Control for Schrödinger Operators on Tori
نویسندگان
چکیده
A well known result of Jaffard states that an arbitrary region on a torus controls, in the L sense, solutions of the free stationary and dynamical Schrödinger equations. In this note we show that the same result is valid in the presence of a potential, that is for Schrödinger operators, −∆ + V , V ∈ C∞.
منابع مشابه
The Essential Spectrum of Schrödinger, Jacobi, and Cmv Operators
We provide a very general result that identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein’s results on orthogonal polynomial...
متن کاملThe Essential Spectrum of Schrödinger, Jacobi, and Cmv Operators Yoram Last and Barry Simon
We provide a very general result that identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein’s results on orthogonal polynomial...
متن کاملOn two-dimensional finite-gap potential Schrödinger and Dirac operators with singular spectral curves
In the present paper we describe a wide class of two-dimensional potential Schrödinger and Dirac operators which are finite-gap on the zero energy level and whose spectral curves at this level are singular and, in particular, may have n-multiple points with n ≥ 3. Dirac operators with such spectral curves are important for the Weierstrass representation of tori in R [1, 2]. A study of finite-ga...
متن کاملContinuity of Measure of the Spectrum for Schrödinger Operators with Potentials Driven by Shifts and Skew-shifts on Tori
We study discrete Schrödinger operators on l2(Z) with γ-Lipschitz potentials defined on higher dimensional torus (Td, T ), where T is a shift or skew-shift with frequency α. We show that under the positive Lyapunov exponent condition, measure of the spectrum at irrational frequency is the limit of measures of spectra at rational approximations.
متن کاملControl for Schrödinger Operators on 2-tori: Rough Potentials
For the Schrödinger equation, (i∂t + ∆)u = 0 on a torus, an arbitrary nonempty open set Ω provides control and observability of the solution: ‖u|t=0‖L2(T2) ≤ KT ‖u‖L2([0,T ]×Ω). We show that the same result remains true for (i∂t + ∆ − V )u = 0 where V ∈ L(T), and T is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for V ∈ C(T) and c...
متن کامل