Geodesic
On the Stability of the Index of Unbounded Nonlocal Operators in Sobolev Spaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 116-135

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Unbounded operators corresponding to nonlocal elliptic problems on a bounded region $G\subset\mathbb R^2$ are considered. The domain of these operators consists of functions in the Sobolev space $W_2^m(G)$ that are generalized solutions of the corresponding elliptic equation of order $2m$ with the right-hand side in $L_2(G)$ and satisfy homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved that lower order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal perturbations on the boundary do not change the index are also formulated. 
@article{TM_2006_255_a8,
     author = {P. L. Gurevich},
     title = {On the {Stability} of the {Index} of {Unbounded} {Nonlocal} {Operators} in {Sobolev} {Spaces}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {116--135},
     publisher = {mathdoc},
     volume = {255},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2006_255_a8/}
}
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P. L. Gurevich. On the Stability of the Index of Unbounded Nonlocal Operators in Sobolev Spaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 116-135. http://geodesic.mathdoc.fr/item/TM_2006_255_a8/