Title: Perspectives on Rank 1 Perturbations
The central question in perturbation theory is as follows. Let A and B be linear transformations; think of (NxN)-matrices or differential operators.
we know the eigenvalues of A and have partial information about B. What can we say about the eigenvalues of A+B?
The perturbation is said to be of rank 1, if we assume that B is a vector projection. This seemingly simple example (a) exhibits a surprisingly rich theory with connections
to harmonic analysis and operator theory; and (b) arises naturally in mathematical physics via a change of boundary conditions of a second order differential operator.