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|.

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

1998 ◽  
Vol 50 (3) ◽  
pp. 538-546 ◽  
Author(s):  
Richard Froese
Keyword(s):  
Simple Proof ◽  
Schrödinger Operator ◽  
Upper Bound ◽  
Counting Function ◽  
Schrödinger Operators ◽  
Upper Bounds ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Odd Dimensions

AbstractThe purpose of this note is to provide a simple proof of the sharp polynomial upper bound for the resonance counting function of a Schrödinger operator in odd dimensions. At the same time we generalize the result to the class of superexponentially decreasing potentials.

On the spectrum and the distribution of singular values of Schrödinger operators with a complex potential

1983 ◽  
Vol 388 (1794) ◽  
pp. 195-218 ◽  
Keyword(s):  
Complex Potential ◽  
Spectral Properties ◽  
Polynomial Growth ◽  
Schrödinger Operators ◽  
Singular Values ◽  
Negative Real Part ◽  
Adjoint Problem ◽  
Schrodinger Operators ◽  
Asymptotic Estimates ◽  
Integrability Conditions

This paper is concerned with spectral properties of the Schrödinger operator ─ ∆+ q with a complex potential q which has non-negative real part and satisfies weak integrability conditions. The problem is dealt with as a genuine non-self-adjoint problem, not as a perturbation of a self-adjoint one, and global and asymptotic estimates are obtained for the corresponding singular values. From these estimates information is obtained about the eigenvalues of the problem. By way of illustration, detailed calculations are given for an example in which the potential has at most polynomial growth.

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