Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-04T21:53:46.805Z Has data issue: false hasContentIssue false

Solvability conditions for some non-Fredholm operators

Published online by Cambridge University Press:  30 November 2010

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, ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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)uau = f where a ≥ 0. The conditions are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2010

References

1.Amrouche, C. and Bonzom, F., Mixed exterior Laplace's problem, J. Math. Analysis Applic. 338 (2008), 124140.CrossRefGoogle Scholar
2.Benkirane, N., Propriété d'indice en théorie Holderienne pour des opérateurs elliptiques dans Rn, C. R. Acad. Sci. Paris Sér. I 307 (1988), 577580.Google Scholar
3.Bolley, P. and Pham, T. L., Propriété d'indice en théorie Holderienne pour des opérateurs différentiels elliptiques dans Rn, J. Math. Pures Appl. 72 (1993), 105119.Google Scholar
4.Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B., Schrödinger operators with application to quantum mechanics and global geometry (Springer, 1987).Google Scholar
5.Ducrot, A., Marion, M. and Volpert, V., Systemes de réaction-diffusion sans propriété de Fredholm, C. R. Acad. Sci. Paris Sér. I 340 (2005), 659664.CrossRefGoogle Scholar
6.Ducrot, A., Marion, M. and Volpert, V., Reaction-diffusion problems with non-Fredholm operators, Adv. Diff. Eqns 13 (2008), 11511192.Google Scholar
7.Kato, T., Wave operators and similarity for some non-selfadjoint operators, Math. Annalen 162 (1966), 258279.CrossRefGoogle Scholar
8.Kryzhevich, S. and Volpert, V., Different types of solvability conditions for differential operators, Electron. J. Diff. Eqns 100 (2006), 124.Google Scholar
9.Lieb, E. and Loss, M., Analysis, Graduate Studies in Mathematics, Volume 14 (American Mathematical Society, Providence, RI, 1997).Google Scholar
10.Lockhart, R. B., Fredholm property of a class of elliptic operators on non-compact manifolds, Duke Math. J. 48 (1981), 289312.CrossRefGoogle Scholar
11.Lockhart, R. B. and McOwen, R. C., On elliptic systems in Rn, Acta Math. 150 (1983), 125135.CrossRefGoogle Scholar
12.Nirenberg, L. and Walker, H. F., The null spaces of elliptic partial differential operators in Rn, J. Math. Analysis Applic. 42 (1973), 271301.CrossRefGoogle Scholar
13.Reed, M. and Simon, B., Methods of modern mathematical physics, Volume III: Scattering theory (Academic Press, 1979).Google Scholar
14.Rodnianski, I. and Schlag, W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (3) (2004), 451513.CrossRefGoogle Scholar
15.Simon, B., Quantum mechanics for Hamiltonians defined as quadratic forms, Princeton Series in Physics (Princeton University Press, 1971).Google Scholar
16.Volpert, A. and Volpert, V., Fredholm property of elliptic operators in unbounded domains, Trans. Moscow Math. Soc. 67 (2006), 127197.CrossRefGoogle Scholar
17.Volpert, V., Kazmierczak, B., Massot, M. and Peradzynski, Z., Solvability conditions for elliptic problems with non-Fredholm operators, Applic. Math. 29 (2) (2002), 219238.Google Scholar
18.Walker, H. F., On the null-space of first-order elliptic partial differential operators in Rn, Proc. Am. Math. Soc. 30 (2) (1971), 278286.Google Scholar
19.Walker, H. F., A Fredholm theory for a class of first-order elliptic partial differential operators in Rn, Trans. Am. Math. Soc. 165 (1972), 7586.Google Scholar