3.8 Complex-Symmetric Operators and Matrices
127
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
0 20 40 60 80 100 120 140 160
Number of scattering states
10
-6
10
-5
10
-4
10
-3
10
-2
rms(Re[u]) (fm -1/2
)
rms(Im[u]) (fm -1/2
)
(i)
(i)
(iii) (ii)
(iii)
(ii)
Fig. 3.6 Real and imaginary parts of the rms deviations (3.97), (3.98) for the 1s 1/2 halo wave
function expanded in different Berggren bases as a function of the number of s 1/2 scattering
wave functions on L + . Cases (i)–(iii) of Table 3.1 are marked by dashed, solid, and dotted lines,
respectively (adapted from Ref. [49])
3.8
Complex-Symmetric Operators and Matrices
In general, the Berggren basis contains complex-energy one-body states, so that
the Hamiltonian matrix in this basis is complex symmetric. This is a fundamental
difference with the hermitian Hamiltonian matrices in standard quantum mechanics,
which are real symmetric and thus can be handled with standard diagonalization
routines.
There are no standard libraries dedicated to the diagonalization of complexsymmetric matrices. Standard numerical routines are available for the general
complex matrix, as in the LAPACK library for example, with which full numerical