Perturbation theory for eigenvalues and resonances of Schrodinger hamiltonians

Abstract

Suppose that e2[epsilon]|x|V [set membership, variant] ReLP(R3) for some p > 2 and for g [set membership, variant] R, H(g) = - [Delta] + g V, H(g) = -[Delta] + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues [lambda](g) as they are absorbed by the continuous spectrum, that is [lambda](g) [NE pointing arrow]6 0 as g [searr]5 g0 > 0. We find a series expansion in powers of (g - g0)1/2, [lambda](g) = [summation operator]n = 2[infinity] an(g - g0)n/2 whose values for g g0 correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues.