Eigenvalues of differential operators between vector bundles

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The Atiyah‒Singer index theorem states that the analytical index of an elliptic pseudo-differential operator from one vector bundle over a manifold to another equals a certain topological index defined from its principal symbol.




Is there any similar formula for the eigenvalues of such an operator, defined in terms of a topological quantity?




I've googled a little and found



https://www.math.uni-bielefeld.de/~grigor/esceps.pdf



where the special case of the Laplacian and a certain Schrödinger operator is treated (apparently there is an old asymptotic formula by Weyl and a new method using so-called "capacitors" which the paper is all about). I'm interested in a method that would be applicable to the whole class of elliptic operators.




operator-theory differential-topology




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    The Atiyah‒Singer index theorem states that the analytical index of an elliptic pseudo-differential operator from one vector bundle over a manifold to another equals a certain topological index defined from its principal symbol.




    Is there any similar formula for the eigenvalues of such an operator, defined in terms of a topological quantity?




    I've googled a little and found



    https://www.math.uni-bielefeld.de/~grigor/esceps.pdf



    where the special case of the Laplacian and a certain Schrödinger operator is treated (apparently there is an old asymptotic formula by Weyl and a new method using so-called "capacitors" which the paper is all about). I'm interested in a method that would be applicable to the whole class of elliptic operators.




    operator-theory differential-topology




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      The Atiyah‒Singer index theorem states that the analytical index of an elliptic pseudo-differential operator from one vector bundle over a manifold to another equals a certain topological index defined from its principal symbol.




      Is there any similar formula for the eigenvalues of such an operator, defined in terms of a topological quantity?




      I've googled a little and found



      https://www.math.uni-bielefeld.de/~grigor/esceps.pdf



      where the special case of the Laplacian and a certain Schrödinger operator is treated (apparently there is an old asymptotic formula by Weyl and a new method using so-called "capacitors" which the paper is all about). I'm interested in a method that would be applicable to the whole class of elliptic operators.




      operator-theory differential-topology




      share|cite|improve this question











      The Atiyah‒Singer index theorem states that the analytical index of an elliptic pseudo-differential operator from one vector bundle over a manifold to another equals a certain topological index defined from its principal symbol.




      Is there any similar formula for the eigenvalues of such an operator, defined in terms of a topological quantity?




      I've googled a little and found



      https://www.math.uni-bielefeld.de/~grigor/esceps.pdf



      where the special case of the Laplacian and a certain Schrödinger operator is treated (apparently there is an old asymptotic formula by Weyl and a new method using so-called "capacitors" which the paper is all about). I'm interested in a method that would be applicable to the whole class of elliptic operators.





      operator-theory differential-topology





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









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