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Solvability conditions for some non-Fredholm operators

Part of: Elliptic equations and systems Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  30 November 2010

Vitali Vougalter  and
Vitaly Volpert
Vitali Vougalter
Affiliation:
University of Toronto, Department of Mathematics, Toronto, ON M5S 2E4, Canada
Vitaly Volpert
Affiliation:
Institut Camille Jordan, UMR 5208 CNRS, Université Claude Bernard Lyon 1, 69622 Villeurbanne, France, (volpert@math.univ-lyon1.fr)
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Abstract

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We obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger-type operators. One of the main results of the paper concerns solvability conditions for the equation –Δu + V(x)u–au = f where a ≥ 0. The conditions are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

Keywords

solvability conditionsnon-Fredholm operatorselliptic problems

MSC classification

Secondary: 35J10: Schrödinger operator 35P10: Completeness of eigenfunctions, eigenfunction expansions 35P25: Scattering theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 249 - 271
Copyright
Copyright © Edinburgh Mathematical Society 2010

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