overcomplete in the Fock space and satisfy a resolution of the identity being an open disc in the complex plane of radius , the radius of convergence of the series (in the case of the CCS, .) The measure is generically of the form (for ), where is related to the through the moment condition. Once again, we see that for an arbitrary vector in the Fock space, the function is of the form , where is an analytic function on the domain . The reproducing kernel associated to these coherent states is Barut–Girardello coherent states By analogy with the CCS case, one can define a generalized annihilation operator by its action on the vectors , and its adjoint operator . These act on the Fock states as Depending on the exact values of the quantities , these two operators, together with the identity and all their commutators, could generate a wide range of algebras including various types of deformed quantum algebras. The term 'nonlinear', as often applied to these generalized coherent states, comes again from quantum optics where many such families of states are used in studying the interaction between the radiation field and atoms, where the strength of the interaction itself depends on the frequency of radiation. Of course, these coherent states will not in general have either the group theoretical or the minimal uncertainty properties of the CCS (there might have more general ones). Operators and of the general type defined above are also known as ladder operators . When such operators appear as generators of representations of Lie algebras, the eigenvectors of are usually called Barut–Girardello coherent states. A typical example is obtained from the representations of the Lie algebra of SU(1,1) on the Fock space. Gazeau–Klauder coherent states A non-analytic extension of the above expression of the non-linear coherent states is often used to define generalized coherent states associated to physical Hamiltonians having pure point spectra. These coherent states, known as Gazeau–Klauder coherent states, are labelled by action-angle variables. Suppose that we are given the physical Hamiltonian , with , i.e., it has the energy eigenvalues and eigenvectors , which we assume to form an orthonormal basis for the Hilbert space of states . Let us write the eigenvalues as by introducing a sequence of dimensionless quantities ordered as: . Then, for all and , the Gazeau–Klauder coherent states are defined as where again is