2
The Discrete Spectrum and the Continuum
One-body Hamiltonians play an important role in the study of many-body quantum
systems. Indeed, the multiple inter-nucleon correlations between the A nucleons
inside the nucleus can be approximated by A one-body Hamiltonians, whose
potentials, acting on a single nucleon, represent the averaged interactions induced
by the remaining A − 1 nucleons. As a zeroth-order approximation, one can also
deem these potentials as spherical, as the bulk properties of nuclei can be accounted
for with spherical potentials.
The three-dimensional Schrödinger equation associated to spherical one-body
potentials can be separated in three one-dimensional equations: two depending
on angular coordinates, whose eigenstates are expressed in terms of spherical
harmonics, and one depending on the radial coordinate, which is the so-called
radial Schrödinger equation. As the latter equation is an integro-differential equation
depending on the radial coordinate r only, it is possible to determine the properties
of its eigenstates using the mathematical apparatus provided by the theory of
analytical functions [1]. In particular, one can demonstrate that the spectrum of onebody Hamiltonians whose potential vanishes for r → +∞ contains a discrete part,
made of bound states of negative energy, and a continuous part of scattering states,
which bear real positive energy. Similarly, one can show that one-body Hamiltonians
whose potential goes to infinity for r → +∞, such as the harmonic oscillator
potential, possess an infinity of bound states.
Radial Schrödinger equation can be solved in closed form in a few particular
cases of the spherically symmetric potentials. This is the case for the well-known
harmonic oscillator potential, of fundamental importance in quantum physics. The
point-particle Coulomb Hamiltonian is also analytically solvable, as the solutions
of its one-body radial Schrödinger equation can be expressed in terms of confluent hypergeometric functions. Eigenfunctions of this Hamiltonian, the so-called
Coulomb wave functions, are the asymptotic solutions of the radial Schrödinger
equation for charged particles. Finally, the Pöschl-Teller-Ginocchio potential, which
is a spherical potential mimicking the nuclear interior is analytically solvable for all
© 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_2
15
The Discrete Spectrum and the Continuum
One-body Hamiltonians play an important role in the study of many-body quantum
systems. Indeed, the multiple inter-nucleon correlations between the A nucleons
inside the nucleus can be approximated by A one-body Hamiltonians, whose
potentials, acting on a single nucleon, represent the averaged interactions induced
by the remaining A − 1 nucleons. As a zeroth-order approximation, one can also
deem these potentials as spherical, as the bulk properties of nuclei can be accounted
for with spherical potentials.
The three-dimensional Schrödinger equation associated to spherical one-body
potentials can be separated in three one-dimensional equations: two depending
on angular coordinates, whose eigenstates are expressed in terms of spherical
harmonics, and one depending on the radial coordinate, which is the so-called
radial Schrödinger equation. As the latter equation is an integro-differential equation
depending on the radial coordinate r only, it is possible to determine the properties
of its eigenstates using the mathematical apparatus provided by the theory of
analytical functions [1]. In particular, one can demonstrate that the spectrum of onebody Hamiltonians whose potential vanishes for r → +∞ contains a discrete part,
made of bound states of negative energy, and a continuous part of scattering states,
which bear real positive energy. Similarly, one can show that one-body Hamiltonians
whose potential goes to infinity for r → +∞, such as the harmonic oscillator
potential, possess an infinity of bound states.
Radial Schrödinger equation can be solved in closed form in a few particular
cases of the spherically symmetric potentials. This is the case for the well-known
harmonic oscillator potential, of fundamental importance in quantum physics. The
point-particle Coulomb Hamiltonian is also analytically solvable, as the solutions
of its one-body radial Schrödinger equation can be expressed in terms of confluent hypergeometric functions. Eigenfunctions of this Hamiltonian, the so-called
Coulomb wave functions, are the asymptotic solutions of the radial Schrödinger
equation for charged particles. Finally, the Pöschl-Teller-Ginocchio potential, which
is a spherical potential mimicking the nuclear interior is analytically solvable for all
© 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_2
15