Abstract
AbstractWe 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=fwhere a ≥ 0. The conditions are formulated in terms of orthogonality of the functionfto the solutions of the homogeneous adjoint equation.
Highlights
Linear elliptic problems in bounded domains with a sufficiently smooth boundary satisfy the Fredholm property if and only if the ellipticity condition, proper ellipticity and the Lopatinskii conditions are satisfied
In the case of unbounded domains, one more condition should be imposed in order to preserve the Fredholm property
This condition can be formulated in terms of limiting operators and requires that all limiting operators should be invertible or that the only bounded solution of limiting problems is trivial [VV06]
Summary
Linear elliptic problems in bounded domains with a sufficiently smooth boundary satisfy the Fredholm property if and only if the ellipticity condition, proper ellipticity and the Lopatinskii conditions are satisfied. The problem can be handled by the method of the Fourier transform in the absence of the potential term V (x) We show that this method can be generalized in the presence of a shallow, short-range V (x) by means of replacing the Fourier harmonics by the functions φk(x), k ∈ R3 of the continuous spectrum of the operator H0, which are the solutions of the Lippmann-Schwinger equation (see (2.1) in Section 2 and the explicit formula (2.2)). P ∈ S√n −ej , IV) When n ≥ 5 and g(x, y) ∈ L2α, x for some α > n + 2 there exists a unique solution u ∈ L2(Rn+m) if and only if: Proving solvability conditions for linear elliptic problems with non-Fredholm operators plays the crucial role in various applications including those to travelling wave solutions of reaction-diffusion systems (see [VKMP02]). Let us first establish several important properties for the functions of the spectrum of the Schrodinder operator in the left side of the equation (1.1) and for the related quantities
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