Lee-friedrichs Model

What is known today as the Lee-Friedrichs model is characterized by a self-adjoint operator H on a Hilbert space H, which is the sum of two self-adjoint operators H0 and V , such that H,H0 and V have common domain; H0 has absolutely continuous spectrum (of uniform multiplicity) except for the end-point of the semi-bounded from below spectrum, and one or more eigenvalues which may or may not be embedded in the continuum. The operator V is compact and of finite rank, and induces a map from the subspace of H spanned by the eigenvectors of H0 to the subspace corresponding to the continuous spectrum (and the reverse). The central idea of the model is that V does not map the subspace corresponding to the continuous spectrum into itself, and, as a consequence, the model becomes solvable in the sense that we shall describe below. In the physical applications of the model, H corresponds to the Hamiltonian operator, the self-adjoint operator (often the self-adjoint completion of an essentially self-adjoint operator) that generates the unitary evolution (through Schrödinger’s equation) of the vector in H representing the state of the physical system in time. The resolvent G(z) = (z − H) generated by the Laplace transform on [0,∞) by e on the Schrödinger evolution operator e (both acting on some suitable f ∈ H) is analytic in the upper half z-plane. Denoting by 〈λ|f) (with Lebesgue measure dλ) the representation of f ∈ H on the continuous spectrum λ of H0 on [0,∞) and φ ∈ H the eigenvector with eigenvalue E0 (assuming for this illustration just one discrete eigenvector), we see that the second resolvent equation