On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes

Abstract The orthogonality and completeness properties of the resonant states as defined by Humblet and Rosenfeld are investigated using a simple regularization method first suggested by Zel'dovich. It is found that at leat for finite-range potentials, the set of bound states and any finite number of proper resonant states can be completed by a set of continuum states. This makes it possible to expand the scattering and reaction amplitudes in such a way that their resonance behabiour is exhibited, and the dependence of the corresponding partial widths on the interaction becomes more explicit.

[1]  A. Siegert.  On the Derivation of the Dispersion Formula for Nuclear Reactions , 1939 .

[2]  C. Bloch,et al.  Configuration mixing in the continuum and nuclear reactions.-I , 1965 .

[3]  R. Newton.  Analytic Properties of Radial Wave Functions , 1960 .

[4]  U. Fano.  Description of States in Quantum Mechanics by Density Matrix and Operator Techniques , 1957 .

[5]  J. Jeukenne.  THEORY OF NUCLEAR REACTIONS. V. PENETRATION FACTORS FOR CHARGED PARTICLES IN LOW ENERGY NUCLEAR REACTIONS , 1964 .

[6]  Murray Gell-Mann,et al.  The formal theory of scattering , 1953 .

[7]  H. Nussenzveig.  The poles of the S-matrix of a rectangular potential well of barrier , 1959 .

[8]  C. Mahaux.  Theory of nuclear reactions: VII. Photon channels , 1965 .

[9]  L. Heller,et al.  OPTICAL ANALYSIS OF POTENTIAL WELL RESONANCES. , 1967 .

[10]  J. Schwinger,et al.  Variational Principles for Scattering Processes. I , 1950 .

[11]  H. Weidenmüller,et al.  On the theory of nuclear reactions. (II). The Jost function, sum rules, interacting continua , 1966 .

[12]  C. Bloch.  Une formulation unifiée de la théorie des réactions nucléaires , 1957 .

[13]  J. Humblet.  THEORY OF NUCLEAR REACTIONS. IV. COULOMB INTERACTIONS IN THE CHANNELS AND PENETRATION FACTORS , 1964 .

[14]  H. Weidenmüller,et al.  Treatment of the continuum-continuum interaction in the shell-model approach to nuclear reactions , 1967 .

[15]  H. Weidenmüller.  On the theory of nuclear reactions: (I). Shell-model calculations in the continuum , 1966 .

[16]  J. Humblet.  Theory of nuclear reactions: III. Physical quantities and channel radii , 1962 .

[17]  L. Rosenfeld.  Theory of nuclear reactions: II. The foundation of the optical model , 1961 .

[18]  L. Rosenfeld,et al.  Theory of nuclear reactions: I. Resonant states and collision matrix , 1961 .

[19]  J. Humblet.  THEORY OF NUCLEAR REACTIONS. VI. UNITARITY, OPEN AND CLOSED CHANNELS , 1964 .