An infinite-dimensional Evans function theory for elliptic boundary value problems

An infinite-dimensional Evans function E(λ) and a stability index theorem are developed for the elliptic eigenvalue problem in a bounded domain Ω Rm. The number of zero points of the Evans function in a bounded, simply connected complex domain D is shown to be equal to the number of eigenvalues of the corresponding elliptic operator in D. When the domain Ω is star-shaped, an associated unstable bundle based on D is constructed, and the first Chern number of also gives the number of eigenvalues of the elliptic operator inside D.