Spectral concentration for self-adjoint operators

If the resolvent of a (not necessarily bounded) self-adjoint operator H converges strongly to the resolvent of a selfadjoint operator H, and if is an isolated eigenvalue of H of multiplicity m < oo, then although H need not have an eigenvalue near , the spectrum of H will in some cases become ''concentrated" near as tc is reduced. In fact, there exist sets C with Lebesgue measure o( p \ p ^ 0, such that the spectral projection assigned by H to C converges strongly as /c->0 to the projection on the -eigenspace of H, if and only if there exist m pairs (j K9 j\ j -1, , m, where j K -> , the JK are nearly-orthogonal unit vectors converging strongly to the -eigenspace, and IKH* j ) j \\ = o( p ). In this case, C may be taken as the union of intervals about the A j , and the j K are essentially the only numbers associated in this way with "pseudoeigenvectors" j of H . The result is applied to the weak-quantization problem in the theory of the Stark effect, where H is the Hamiltonian operator for the hydrogen atom, and H is the same for the atom in a uniform electric field which vanishes with tc. E[X, 0] is zero unless X is an eigenvalue of H, in which case E[X] is the projection on the -eigenspace.