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Elliptic translators on manifolds with point singularities. (English. Russian original) Zbl 1267.35269
Differ. Equ. 48, No. 12, 1577-1585 (2012); translation from Differ. Uravn 48, No. 12, 1612-1620 (2012).
Consider a smooth closed manifold \(M\) and two closed submanifolds \(Y^p\), \(p= 1,2\), transversally intersecting along a smooth submanifold \(X\), \(\dim X= 0\), \(X\) being the set of singularities of \(Y= Y^1\cup Y^2\). Following the second author, the authors define a pseudodifferential translator \(T_{12}\) acting from \(Y^2\) to \(Y^1\) and develop an elliptic theory for the operators of the form \[ 1+ T: H^s(Y)\to H^s(Y),\quad H^2(Y)= H^{s_1}(Y^1)\oplus H^{s_2}(Y^2). \] The elliptic operators are of Fredholm type (Th. 1) and an index formula for them is found in Th. 2. The index of \((1+T)\) is expressed by the winding number of the invertible on the weight line \(\text{Re\,}z= \alpha\) function \(1+\sigma(T)(z)\), \(\sigma(T)(z)\) being the symbol of corresponding operator family.

MSC:
35S15 Boundary value problems for PDEs with pseudodifferential operators
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
58J32 Boundary value problems on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
47A53 (Semi-) Fredholm operators; index theories
58B15 Fredholm structures on infinite-dimensional manifolds
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References:
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