Symplectic Operator–extension Techniques and Zero–range Quantum Models

F. Berezin and L. Faddeev have interpreted the Fermi zero-range model as a self-adjoint extension of the Laplacian. Various modifications of this model possess rich spectral properties, but unavoidably have negative effective radius and contain numerous parameters which do not have a direct physical meaning. We suggest, for spherically-symmetric scattering, a generalization of the Fermi zero-range model supplied with an indefinite metric in the inner space and a Hamiltonian of the inner degrees of freedom. The effective radius of this model may be positive or negative. We also propose a general principle of analyticity, formulated in terms of kcotδ(k) as a function of the scattering phase shift δ(k) depending on the wave-number k. This principle allows us to evaluate all parameters of the model, including the indefinite metric tensor of the inner space, once the basic parameters of the model: that is the spectrum of the inner Hamiltonian, the scattering length and the effective radius, are fixed, so that the sign of the effective radius is connected with the spectrum of the inner Hamiltonian by a consistency condition.