Balslev-Combes theorem

B. Simon, Quadratic form techniques and the Balslev-Combes theorem, Commun. Math. Phys. 27 (1972) 1. 

The underlying idea behind the complex coordinate rotation (CCR) method " that is suggested by the Balslev-Combes theorem is a complex scaling of the Cartesian coordinates in the Hamiltonian operator, each by the same complex phase factor x xe. This transformation defines a new, complex-scaled Hamiltonian, H H 0). In one dimension (for simplicity), the complex-scaled Hamiltonian is... 

With these examples in hand, a pedagogical version of the Balslev-Combes theorem can be stated as follows. 

Figure 33 Pictorial illustration of the Balslev-Combes theorem and the complex coordinate rotation method. Horizontal and vertical axes represent the real and imaginary parts of the complex energy, respectively. Application of the complex-scaling transformation X xe rotates the continuum hy an angle of —26 in the complex plane, leaving the resonances exposed as discrete states with square-integrahle wave functions and complex energies. Bound states remain on the real axis. Adapted with permission from Ref. 187 copyright 2013 American Institute of Physics.

The Aquilar, Balslev, Combes and Simon (ABCS) theorems concerning spectrum of analytically dilated (complex rotated) Hamiltonians were published in the early seventies (1-3). They caused an explosive development of techniques using complex coordinates and their applications, which have been reviewed in numerous articles (4-13). 

The theorems of Aguilar, Balslev and Combes /27-30/ prescribe that the complex scaling of all the electronic coordinates in the Hamiltonian modifies its spectrum as depicted in fig. 1. 

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