) (in fact, σ(A) = σe(A )). The Gelfand transformation [3] (1.4) u(x) → U(x; η) = 1 √ 2π ∑

On the spectrum of the Laplace operator on the infinite Dirichlet ladder