2.4 Pöschl-Teller-Ginocchio Potential
31
Notice that the Coulomb wave functions can vary by several orders of
magnitude in a small region of the complex plane. Check their precision in the
considered region of the complex plane with the same code.
Notice the discontinuity of Coulomb wave functions at their cut on the
negative real axis. Explain the appearance of discontinuities using the analytic
properties of Coulomb wave functions.
2.4
Pöschl-Teller-Ginocchio Potential
The solvable Hamiltonians are worth considering, as they provide physical insight
that is difficult to access with numerical calculations only. Analytical solutions of
the Schrödinger equation have been derived for several nonrelativistic Hamiltonians
in Refs. [14–20]. Such solutions exist also for the Klein-Gordon [21] and Dirac
equations [22–24]. Solvable Hamiltonians corresponds mainly to one-dimensional
potentials. The harmonic oscillator Hamiltonian is the only one-body Hamiltonian
which is solvable in three-dimensions for all partial waves. However, if one removes
artificially the centrifugal and Coulomb barriers of the potential at infinity, it is
possible to devise several Hamiltonians which are fully solvable in the threedimensional case as well. In the nonrelativistic case, the hypergeometric potentials
of Natanzon type are particularly important due to their six defining parameters,
allowing these potentials to model a wide range of physical situations [19, 25–27].
The most interesting potential of this kind is the Pöschl-Teller-Ginocchio
potential [28–30]. This potential resembles typical nuclear potential and its bound
or unbound eigenstates are solvable analytically for all energies [31–34]. Hence,
contrary to the harmonic oscillator potential, the Pöschl-Teller-Ginocchio potential
yields bound, resonance, and scattering states, so that it can provide with important
information about the physics of the continuum.
Therefore, before dealing with the general case of spherical potential introduced
in Sect. 2.1, one will study the one-body eigenstates generated by a PöschlTeller-Ginocchio potential. As all wave functions and resonant eigenenergies are
analytical, the different nature of resonant and scattering wave functions is explicitly
from in the formulas providing with radial solutions of the Schrödinger equation. As
resonant energies are simple functions of the parameters entering the Pöschl-TellerGinocchio potential, one can directly see how bound states can be generated in the
presence of a sufficiently binding potential. Added to that, resonance eigenstates
can exist only for a certain class of Pöschl-Teller-Ginocchio potentials, which is
embedded in a simple inequality involving the parameters of the Pöschl-TellerGinocchio potential. Hence, the complex conditions leading to resonance formation
can be modeled in a simple analytical manner within the Pöschl-Teller-Ginocchio
potential. Hence, as many properties related to bound and continuum states can
be analytically expressed using a Pöschl-Teller-Ginocchio potential in Eq. (2.2), its
study provides an interesting introduction to the physics of continuum at one-body
31
Notice that the Coulomb wave functions can vary by several orders of
magnitude in a small region of the complex plane. Check their precision in the
considered region of the complex plane with the same code.
Notice the discontinuity of Coulomb wave functions at their cut on the
negative real axis. Explain the appearance of discontinuities using the analytic
properties of Coulomb wave functions.
2.4
Pöschl-Teller-Ginocchio Potential
The solvable Hamiltonians are worth considering, as they provide physical insight
that is difficult to access with numerical calculations only. Analytical solutions of
the Schrödinger equation have been derived for several nonrelativistic Hamiltonians
in Refs. [14–20]. Such solutions exist also for the Klein-Gordon [21] and Dirac
equations [22–24]. Solvable Hamiltonians corresponds mainly to one-dimensional
potentials. The harmonic oscillator Hamiltonian is the only one-body Hamiltonian
which is solvable in three-dimensions for all partial waves. However, if one removes
artificially the centrifugal and Coulomb barriers of the potential at infinity, it is
possible to devise several Hamiltonians which are fully solvable in the threedimensional case as well. In the nonrelativistic case, the hypergeometric potentials
of Natanzon type are particularly important due to their six defining parameters,
allowing these potentials to model a wide range of physical situations [19, 25–27].
The most interesting potential of this kind is the Pöschl-Teller-Ginocchio
potential [28–30]. This potential resembles typical nuclear potential and its bound
or unbound eigenstates are solvable analytically for all energies [31–34]. Hence,
contrary to the harmonic oscillator potential, the Pöschl-Teller-Ginocchio potential
yields bound, resonance, and scattering states, so that it can provide with important
information about the physics of the continuum.
Therefore, before dealing with the general case of spherical potential introduced
in Sect. 2.1, one will study the one-body eigenstates generated by a PöschlTeller-Ginocchio potential. As all wave functions and resonant eigenenergies are
analytical, the different nature of resonant and scattering wave functions is explicitly
from in the formulas providing with radial solutions of the Schrödinger equation. As
resonant energies are simple functions of the parameters entering the Pöschl-TellerGinocchio potential, one can directly see how bound states can be generated in the
presence of a sufficiently binding potential. Added to that, resonance eigenstates
can exist only for a certain class of Pöschl-Teller-Ginocchio potentials, which is
embedded in a simple inequality involving the parameters of the Pöschl-TellerGinocchio potential. Hence, the complex conditions leading to resonance formation
can be modeled in a simple analytical manner within the Pöschl-Teller-Ginocchio
potential. Hence, as many properties related to bound and continuum states can
be analytically expressed using a Pöschl-Teller-Ginocchio potential in Eq. (2.2), its
study provides an interesting introduction to the physics of continuum at one-body