| F. Berezin and L. Faddeev interpreted Fermi zero-range model as a self-adjoint extension of the Laplacian. Various modifications of this model in conventional Hilbert space possess rich spectral properties, but unavoidably have the 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. Effective radius of this model may be both positive or negative. We propose also a general {it principle of analyticity} formulated in terms of $k{rm cot}delta (k)$ as a function of the scattering phase shift $delta(k)$ 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: the spectrum $sigma_{_{p}}$ of the inner Hamiltonian, the scattering length and the effective radius, are fixed, such that the sign of the effective radius is connected with the spectrum $sigma_{_{p}}$ by an appropriate consistency condition. The absolutely-continuous part of the extension plays a role of the quantum Hamiltonian of the special model. |
Keywords
zero-range potential
Math Review Classification
Primary PACS numbers : 03.65.Nk, 82.20.FD, 28.20.Cz
Last Updated
25 August 2004
Length
17 pages
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