3.5 Completeness Relation Involving Single-Particle Gamow States
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scattering states generated by a complex potential of real and imaginary parts equal
to V R (r) and V I (i), respectively, are denoted as u(k, r).
One will write the potential generating the basis as a function of the complex
variable λ:
V (λ, r) = V R (r) + λV I (r) .
(3.76)
Obviously, the potential in Eq. (3.76) is real for real values λ. Equation (3.75)
then vanishes identically in this case, as it is the difference of two completeness
relations generated by two real potentials. However, as V R (r) and V I (i) are the real
and imaginary parts of the complex potential of interest, one has to have λ = i
in Eq. (3.76). One then has to recur to analytical continuation in λ to recover the
complex-potential case.
Equation (2.2) implies that u(k, r) functions are defined for complex values of λ.
However, the resonant states u n (r) of Eq. (3.75) can no longer be normalized to one
if the norm provided by complex scaling in Eq. (3.58) is equal to zero or diverges.
Moreover, scattering states cannot be normalized with a Dirac delta if C ± = 0 in
Eq. (3.18). Consequently, one will assume for the moment that all wave functions
remain normalizable when λ varies in the complex k-plane. The special conditions
leading to diverging u n (r) and u(k, r) functions in Eq. (3.75) will be dealt with in
the following.
I (r, r , λ) in Eq. (3.75), which is defined by an absolutely converging integral,
is then an analytical function of λ, hence vanishing identically. The Berggren
completeness relation is then obtained for λ = i by an analytical continuation with
respect to λ. Consequently, the bound, resonances, and scattering states generated
by a complex potential form a complete set as long as its bound and resonance states
can be normalized with complex scaling and the integral of scattering states is well
defined.
A situation where analytic continuation must be used with caution is when
one has spectral singularities [24], that is, resonances on the real axis, which thus
generate nonanalytic points on the real-energy contour. This situation can be quickly
fixed by deforming the contour in the complex plane. Indeed, spectral singularities
can be normalized with complex scaling, as they bear the same properties as the
resonance states generated by real potentials.
Let us consider Eq. (3.76) with real potentials, that is, with λ real. One demands
therein that Eq. (3.76) is defined with resonance u n (r) states and a complex contour
of u(k, r) scattering states encompassing these resonances. When λ goes from
the real axis to the demanded value of λ = i, resonance u n (r) states will
become spectral singularities. Moreover, as one always integrates along an L +
complex contour, the integral of Eq. (3.75) is always well defined. Hence, analytic
continuation can be used when λ goes from the real axis to λ = i. Therefore,
I (r, r , λ) = 0 for λ = i, and the Berggren completeness relation also holds in
the presence of spectral singularities.
Another situation where analytic continuation cannot be used arises when one
obtains exceptional points [25–30]. Exceptional points arise when two resonances
107
scattering states generated by a complex potential of real and imaginary parts equal
to V R (r) and V I (i), respectively, are denoted as u(k, r).
One will write the potential generating the basis as a function of the complex
variable λ:
V (λ, r) = V R (r) + λV I (r) .
(3.76)
Obviously, the potential in Eq. (3.76) is real for real values λ. Equation (3.75)
then vanishes identically in this case, as it is the difference of two completeness
relations generated by two real potentials. However, as V R (r) and V I (i) are the real
and imaginary parts of the complex potential of interest, one has to have λ = i
in Eq. (3.76). One then has to recur to analytical continuation in λ to recover the
complex-potential case.
Equation (2.2) implies that u(k, r) functions are defined for complex values of λ.
However, the resonant states u n (r) of Eq. (3.75) can no longer be normalized to one
if the norm provided by complex scaling in Eq. (3.58) is equal to zero or diverges.
Moreover, scattering states cannot be normalized with a Dirac delta if C ± = 0 in
Eq. (3.18). Consequently, one will assume for the moment that all wave functions
remain normalizable when λ varies in the complex k-plane. The special conditions
leading to diverging u n (r) and u(k, r) functions in Eq. (3.75) will be dealt with in
the following.
I (r, r , λ) in Eq. (3.75), which is defined by an absolutely converging integral,
is then an analytical function of λ, hence vanishing identically. The Berggren
completeness relation is then obtained for λ = i by an analytical continuation with
respect to λ. Consequently, the bound, resonances, and scattering states generated
by a complex potential form a complete set as long as its bound and resonance states
can be normalized with complex scaling and the integral of scattering states is well
defined.
A situation where analytic continuation must be used with caution is when
one has spectral singularities [24], that is, resonances on the real axis, which thus
generate nonanalytic points on the real-energy contour. This situation can be quickly
fixed by deforming the contour in the complex plane. Indeed, spectral singularities
can be normalized with complex scaling, as they bear the same properties as the
resonance states generated by real potentials.
Let us consider Eq. (3.76) with real potentials, that is, with λ real. One demands
therein that Eq. (3.76) is defined with resonance u n (r) states and a complex contour
of u(k, r) scattering states encompassing these resonances. When λ goes from
the real axis to the demanded value of λ = i, resonance u n (r) states will
become spectral singularities. Moreover, as one always integrates along an L +
complex contour, the integral of Eq. (3.75) is always well defined. Hence, analytic
continuation can be used when λ goes from the real axis to λ = i. Therefore,
I (r, r , λ) = 0 for λ = i, and the Berggren completeness relation also holds in
the presence of spectral singularities.
Another situation where analytic continuation cannot be used arises when one
obtains exceptional points [25–30]. Exceptional points arise when two resonances