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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

12 votes
1 answer
1k views

Path integrals, localisation

Physicists use the "Atiyah-Bott formula" for path "integrals" (for instance the supersymmetric proof of the Atiyah-Singer index theorem. Is there some way to make atleast some of these ideas rigorous? …
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1 vote
1 answer
309 views

References for weak ellipticity

There are good books (like Evans) for strongly elliptic second order linear PDE. I want to learn about weakly elliptic PDE (of any order). Are there any good books for the same? I am very curious as t …
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  • 3,383
1 vote
2 answers
1k views

Lipschitz continuity of eigenvalues and eigenvectors of Hermitian matrices

It is well-known that the eigenvalues (in decreasing order) of a Hermitian matrix $A$ are Lipschitz continuous functions of $A$. Do there exist orthonormal eigenvectors that vary in a Lipschitz contin …
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3 votes
1 answer
461 views

Regarding Discrete Eigenvalues

For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete. But, supp …
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