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Results tagged with differential-operators
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user 18850
Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
1
vote
Elliptic operator becomes Fredholm
Assume $L$ acting on functional space over the cylinderal end is Fredholm, then restricted to this space it has finite dimensional kernel and cokernel. The kernel of functions $L$ acting on the functi …
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2
votes
Hochschild Homology and Formal Geometry
I do not really know the six-functor proof. I assume you are talking about the paper by Wodzicki. The paper is rather dense and the proof is purely technical.
If you prefer, you may start with by lo …
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2
votes
Accepted
Reference for Weyl's law for higher order operators on closed Riemannian manifolds
One possible reference is Seeley's paper on Complex powers of Elliptic Operators, where Seeley did it for the Laplacian (page 6). But the discussion carries over to all elliptic $\Psi DO$s without muc …
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2
votes
Accepted
Atiyah-Patodi-Singer for manifolds with cusps
This type of questions has been investigated systematically by Melrose in the framework of 'c-calculus', where $c$ stands for the cusp. The basic idea, if I recall correctly is to blow up the boundary …
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24
votes
3
answers
3k
views
How we do actually compute the topological index in Atiyah-Singer?
This is migrated by math.stackexchange as I did not receive an answer. I do not know if it is too naive for this site.
I am taking a lectured class in Atiyah-Singer this semester. While the class is …
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4
votes
Symbol of the Laplace-Beltrami on $\mathbb{S}^2$
The "chart problem" you mentioned is not really a problem, because principal symbols can be defined via local coordinates and glue them together. After all we obtain pseudo-differential operators this …
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2
votes
What does the flow of the principal symbol of the differential operator tell us about the PDE?
I learned the following result from Sogge's book:
Let $P$ be a first order self-adjoint elliptic operator on $M$. Let $Q\in \Psi^{m}(M)$ be a classical $\Psi DO$ of order $m$. Then there exists a one …
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