3.3 Normalization and Orthogonality of Gamow States and One-Body Matrix. . .
101
If u i (r) and u f (r) are wave functions of bound or decaying states, one can write
k i = |k i |e −iα i and k f = |k f |e −iα f . As u i ∼ a i (z)e ik i z and u f (z) ∼ a f (z)e ik f z
when (z) → +∞, with a i and a f the algebraic increasing functions, the integrals
defining F (V o ) converge if one takes:
θ > α f + α i .
(3.61)
In addition, the expression for F (V o ) is analytical because F (V o ) is a function of
converging integrals of analytical functions. Square root in Eq. (3.57) causes no
problems because N 2
i and N 2
f never cross the negative real axis when V o varies.
Consequently, following the theorem of analytic continuation, Eq. (3.54) defines
also F (V o ) for V o > V lim . In this way, one may calculate the radial matrix elements
of resonance states which a priori are not normalizable.
Complex scaling in Eq. (3.58) can be readily applied to Eq. (2.174) when u(k n , r)
is unbound, so that Eq. (2.174) can be immediately generalized to resonance u(k n , r)
states. Similarly, one can show that u(k n , r) bound and resonance states are orthogonal to each other, by using the method explained in Sect. (2.5.1) and by integrating
the complex-rotated functions. The fundamental features of complex scaling at
theoretical level are depicted in Exercise VII, while the numerical accuracy of
complex scaling in practical applications is demonstrated in Exercise VIII.
Exercise VIII
One will now describe how to use the method of complex scaling numerically
and show its usefulness in practice.
A. In the Gamow shell model, the radial matrix elements are calculated with
complex scaling numerically. Numerical implementation involves the use of
the Gauss-Legendre quadrature to evaluate integrals.
One will now present how complex scaling is numerically implemented. For
this, one will calculate numerically the integral of B in Exercise VII.
Show that the latter integral is equal to:
+∞
0
exp(ik(R + xe
iθ ))e
iθ dx = 4e
iθ
R −1/4
0
exp(ik(R + x
−4 e
iθ ))x
−5 dx .
(3.62)
Explain why the integral on the right-hand side can be discretized with GaussLegendre quadrature and why this is impossible on the left-hand side.
Show that the change of variable in x → x −4 provides with a quickly
converging integral even for very small k. Notice that in practice, one has
sin(θ − θ k ) > 0.1, |k| > 10 −5 fm −1 and the first Gaussian point on [0 : R −1/4 ]
is about 0.01 (see Exercise VII for the definition of θ k ).
101
If u i (r) and u f (r) are wave functions of bound or decaying states, one can write
k i = |k i |e −iα i and k f = |k f |e −iα f . As u i ∼ a i (z)e ik i z and u f (z) ∼ a f (z)e ik f z
when (z) → +∞, with a i and a f the algebraic increasing functions, the integrals
defining F (V o ) converge if one takes:
θ > α f + α i .
(3.61)
In addition, the expression for F (V o ) is analytical because F (V o ) is a function of
converging integrals of analytical functions. Square root in Eq. (3.57) causes no
problems because N 2
i and N 2
f never cross the negative real axis when V o varies.
Consequently, following the theorem of analytic continuation, Eq. (3.54) defines
also F (V o ) for V o > V lim . In this way, one may calculate the radial matrix elements
of resonance states which a priori are not normalizable.
Complex scaling in Eq. (3.58) can be readily applied to Eq. (2.174) when u(k n , r)
is unbound, so that Eq. (2.174) can be immediately generalized to resonance u(k n , r)
states. Similarly, one can show that u(k n , r) bound and resonance states are orthogonal to each other, by using the method explained in Sect. (2.5.1) and by integrating
the complex-rotated functions. The fundamental features of complex scaling at
theoretical level are depicted in Exercise VII, while the numerical accuracy of
complex scaling in practical applications is demonstrated in Exercise VIII.
Exercise VIII
One will now describe how to use the method of complex scaling numerically
and show its usefulness in practice.
A. In the Gamow shell model, the radial matrix elements are calculated with
complex scaling numerically. Numerical implementation involves the use of
the Gauss-Legendre quadrature to evaluate integrals.
One will now present how complex scaling is numerically implemented. For
this, one will calculate numerically the integral of B in Exercise VII.
Show that the latter integral is equal to:
+∞
0
exp(ik(R + xe
iθ ))e
iθ dx = 4e
iθ
R −1/4
0
exp(ik(R + x
−4 e
iθ ))x
−5 dx .
(3.62)
Explain why the integral on the right-hand side can be discretized with GaussLegendre quadrature and why this is impossible on the left-hand side.
Show that the change of variable in x → x −4 provides with a quickly
converging integral even for very small k. Notice that in practice, one has
sin(θ − θ k ) > 0.1, |k| > 10 −5 fm −1 and the first Gaussian point on [0 : R −1/4 ]
is about 0.01 (see Exercise VII for the definition of θ k ).