6.6 Comparison of Gamow Shell Model and Hamiltonian Complex Scaling. . .
293
The complex-scaled Hamiltonian arises from a similarity transformation using
the ˆ
U θ operator of Eq. (6.66):
ˆ
H θ = ˆ
U θ ˆ
H ˆ
U
−1
θ .
(6.67)
The basic properties of ˆ
U θ and ˆ
H θ in Eqs. (6.66) and (6.67) in the one-body case,
necessary in order to use complex-scaled Hamiltonians in practical applications, are
studied in Exercise III.
One can write the general local one-body complex-scaled spherical Hamiltonian
ˆ
H θ (see Exercise III):
ˆ
H θ = e
−2iθ p 2
2m
+ U(r e
iθ ) ,
(6.68)
where m is the mass of the nucleon and ˆ
U (r) is a one-body potential. Let us consider
the potentials which can give rise to complex-scaled Hamiltonians. The fundamental
issue of Eq. (6.68) is evidently to be able to define ˆ
U(r e iθ ) with complex values for
the θ angles of interest. In fact, complex scaling of potentials is well defined only
in the case of dilation analytic potentials, i.e. potentials which admit an analytic
continuation for |θ | < π/4, as demanded in the ABC theorem [139]. This is a rather
restrictive condition, because potentials as pervasive as Woods–Saxon potentials do
not belong to this category. Indeed, a Woods–Saxon potential possesses an infinity
of poles in the complex r-plane. They are equal to
z n = R 0 + iπd(2n + 1) ,
where R 0 and d are the radius and diffuseness of the potential, respectively, and n
is an integer. Thus, Hamiltonian complex scaling cannot be applied if |θ | ≥ arg(z 0 )
because of the appearance of poles. In particular, the requirements of the ABC
theorem are not fulfilled by a Woods–Saxon potential bearing R 0 = 3 fm and
d = 0.65 fm, which are typical values for light nuclei, as one has arg(z 0 ) ∼ 0.2 in
this case. Therefore, it is often preferred to use other potentials than Woods–Saxon
potentials to mimic the effect of the core.
A popular potential is the Kanada–Kaneko–Nagata–Nomoto potential [155,156],
or shortly the KKNN potential [155]). The KKNN potential is built from Gaussian
functions, so that it is dilation analytic, and is of physical interest as it resembles a
Woods–Saxon potential on the real r-axis [155,156]. Let us determine the spectrum
of ˆ
H θ (see Eq. (6.68)). For this, one will consider the operator e 2iθ ˆ
H θ . Clearly,
e 2iθ ˆ
H θ has the form of a non-rotated one-body Hamiltonian bearing a complex
potential. Hence, the bound and scattering eigenstates of e 2iθ ˆ
H θ form a Newton
completeness relation (see Sect. 3.5.2). Consequently, the resonant states of e 2iθ ˆ
H θ
consist of localized wave functions on the real r-axis, hence whose linear momenta
belong to the upper complex plane. The scattering eigenstates of e 2iθ ˆ
H θ bear
real positive energies (see Sect. 3.5.2). Thus, the ˆ
H θ operator of Eq. (6.68) has a
293
The complex-scaled Hamiltonian arises from a similarity transformation using
the ˆ
U θ operator of Eq. (6.66):
ˆ
H θ = ˆ
U θ ˆ
H ˆ
U
−1
θ .
(6.67)
The basic properties of ˆ
U θ and ˆ
H θ in Eqs. (6.66) and (6.67) in the one-body case,
necessary in order to use complex-scaled Hamiltonians in practical applications, are
studied in Exercise III.
One can write the general local one-body complex-scaled spherical Hamiltonian
ˆ
H θ (see Exercise III):
ˆ
H θ = e
−2iθ p 2
2m
+ U(r e
iθ ) ,
(6.68)
where m is the mass of the nucleon and ˆ
U (r) is a one-body potential. Let us consider
the potentials which can give rise to complex-scaled Hamiltonians. The fundamental
issue of Eq. (6.68) is evidently to be able to define ˆ
U(r e iθ ) with complex values for
the θ angles of interest. In fact, complex scaling of potentials is well defined only
in the case of dilation analytic potentials, i.e. potentials which admit an analytic
continuation for |θ | < π/4, as demanded in the ABC theorem [139]. This is a rather
restrictive condition, because potentials as pervasive as Woods–Saxon potentials do
not belong to this category. Indeed, a Woods–Saxon potential possesses an infinity
of poles in the complex r-plane. They are equal to
z n = R 0 + iπd(2n + 1) ,
where R 0 and d are the radius and diffuseness of the potential, respectively, and n
is an integer. Thus, Hamiltonian complex scaling cannot be applied if |θ | ≥ arg(z 0 )
because of the appearance of poles. In particular, the requirements of the ABC
theorem are not fulfilled by a Woods–Saxon potential bearing R 0 = 3 fm and
d = 0.65 fm, which are typical values for light nuclei, as one has arg(z 0 ) ∼ 0.2 in
this case. Therefore, it is often preferred to use other potentials than Woods–Saxon
potentials to mimic the effect of the core.
A popular potential is the Kanada–Kaneko–Nagata–Nomoto potential [155,156],
or shortly the KKNN potential [155]). The KKNN potential is built from Gaussian
functions, so that it is dilation analytic, and is of physical interest as it resembles a
Woods–Saxon potential on the real r-axis [155,156]. Let us determine the spectrum
of ˆ
H θ (see Eq. (6.68)). For this, one will consider the operator e 2iθ ˆ
H θ . Clearly,
e 2iθ ˆ
H θ has the form of a non-rotated one-body Hamiltonian bearing a complex
potential. Hence, the bound and scattering eigenstates of e 2iθ ˆ
H θ form a Newton
completeness relation (see Sect. 3.5.2). Consequently, the resonant states of e 2iθ ˆ
H θ
consist of localized wave functions on the real r-axis, hence whose linear momenta
belong to the upper complex plane. The scattering eigenstates of e 2iθ ˆ
H θ bear
real positive energies (see Sect. 3.5.2). Thus, the ˆ
H θ operator of Eq. (6.68) has a