Absolute Continuity of the Spectrum of a Schrödinger Operator with a Potential Which is Periodic in Some Directions and Decays in Others

We prove that the spectrum of a Schrödinger operator with a potential which is periodic in certain directions and superexponentially decaying in the others is purely absolutely continuous. Therefore, we reduce the operator using the Bloch-Floquet-Gelfand transform in the periodic variables, and show that, except for at most a set of quasi-momenta of measure zero, the reduced operators satisfies a limiting absorption principle. 2000 Mathematics Subject Classification: 35J10, 35Q40, 81C10 1 Formulation of the result There are many papers (see, for example, [1, 9]) devoted to the question of the absolute continuity of the spectrum of differential operators with coefficients periodic in the whole space. In the present article, we consider the situation where the coefficients are periodic in some variables and decay very fast (super-exponentially) when the other variables tend to infinity. The corresponding operator describes the scattering of waves on an infinite membrane or filament. Recently, quite a few studies have been devoted to similar problems, for periodic, quasi-periodic or random surface Hamiltonians (see, e.g. [3, 7, 2]). N.F.’s research was partially supported by the FNS 2000 “Programme Jeunes Chercheurs”. F.K.’s research was partially supported by the program RIAC 160 at Université Paris 13 and by the FNS 2000 “Programme Jeunes Chercheurs”. Documenta Mathematica 9 (2004) 107–121 108 N. Filonov and F. Klopp Let (x, y) denote the points of the space R. Define Ω = R × (0, 2π) and 〈x〉 = √ x2 + 1. For a ∈ R, introduce the spaces Lp,a = {f : ef ∈ Lp(Ω)}, H a = {f : ef ∈ H(Ω)}, where 1 ≤ p ≤ ∞ and H(Ω) is the Sobolev space. Our main result is Theorem 1.1. Consider in L2(R ) the self-adjoint operator Hu = −div(g∇u) + V u (1) and assume that the functions g : R → R and V : R → R satisfy following conditions: 1. ∀l ∈ Z, ∀(x, y) ∈ R, g(x, y + 2πl) = g(x, y), V (x, y + 2πl) = V (x, y); 2. there exists g0 > 0 such that (g − g0), ∆g, V ∈ L∞,a for any a > 0; 3. there exists c0 > 0 such that ∀(x, y) ∈ R, g(x, y) ≥ c0. Then, the spectrum of H is purely absolutely continuous. Remark 1.1. Operators with different values of g0 differ from one another only by multiplication by a constant; so, without loss of generality, we can and, from now on, do assume that g0 = 1. Remark 1.2. If V ≡ 0, (1) is the acoustic operator. If g ≡ 1, it is the Schrödinger operator with electric potential V . The basic philosophy of our proof is the following. To prove the absolute continuity of the spectrum for periodic operators (i.e., periodic with respect to a non degenerate lattice in R), one applies the Floquet-Bloch-Gelfand reduction to the operator and one is left with proving that the Bloch-Floquet-Gelfand eigenvalues must vary with the quasi-momentum i.e., that they cannot be constant on sets of positive measure (see e.g. [9]). If one tries to follow the same line in the case of operators that are only periodic with respect to a sub-lattice, the problem one encounters is that, as the resolvent of the Bloch-Floquet-Gelfand reduction of the operator is not compact, its spectrum may contain continuous components and some Bloch-Floquet-Gelfand eigenvalues may be embedded in these continuous components. The perturbation theory of such embedded eigenvalues (needed to control their behavior in the Bloch quasi-momentum) is more complicated than that of isolated eigenvalues. To obtain a control on these eigenvalues, we use an idea of the theory of resonances (see e.g. [13]): if one analytically dilates Bloch-Floquet-Gelfand reduction of the operator, these embedded eigenvalues become isolated eigenvalues, and thus can be controlled in the usual way. Documenta Mathematica 9 (2004) 107–121 Absolute Continuity of the Spectrum 109 Let us now briefly sketch our proof. We make the Bloch-Floquet-Gelfand transformation with respect to the periodic variables (see section 3) and get a family of operators H(k) in the cylinder Ω. Then, we consider the corresponding resolvent in suitable weighted spaces. It analytically depends on the quasimomentum k and the spectral (non real) parameter λ. It turns out that we can extend it analytically with respect to λ from the upper half-plane to the lower one (see Theorem 5.1 below) and thus establish the limit absorption principle. This suffices to prove the absolute continuity of the initial operator (see section 7). Note that an analytic extension of the resolvent of the operator (1) with coefficients g and V which decay in all directions is constructed in the paper [4] (with m = 3, d = 0; see also [10] for g ≡ 1). In the case of a potential decaying in all directions but one (i.e., if d = 1), the analytic extension of the resolvent of the whole operator (1) (not only for the operator H(k) (see section 3)) is investigated in [6] when g ≡ 1. Note also that our approach has shown to be useful in the investigation of the perturbation of free operator in the half-plane by δ-like potential concentrated on a line (see [5]); the wave operators are also constructed there. In section 2, we establish some auxiliary inequalities. In section 3, we define the Floquet-Gelfand transformation and construct an analytic extension of the resolvent of free operator in the cylinder Ω. In sections 4 and 5, we prove a limiting absorption principle for the initial operator in the cylinder. An auxiliary fact from theory of functions is established in section 6. Finally, the proof of Theorem 1.1 is completed in section 7. We denote by Bδ(k0) a ball in real space Bδ(k0) = {k ∈ R : |k − k0| < δ} and by k1 the first coordinate of k, k = (k1, k ). We will use the spaces of function in Ω with periodic boundary conditions,