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The rigged Hilbert space approach to the Gamow states
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88.Interestingly, the space of test functions we will use is very similar to the space of test functions of A. van Tonder, and M. Dorca, “Non-perturbative quantization of phantom, and ghost theories: Relating definite, and indefinite representations,” Int. J. Mod. Phys. A 22, 2563–2608 (2007);
and of A. van Tonder
, “Unitarity, Lorentz invariance and causality in Lee-Wick theories: An asymptotically safe completion of QED
,” e-print arXiv:0810.1928
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97.Instead of “complex Hermitian conjugation,” one can use the term “Schwartz complex conjugation,” since such operation actually originates from the Schwarz reflection principle. This principle states that for a holomorphic function in the upper complex plane, continuous and real valued on the real line, one can write an analytic continuation for the whole plane such that f(z*) = f*(z).
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100.Strictly speaking, we should have demanded that our test functions fall off at infinity in the position representation not like but like , where ε is any small positive number. This way, the exponential blowup of the test functions in the energy representation would had been given by , δ = ε/(1 + ε) > 0 and small, rather than by Eqs. (8.5) and (8.6). For the sake of clarity, we will ignore this technicality.
101. M. C. Rocca, private communication (2001).
102.Examples of square-integrable wave functions that approximate a Gamow state very closely can be found in Refs. 49 and 79.
103. V. V. Nesterenko
, A. Feoli
, G. Lambiase
, and G. Scarpetta
, “Quasi-normal modes of a dielectric ball and some of their implications
,” e-print arXiv:hep-th/0512340
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