On the singular spectrum of Schrödinger operators with decaying potential

Authors:
S. Denisov and S. Kupin

Journal:
Trans. Amer. Math. Soc. **357** (2005), 1525-1544

MSC (2000):
Primary 34L05

DOI:
https://doi.org/10.1090/S0002-9947-04-03553-6

Published electronically:
October 5, 2004

MathSciNet review:
2115375

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Abstract: The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the so-called Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling and Christ-Kiselev, are optimal.

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Additional Information

**S. Denisov**

Affiliation:
Department of Mathematics, 253-37, Caltech, Pasadena, California 91125

Email:
denissov@its.caltech.edu

**S. Kupin**

Affiliation:
Department of Mathematics, 253-37, Caltech, Pasadena, California 91125

Address at time of publication:
CMI, Université de Provence, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France

Email:
kupin@its.caltech.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03553-6

Keywords:
Schr\"odinger operators,
Dirac operators,
Krein systems,
singular part of the spectral measure

Received by editor(s):
February 27, 2002

Received by editor(s) in revised form:
November 4, 2003

Published electronically:
October 5, 2004

Article copyright:
© Copyright 2004
American Mathematical Society