Essential spectra of pseudodifferential operators and exponential decay of their solutions. Applications to Schrödinger operators (original) (raw)
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Spectral properties of a limit-periodic Schr\"odinger operator in dimension two
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We study Schrödinger operator H = −∆ + V (x) in dimension two, V (x) being a limit-periodic potential. We prove that the spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves e i k, x at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous. *. Considering that ϕ is close to ϕ b ± π/2, we readily obtain: b, ν * = o(b 0), b, µ * = ±b 0 (1 + o(1)). Using also estimates (2.70) for h 1 , we get T 1 = ±2b 0 k(1 + o(1)). By (2.63), ∇f 1 y (1) (ϕ) − ∇f 1 y (1) (ϕ) − b = O(b 0 k −2+36s 1 +24δ). Hence, T 2 = o (b 0 k). Adding the estimates for T 1 , T 2 , we get (2.132).
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