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THE FINITE SECTION METHOD FOR DISSIPATIVE OPERATORS

Part of: Ordinary differential operators Special classes of linear operators

Published online by Cambridge University Press:  14 May 2014

Marco Marletta  and
Sergey Naboko
Marco Marletta
Affiliation:
Wales Institute of Mathematical and Computational Sciences, School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG,U.K. email marlettam@cf.ac.uk
Sergey Naboko
Affiliation:
School of Mathematics, Statistics and Actuarial Science, The University of Kent, Canterbury, Kent CT2 7NZ,U.K. email sergey.naboko@gmail.com

Abstract

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We show that for self-adjoint Jacobi matrices and Schrödinger operators, perturbed by dissipative potentials in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\ell ^1({\mathbb{N}})$ and $L^1(0,\infty )$ respectively, the finite section method does not omit any points of the spectrum. In the Schrödinger case two different approaches are presented. Many aspects of the proofs can be expected to carry over to higher dimensions, particularly for absolutely continuous spectrum.

MSC classification

Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 34L05: General spectral theory 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)

Information

Type
Research Article
Information
Mathematika , Volume 60 , Issue 2 , July 2014 , pp. 415 - 443
Copyright
Copyright © University College London 2014 

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