Narrow Results

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on ℝ with multiplicity space , , is constructed such that exactly the resonances (poles of the inverse of the Livšic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Möller) operators the corresponding eigen(anti)linear forms on the Schwartz space for the unperturbed Hamiltonian H₀ are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector λ → k/(ζ₀ - λ)^-1, ζ₀ resonance, , which is uniquely determined by restriction of to , where denotes the Hardy space of the upper half-plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t ≥ 0 of the Toeplitz type on . That is: Exactly those pre-Gamov vectors λ → k/(ζ - λ)^-1, ζ from the lower half-plane, , have an extension to a generalized eigenvector of H if ζ is a resonance and if k is from that subspace of which is uniquely determined by its corresponding Dirac type antilinear form.

The addendum refers mainly to Sec. 5 of the paper (Friedrichs model on the positive half line). The "Schwartz space framework" is omitted because it is dispensable for the results. Improvements of the proofs are indicated. A supplement presents in the special case G₀ := ℂ\(-∞,0] a surprising implication: the scattering matrix has only simple poles and its "main part" is a linear combination of all Gamov vectors.

In this paper, we discuss the structure of Gamow states as solutions of Lippmann–Schwinger equation. The Friedrichs model is used to demonstrate it, both analytically and by applying perturbation theory to the extended spectrum of the Hamiltonian. The method presented here may be relevant to the inclusion of resonances in discrete basis, without the need of numerical constructions to define Gamow states, as entities depending on the choice of integrals contours, or as states resulting from ad hoc discretizations of the continuum.

We present that the X(3872) could be represented as a dynamically generated state in the extended Friedrichs scheme, in which the ratio of “elementariness” and “compositeness” of the different components in the X(3872) is about $Z_{c\overline{c}}:X_{{\overline{D}}^0{D}^{0*}}:X_{D^{+}{D}^{-*}}:X_{{\overline{D}}^{*}{D}^{*}}=1:\left(2.67\sim 8.85\right):\left(0.45\sim 0.46\right):0.04$. Furthermore, its decays to π⁰ and a P-wave charmonium χ_cJ state with J = 0; 1, or 2, J/ψπ⁺π⁻, and J/ψπ⁺π⁻π⁰ could be calculated out with the help of Barnes-Swanson model. The isospin breaking effects is easily understood in this scheme. This calculation also shows that the decay rate of X(3872) to χ_c1π⁰ is much smaller than its decay rate to J/ψπ⁺π⁻.