  We present broadly applicable nonperturbative results on the behavior of
eigenvalues and eigenvectors under the addition of self-adjoint operators and
under the multiplication of unitary operators, in finite-dimensional Hilbert
spaces. To this end, we decompose these operations into elementary 1-parameter
processes in which the eigenvalues move similarly to the spheres in Newton's
cradle. As special cases, we recover level repulsion and Cauchy interlacing. We
discuss two examples of applications. Applied to adiabatic quantum computing,
we obtain new tools to relate algorithmic complexity to computational slowdown
through gap narrowing. Applied to information theory, we obtain a
generalization of Shannon sampling theory, the theory that establishes the
equivalence of continuous and discrete representations of information. The new
generalization of Shannon sampling applies to signals of varying information
density and finite length.
