100
3 Berggren Basis and Completeness Relations
Hence, on the interval (−∞; V lim ), one can define F (V o ) by Eq. (3.54), with the
norm (3.51) given by:
N i =
R
0
u 2
i (r) dr +
+∞
0
u 2
i (R + x · e iθ ) e iθ dx
(3.57)
and with the matrix element (3.52) of the form:
O if =
R
0
u f (r) ˆ
O(r) u i (r) dr
+
+∞
0
u f (R + x · e
iθ ) ˆ
O(R + x · e
iθ )u i (R + x · e
iθ )e
iθ dx
. (3.58)
Exercise VII
One will present the fundamental features of the complex scaling method
from a theoretical point of view.
A. In the Gamow shell model, analytical calculations of integrals are not feasible,
and all integrals involving complex scaling must be computed numerically.
Consequently, in order to prove the efficiency of complex scaling on formal
grounds, one will concentrate on a simple case where all integrals can be
calculated analytically. For this, the integral of exp(ikr) will be calculated
analytically and numerically using complex rotation on the interval [R : +∞[.
Explain why this situation qualitatively corresponds to those encountered in
practical cases.
B. Prove that:
Reg
+∞
R
exp(ikz)dz =
+∞
0
exp(ik(R + xe
iθ ))e
iθ dx ,
(3.59)
where Reg means that the diverging integral is regularized with complex
scaling and where θ is the rotation angle, chosen so that the integral converges.
C. Show that one must have θ ∈]θ k : θ k + π[ with k = |k| exp(−iθ k ), for the
integral of B to converge. Note that θ is defined modulo 2π. Show that:
Reg
+∞
R
exp(ikz)dz = −
exp(ikR)
ik
(3.60)
and explain the θ -independence of the integral using the arguments of analytic
continuation.