6
1 Introduction: From Bound States to the Continuum
dynamics (in Q) is given by the energy-dependent effective Hamiltonian which
includes couplings to the environment (in P ). Below the lowest reaction threshold,
the effective Hamiltonian is Hermitian, whereas above the first threshold, the nonHermitian part describes the irreversible decay from the system to the environment.
Coupling to the continuum generates effective many-body interactions in Q, even if
in the combined system-plus-environment it is the two-body interaction. One should
stress that the combined system-plus-environment remains the closed quantum
system which is described by the Hermitian Hamiltonian.
In this formulation of continuum shell model, a direct consequence of the
opening of quantum system is the replacement of Hermitian Hamiltonian of
the closed quantum system by the non-Hermitian, complex-symmetric effective
Hamiltonian. The proper framework for the continuum shell model in Hilbert space
is provided by non-Hermitian quantum mechanics which became an important
alternative to the standard Hermitian quantum mechanics [16, 44]. Eigenvalues and
eigenfunctions of this Hamiltonian may be identified with physical quantities if and
only if the subspaces of the Hilbert space describing the system and the environment
are defined adequately. The subspace of the environment contains scattering wave
functions with the asymptotic conditions of the scattering matrix, whereas the
subspace of the system is defined by shell model wave functions constructed from
bound single-particle states having large amplitude inside the nucleus. Adding them
up, one immediately notice a conceptual difficulty which concerns single-particle
resonances which may have large amplitude inside the nucleus and, at the same
time, their well-defined asymptotic behavior make them a part of the environment.
Hence, the environment should be redefined by excluding from it a large amplitude
part of the single-particle resonances, which are localized in the nucleus. The so
redefined environment contains only the scattering states and the tails of the singleparticle resonances, whereas the system includes the bound states and the localized
parts of resonance wave functions inside the nucleus. In this way, one obtains the
resonance-free single-particle basis which can be used for the construction of a
complete many-body basis of the open quantum system.
For appropriately defined subspaces of the system and the environment, the
continuum shell model in Hilbert space provides the unitary description of weakly
bound and unbound states. In practice, however, the unitarity can be hardly
guaranteed in view of both the large number of reaction channels, which involve also
complicate cluster channels, and the intricate conditions, which physical particle
continua should abide. For that reason, most applications were restricted to the oneor two-nucleon scattering continuum, severely limiting the number of considered
physical cases.
The origin of difficulties encountered in Hilbert space formulation of continuum
shell model is related to how resonances are considered in Hilbert space. In
this respect, the mathematical formulation of rigged Hilbert space [2] opens new
perspectives [45–48]. The breakthrough in this field was a result of a series of
unrelated developments in mathematics and theoretical physics over more than 40
years. The theory of rigged Hilbert space, also called a Gel’fand triplet, has been
developed in the works of Gel’fand and Vilenkin [2] (see also e.g., Refs. [4, 8]) and
1 Introduction: From Bound States to the Continuum
dynamics (in Q) is given by the energy-dependent effective Hamiltonian which
includes couplings to the environment (in P ). Below the lowest reaction threshold,
the effective Hamiltonian is Hermitian, whereas above the first threshold, the nonHermitian part describes the irreversible decay from the system to the environment.
Coupling to the continuum generates effective many-body interactions in Q, even if
in the combined system-plus-environment it is the two-body interaction. One should
stress that the combined system-plus-environment remains the closed quantum
system which is described by the Hermitian Hamiltonian.
In this formulation of continuum shell model, a direct consequence of the
opening of quantum system is the replacement of Hermitian Hamiltonian of
the closed quantum system by the non-Hermitian, complex-symmetric effective
Hamiltonian. The proper framework for the continuum shell model in Hilbert space
is provided by non-Hermitian quantum mechanics which became an important
alternative to the standard Hermitian quantum mechanics [16, 44]. Eigenvalues and
eigenfunctions of this Hamiltonian may be identified with physical quantities if and
only if the subspaces of the Hilbert space describing the system and the environment
are defined adequately. The subspace of the environment contains scattering wave
functions with the asymptotic conditions of the scattering matrix, whereas the
subspace of the system is defined by shell model wave functions constructed from
bound single-particle states having large amplitude inside the nucleus. Adding them
up, one immediately notice a conceptual difficulty which concerns single-particle
resonances which may have large amplitude inside the nucleus and, at the same
time, their well-defined asymptotic behavior make them a part of the environment.
Hence, the environment should be redefined by excluding from it a large amplitude
part of the single-particle resonances, which are localized in the nucleus. The so
redefined environment contains only the scattering states and the tails of the singleparticle resonances, whereas the system includes the bound states and the localized
parts of resonance wave functions inside the nucleus. In this way, one obtains the
resonance-free single-particle basis which can be used for the construction of a
complete many-body basis of the open quantum system.
For appropriately defined subspaces of the system and the environment, the
continuum shell model in Hilbert space provides the unitary description of weakly
bound and unbound states. In practice, however, the unitarity can be hardly
guaranteed in view of both the large number of reaction channels, which involve also
complicate cluster channels, and the intricate conditions, which physical particle
continua should abide. For that reason, most applications were restricted to the oneor two-nucleon scattering continuum, severely limiting the number of considered
physical cases.
The origin of difficulties encountered in Hilbert space formulation of continuum
shell model is related to how resonances are considered in Hilbert space. In
this respect, the mathematical formulation of rigged Hilbert space [2] opens new
perspectives [45–48]. The breakthrough in this field was a result of a series of
unrelated developments in mathematics and theoretical physics over more than 40
years. The theory of rigged Hilbert space, also called a Gel’fand triplet, has been
developed in the works of Gel’fand and Vilenkin [2] (see also e.g., Refs. [4, 8]) and