Symplectic operator-extension techniques and zero-range quantum models

Boris S. Pavlov and Vladimir I. Kruglov


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.

zero-range potential

Math Review Classification
Primary PACS numbers : 03.65.Nk, 82.20.FD, 28.20.Cz

Last Updated
25 August 2004

17 pages

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