Schrödinger Operators with Sparse Potentials: Asymptotics of the Fourier Transform¶of the Spectral Measure

2001 ◽  
Vol 223 (3) ◽  
pp. 509-532 ◽  
Author(s):  
Denis Krutikov ◽  
Christian Remling
Keyword(s):  
Fourier Transform ◽  
Spectral Measure ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Sparse Potentials ◽  
The Fourier Transform

Correction to: Upper Bounds for the Resonance Counting Function of Schrödinger Operators in Odd Dimensions

2001 ◽  
Vol 53 (4) ◽  
pp. 756-757 ◽  
Author(s):  
Richard Froese
Keyword(s):  
Fourier Transform ◽  
Counting Function ◽  
Schrödinger Operators ◽  
Singular Values ◽  
Upper Bounds ◽  
Schrodinger Operators ◽  
The Fourier Transform ◽  
Odd Dimensions

AbstractThe proof of Lemma 3.4 in [F] relies on the incorrect equality μj(AB) = μj(BA) for singular values (for a counterexample, see [S, p. 4]). Thus, Theorem 3.1 as stated has not been proven. However, with minor changes, we can obtain a bound for the counting function in terms of the growth of the Fourier transform of |V|.

THE EXPONENTIAL LOCALIZATION AND STRUCTURE OF THE SPECTRUM FOR 1D QUASI-PERIODIC DISCRETE SCHRÖDINGER OPERATORS

1991 ◽  
Vol 03 (03) ◽  
pp. 241-284 ◽  
Author(s):  
V. A. CHULAEVSKY ◽  
YA. G. SINAI
Keyword(s):  
Spectral Measure ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Discrete Schrödinger Operators ◽  
Exponential Localization

We discuss main mechanisms of the exponential localization of the eigenfunctions for one-dimensional quasi-periodic Schrödinger operators with the potential of the form V(α + nω), where V(α) is a non-degenerate C2-function on the d-dimensional torus, and ω ∈ ℝd is a typical vector with rationally incommensurate components. The exponential localization is proved so far for d ≤ 2. We emphasize the different nature of the support of the spectral measure for d = 1 and for d > 1.

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