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To asymptotic complete scattering systems {M(+) + V, M(+)} on H(+) := L(2)(R(+), K, d lambda), where M(+) is the multiplication operator on H(+) and V is a trace class operator with analyticity conditions, a decay semigroup is associated such that the spectrum of the generator of this semigroup coincides with the set of all resonances (poles of the analytic continuation of the scattering matrix into the lower half plane across the positive half line), i.e. the decay semigroup yields a "time-dependent" characterization of the resonances. As a counterpart a "spectral characterization" is mentioned which is due to the "eigenvalue-like" properties of resonances.
The paper presents a classification of the basic types of admissible solutions of the general Friedmann equation with non-vanishing cosmological constant and for the case that radiation and matter do not couple. There are four distinct types. The classification uses first the discriminant of a polynomial of the third degree, closely related to the right hand side of the Friedmann equation. The decisive term is then a critical radiation density which can be calculated explicitly.
Generalized eigenvectors for resonances in the Friedrichs model and their associated Gamov vectors
(2006)
A Gelfand triplet for the Hamiltonian H of the Priedrichs model on R with multiplicity space K, dim K < infinity, is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Moller) operators the corresponding eigen(anti)linear forms on the Schwartz space S for the unperturbed Hamiltonian Ho are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector lambda -> k/(zeta(0)-lambda)(-1), zeta(0) resonance, k epsilon K, which is uniquely determined by restriction of S to S boolean AND H-+(2), where H-+(2) 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 H-+(2). That is: Exactly those pre-Gamov vectors a lambda -> k/(zeta-lambda)(-1), ( from the lower half-plane, k epsilon K., have an extension to a generalized eigenvector of H if zeta is a resonance and if k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinear form
Laudatio Hans Kaiser
(1993)