1 Introduction: From Bound States to the Continuum
9
Let us now review different chapters in some details.
Chapter 2 Here the essential mathematical formalism for the discussion of
discrete states and continuum is presented along with the examples of various
useful potentials. The emphasis is put on the analytical properties of the wave
functions, their asymptotic behavior, and analyticity of the complex momentum
wave functions. These properties are used in a subsequent chapter to prove the
one-body completeness relations in general case. Much attention is devoted to
a discussion of the Coulomb potential and Coulomb wave functions. This basic
material is then referred to in different contexts all over this textbook.
Chapter 3 This chapter is devoted to the derivation of the one-body completeness
relation in different special cases, and in the general case of the Coulomb wave
function and the finite angular momentum. In particular, the Berggren completeness
relation for Gamow states will be derived and discussed for real and complex
local potentials, as well as for the nonlocal potentials. Important practical question
concerns the domains of applicability of the Berggren completeness relation. This
discussion, which begins in this chapter, will be further continued in Chap. 5, where
the many-body completeness relation is formulated.
Different aspects of Gamow states, necessary for proving the completeness
relation or perform practical calculations with Gamow functions, such as the
normalization and orthogonality of Gamow states, and the calculation matrix
elements of one- and two-body potentials are discussed as well. Together with
the analytical discussion of completeness relations, we will also analyze different
aspects of the numerical implementation of the Berggren relation, in a standard case
which includes resonances, and also in the special case of antibound neutron state.
The last part of this chapter includes the presentation of basic properties of complexsymmetric matrices which replace Hermitian matrices in the general Gamow shell
model problem.
Chapter 4 Various physically important two-particle systems are discussed in this
chapter. These include unbound dineutron, diproton, and weakly bound deuteron,
which will be described in Berggren basis using a realistic chiral interaction. In this
context, it is instructive to see a qualitative difference between singlet (S = 0) and
triplet (S = 1) two-nucleon channels. The second class of two-particle systems is
the dipolar and quadrupolar anions which are described using the extension of the
particle-plus-rotor model in Berggren ensemble. For certain ranges of dipole and/or
quadrupole moments, the anions form extremely extended, weakly bound systems.
The rich spectrum of resonances with a characteristic band structure is foreseen in
these anions. The resonance spectra depend strongly on the asymptotic properties
of the pseudo-potentials in each studied case. The particle-plus-rotor model is also
used to describe the spectra of favored and unfavored resonance bands in a oneneutron halo nucleus of 11 Be.
9
Let us now review different chapters in some details.
Chapter 2 Here the essential mathematical formalism for the discussion of
discrete states and continuum is presented along with the examples of various
useful potentials. The emphasis is put on the analytical properties of the wave
functions, their asymptotic behavior, and analyticity of the complex momentum
wave functions. These properties are used in a subsequent chapter to prove the
one-body completeness relations in general case. Much attention is devoted to
a discussion of the Coulomb potential and Coulomb wave functions. This basic
material is then referred to in different contexts all over this textbook.
Chapter 3 This chapter is devoted to the derivation of the one-body completeness
relation in different special cases, and in the general case of the Coulomb wave
function and the finite angular momentum. In particular, the Berggren completeness
relation for Gamow states will be derived and discussed for real and complex
local potentials, as well as for the nonlocal potentials. Important practical question
concerns the domains of applicability of the Berggren completeness relation. This
discussion, which begins in this chapter, will be further continued in Chap. 5, where
the many-body completeness relation is formulated.
Different aspects of Gamow states, necessary for proving the completeness
relation or perform practical calculations with Gamow functions, such as the
normalization and orthogonality of Gamow states, and the calculation matrix
elements of one- and two-body potentials are discussed as well. Together with
the analytical discussion of completeness relations, we will also analyze different
aspects of the numerical implementation of the Berggren relation, in a standard case
which includes resonances, and also in the special case of antibound neutron state.
The last part of this chapter includes the presentation of basic properties of complexsymmetric matrices which replace Hermitian matrices in the general Gamow shell
model problem.
Chapter 4 Various physically important two-particle systems are discussed in this
chapter. These include unbound dineutron, diproton, and weakly bound deuteron,
which will be described in Berggren basis using a realistic chiral interaction. In this
context, it is instructive to see a qualitative difference between singlet (S = 0) and
triplet (S = 1) two-nucleon channels. The second class of two-particle systems is
the dipolar and quadrupolar anions which are described using the extension of the
particle-plus-rotor model in Berggren ensemble. For certain ranges of dipole and/or
quadrupole moments, the anions form extremely extended, weakly bound systems.
The rich spectrum of resonances with a characteristic band structure is foreseen in
these anions. The resonance spectra depend strongly on the asymptotic properties
of the pseudo-potentials in each studied case. The particle-plus-rotor model is also
used to describe the spectra of favored and unfavored resonance bands in a oneneutron halo nucleus of 11 Be.