Redundant poles of the S -matrix for the one-dimensional Morse potential
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Redundant poles of the S-matrix for the one-dimensional Morse potential M. Gadella1, A. Hernández-Ortega1, S. ¸ Kuru2 , J. Negro1,a 1 Dept. Física Teórica, Atómica y Optica and IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain 2 Department of Physics, Faculty of Science, Ankara University, 06100 Ankara, Turkey
Received: 26 August 2020 / Accepted: 5 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We analyze the structure of the scattering matrix, S(k), for the one-dimensional Morse potential. We show that, in addition to a finite number of bound state poles and an infinite number of antibound poles, there exist infinite redundant poles, on the positive imaginary axis, which do not correspond to either of the other types. We explain in detail the role of these redundant poles, in particular when they coincide with the bound poles. This can be solved analytically and exactly. In addition, we obtain wave functions for all these poles and ladder operators connecting them.
1 Introduction As is well known when we deal with non-relativistic quantum scattering, and under some causality conditions [1], the scattering matrix in the momentum representation, S(k), has an analytic continuation to a meromorphic function on the complex plane. Its isolated singularities are poles, which are classified as bound state poles, in one to one correspondence with the bound states, antibound poles and resonance poles. However, for some types of potentials other kinds of singularities may arise, like branch cuts and redundant poles. The latter do not correspond to physical states and have been studied already long ago [2–4]. This work has been continued by some authors and has inspired a bunch of result in scattering theory either for Hermitian or for non-Hermitian Hamiltonians [5–11]. Very recently, Moroz and Miroshnichenko [12,13] had exhaustively studied the analytic behavior of the scattering matrix S(k) corresponding to the radial Schrödinger equation with potential V (r ) = ±V0 e−r/a , (1.1) with V0 > 0 and a > 0. The authors find a series of redundant poles of S(k). These results have in part motivated the discussion presented in this paper. We have started with an exactly solvable one-dimensional potential, which is the Morse potential, and study the properties of the scattering matrix, S(k), produced by it. In this case, S(k) can be analytically continued to a meromorphic function on the whole complex plane with an infinite number of simple poles as the only singularities. These poles can be classified into three kinds: (i) A finite number of bound state poles located on the positive imaginary
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semi axis. (ii) An infinite number of poles on the negative imaginary semiaxis, which are usually called the virtual or antibound poles. (iii) Finally, an infinite number of simple poles located along the positive imaginary semiaxis
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