3
Berggren Basis and Completeness Relations
The main interest of the one-body states introduced in Chap. 2 is that they form
a complete set of states. The complete character of the solutions of differential
equations defined in a finite radial interval is straightforward to demonstrate within
the regular Sturm-Liouville theory [1, 2]. The regular Sturm-Liouville problem
with nonlocal potentials is evidently more complex and was treated in Refs. [3, 4].
Extension to the singular case of infinite intervals in the frame of differential
equations only has been considered in Refs. [2, 5]. Demonstrating the general
completeness relation provided by the spectral decomposition of a self-adjoint
operator, that is, of its set of bound and scattering eigenstates, however, demands
the use of Lebesgue measure and the Riesz representation theorem [6].
The Hamiltonian h of Eq. (2.3) is self-adjoint and defined in a dense subset of the
Hilbert space of square-integrable functions. Thus, it can be written as an integral
involving its spectral decomposition, that is, the bound and scattering u(k, r) states
of Eq. (2.2), embedded in a Stieltjes-Lebesgue measure, denoted as d(u(k, r)) [6].
The existence and unicity of u(k, r) functions are a direct consequence of the
existence of the Stieltjes-Lebesgue measure [6]. One has:
h =
k
k
2 d(u(k, r)) .
The Stieltjes-Lebesgue measure d(u(k, r)) also gives rise to the resolution of
identity:
k
d(u(k, r)) = ˆ
1 .
The resolution of identity is, in fact, equivalent to the completeness of u(k, r)
eigenstates. However, the nature of the spectrum of the considered operator is
implicit therein, so that one cannot assess the convergence properties associated with
the expansion in u(k, r) eigenstates for a given physical situation. Consequently, it
© Springer International Publishing AG 2021
N. Michel, M. Płoszajczak, Gamow Shell Model, Lecture Notes in Physics 983,
https://doi.org/10.1007/978-3-030-69356-5_3
81
Berggren Basis and Completeness Relations
The main interest of the one-body states introduced in Chap. 2 is that they form
a complete set of states. The complete character of the solutions of differential
equations defined in a finite radial interval is straightforward to demonstrate within
the regular Sturm-Liouville theory [1, 2]. The regular Sturm-Liouville problem
with nonlocal potentials is evidently more complex and was treated in Refs. [3, 4].
Extension to the singular case of infinite intervals in the frame of differential
equations only has been considered in Refs. [2, 5]. Demonstrating the general
completeness relation provided by the spectral decomposition of a self-adjoint
operator, that is, of its set of bound and scattering eigenstates, however, demands
the use of Lebesgue measure and the Riesz representation theorem [6].
The Hamiltonian h of Eq. (2.3) is self-adjoint and defined in a dense subset of the
Hilbert space of square-integrable functions. Thus, it can be written as an integral
involving its spectral decomposition, that is, the bound and scattering u(k, r) states
of Eq. (2.2), embedded in a Stieltjes-Lebesgue measure, denoted as d(u(k, r)) [6].
The existence and unicity of u(k, r) functions are a direct consequence of the
existence of the Stieltjes-Lebesgue measure [6]. One has:
h =
k
k
2 d(u(k, r)) .
The Stieltjes-Lebesgue measure d(u(k, r)) also gives rise to the resolution of
identity:
k
d(u(k, r)) = ˆ
1 .
The resolution of identity is, in fact, equivalent to the completeness of u(k, r)
eigenstates. However, the nature of the spectrum of the considered operator is
implicit therein, so that one cannot assess the convergence properties associated with
the expansion in u(k, r) eigenstates for a given physical situation. Consequently, it
© Springer International Publishing AG 2021
N. Michel, M. Płoszajczak, Gamow Shell Model, Lecture Notes in Physics 983,
https://doi.org/10.1007/978-3-030-69356-5_3
81