On inverse spectral theory for singularly perturbed operators: point spectrum

Let A be a self-adjoint operator in a separable Hilbert space H. Let <{psi(k) : k >= 1}under bar> be a given (finite or infinite) orthonormal system such that span {psi(k) : k >= 1} boolean AND dom (A) = {0} and let Lambda : = {lambda(k) : k >= 1} be an arbitrary sequence of real numbers. Conditions for the existence of a pure singular perturbation (A) over tilde of A solving the inverse eigenvalue problem (A) over tilde psi(k) = lambda(k)psi(k), k >= 1 are given.