Is Complex Scaling in Quantum Mechanics Fully Understood?

[Quantum River]

Could anyone tell me what complex scaling is?

Do the eigenfunctions of the complex scaled Hamiltonian form a complete basis set? Some articles say the theoretical basis of the complex scaling method has been understood well by the ABC (Aguilar-Balslev-Combes) theorm. But the ABC article of these three guys is in Commun. Math. Phys. of 1971 year (I could not get access to springerlink.com).

I am aware of that this kind of technical problems should not be brought out here. But I just could not understand this method well. Could anyone give me a simple mathematical argument why the complex scaled basis set is complete? (Or give me some internet link that prove its mathematical basis.) The help will be greatly appreciated.

Thanks!

Quantum River

 

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[denian]

:

Complex scaling is a mathematical technique used in quantum mechanics to simplify the description of complex systems. It involves rotating the coordinate system in a complex plane, which allows for easier calculation of the system's behavior. The eigenfunctions of the complex scaled Hamiltonian do indeed form a complete basis set, as shown by the ABC theorem. This theorem, proposed by Aguilar, Balslev, and Combes in 1971, proves the completeness of the complex scaled basis set in the context of quantum mechanics. While I cannot provide a simple mathematical argument in this response, there are several resources available online that explain the mathematical basis of complex scaling in detail. I recommend searching for "ABC theorem" or "complex scaling in quantum mechanics" for more information.