The Analytic Continuation of the Lippmann–Schwinger Eigenfunctions, and Antiunitary Symmetries

Rafael de la Madrid

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

We review the way to analytically continue the Lippmann–Schwinger bras and kets into the complex plane. We will see that a naive analytic continuation leads to nonsensical results in resonance theory, and we will explain how the non-obvious but correct analytical continuation is done. We will see that the physical basis for the non-obvious but correct analytic continuation lies in the invariance of the Hamiltonian under anti-unitary symmetries such as time reversal or $\mathcal{PT}$.

Détail

Keywords: Lippmann–Schwinger equation; resonances; Gamow states; resonant expansions; time reversal; $\mathcal{PT}$ symmetry.

Bibliographie

[1] Lippmann B. A., Schwinger J., “Variational principles for scattering processes. I”, Phys. Rev., 79 (1950), 469–480 | DOI | MR | Zbl

[2] Newton R. G., Scattering theory of waves and particles, 2nd ed., Springer Verlag, New York, 1982 | MR | Zbl

[3] Taylor J. R., Scattering theory, John Wiley Sons, Inc., New York, 1972

[4] de la Madrid R., “The rigged Hilbert space approach to the Lippmann–Schwinger equation. Part II: The analytic continuation of the Lippmann–Schwinger bras and kets”, J. Phys. A: Math. Gen., 39 (2006), 3981–4009 ; quant-ph/0603177 | DOI | MR | Zbl

[5] de la Madrid R., “The resonance amplitude associated with the Gamow states”, Nuclear Phys. A, 812 (2008), 13–27 ; arXiv:0810.0876 | DOI

[6] de la Madrid R., “Rigged Hilbert space approach to the Schrödinger equation”, J. Phys. A: Math. Gen., 35 (2002), 319–342 ; quant-ph/0110165 | DOI | MR | Zbl

[7] Gamow G., “Zur Quantentheorie de Atomkernes”, Z. Phys., 51 (1928), 204–212 | DOI | Zbl

[8] Siegert A. F. J., “On the derivation of the dispersion formula for nuclear reactions”, Phys. Rev., 56 (1939), 750–752 | DOI

[9] Peierls R. E., “Complex eigenvalues in scattering theory”, Proc. Roy. Soc. London. Ser. A, 253 (1959), 16–36 | DOI | MR

[10] Humblet J., Rosenfeld L., “Theory of nuclear reactions. I. Resonant states and collision matrix”, Nuclear Phys., 26 (1961), 529–578 | DOI | Zbl

[11] Zeldovich Ya. B., “On the theory of unstable states”, Sov. Phys. JETP, 12 (1961), 542–545 | MR

[12] Berggren T., “On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes”, Nuclear Phys. A, 109 (1968), 265–287 | DOI

[13] García-Calderón G., Peierls R., “Resonant states and their uses”, Nuclear Phys. A, 265 (1976), 443–460 | DOI | MR

[14] Berggren T., “On the interpretation of complex cross sections for production of resonant final states”, Phys. Lett. B, 73 (1978), 389–392 | DOI

[15] Parravicini G., Gorini V., Sudarshan E. C. G., “Resonances, scattering theory, and rigged Hilbert spaces”, J. Math. Phys., 21 (1980), 2208–2226 | DOI | MR | Zbl

[16] Hernández E., Mondragón A., “Resonant states in momentum representation”, Phys. Rev. C, 29 (1984), 722–738 | DOI

[17] Curutchet P., Vertse T., Liotta R. J., “Resonant random phase approximation”, Phys. Rev. C, 39 (1989), 1020–1031 | DOI

[18] Berggren T., Lind P., “Resonant state expansion of the resolvent”, Phys. Rev. C, 47 (1993), 768–778 | DOI

[19] Lind P., “Completeness relations and resonant state expansions”, Phys. Rev. C, 47 (1993), 1903–1920 | DOI

[20] Berggren T., “Expectation value of an operator in a resonant state”, Phys. Lett. B, 373 (1996), 1–4 | DOI | MR

[21] Bollini C. G., Civitarese O., De Paoli A. L., Rocca M. C., “Physical representations of Gamow states in a rigged Hilbert space”, Phys. Lett. B, 382 (1996), 205–208 | DOI

[22] Ferreira L. S., Maglione E., Liotta R. J., “Nucleon resonances in deformed nuclei”, Phys. Rev. Lett., 78 (1997), 1640–1643 | DOI

[23] Cavalcanti R. M., de Carvalho C. A. A., “On the effectiveness of Gamow's method for calculating decay rates”, Rev. Bras. Ens. Fis., 21 (1999), 464–468; quant-ph/9711037

[24] de la Madrid R., Quantum mechanics in rigger Hilbert space languare, Ph.D. Thesis, Universidad de Valladolid, 2001; available at http://www.physics.ohio-state.edu/~rafa/

[25] Id Betan R., Liotta R. J., Sandulescu N., Vertse T., “Two-particle resonant states in a many-body mean field”, Phys. Rev. Lett., 89 (2002), 042501, 4 pp., ages ; nucl-th/0201077 | DOI

[26] Michel N., Nazarewicz W., Ploszajczak M., Bennaceur K., “Gamow shell model description of neutron-rich nuclei”, Phys. Rev. Lett., 89 (2002), 042502, 4 pp., ages ; nucl-th/0201073 | DOI

[27] de la Madrid R., Gadella M., “A pedestrian introduction to Gamow vectors”, Amer. J. Phys., 70 (2002), 626–638 ; quant-ph/0201091 | DOI | MR

[28] Kapuscik E., Szczeszek P., “Model-independent normalization condition for Gamow vectors”, Czech. J. Phys., 53 (2003), 1053–1056 | DOI | MR

[29] Hernández E., Jáuregui A., Mondragón A., “Jordan blocks and Gamow–Jordan eigenfunctions associated with a degeneracy of unbound states”, Phys. Rev. A, 67 (2003), 022721, 15 pp., ages ; quant-ph/0204084 | DOI

[30] Michel N., Nazarewicz W., Ploszajczak M., Okolowicz J., “Gamow shell model description of weakly bound nuclei and unbound nuclear states”, Phys. Rev. C, 67 (2003), 054311, 17 pp., ages ; nucl-th/0302060 | DOI

[31] Kapuscik E., Szczeszek P., “The physical mechanism of formation of quantum mechanical Gamow states”, Found. Phys. Lett., 18 (2005), 573–580 | DOI | Zbl

[32] de la Madrid R., Garcia-Calderon G., Muga J. G., “Resonance expansions in quantum mechanics”, Czech. J. Phys., 55 (2005), 1141–1150 ; quant-ph/0512242 | DOI | MR

[33] Michel N., Nazarewicz W., Ploszajczak M., Rotureau J., “Antibound states and halo formation in the Gamow shell model”, Phys. Rev. C, 74 (2006), 054305, 6 pp., ages ; nucl-th/0609016 | DOI

[34] Rotureau J., Michel N., Nazarewicz W., Ploszajczak M., Dukelsky J., “Density matrix renormalization group approach for many-body open quantum systems”, Phys. Rev. Lett., 97 (2006), 110603, 4 pp., ages ; nucl-th/0603021 | DOI

[35] Michel N., Nazarewicz W., Ploszajczak M., “Threshold effects in multichannel coupling and spectroscopic factors in exotic nuclei”, Phys. Rev. C, 75 (2007), 031301, 5 pp., ages ; nucl-th/0702021 | DOI

[36] Julve J., de Urríes F. J., “Tunnelling of plane waves through a square barrier”, J. Phys. A: Math. Theor., 41 (2008), 304010, 14 pp., ages ; quant-ph/0701213 | DOI | MR | Zbl

[37] Michel N., Nazarewicz W., Ploszajczak M., “Continuum coupling and single-nucleon overlap integrals”, Nuclear Phys. A, 794 (2007), 29–46 ; arXiv:0707.0767 | DOI

[38] Hatano N., Sasada K., Nakamura H., Petrosky T., “Some properties of the resonant state in quantum mechanics and its computation”, Progr. Theor. Phys., 119 (2008), 187–222 ; arXiv:0705.1388 | DOI

[39] Costin O., Lebowitz J. L., Stucchio C., “Ionization in a 1-dimensional dipole model”, Rev. Math. Phys., 20 (2008), 835–872 ; math-ph/0609069 | DOI | MR | Zbl

[40] Fernandez-Garcia N., Rosas-Ortiz O., “Gamow–Siegert functions and Darboux-deformed short range potentials”, Ann. Physics, 323 (2008), 1397–1414 ; arXiv:0810.5597 | DOI | MR | Zbl

[41] Michel N., Nazarewicz W., Ploszajczak M., Vertse T., “Shell model in the complex energy plane”, J. Phys. G: Nucl. Part. Phys., 36 (2009), 013101, 40 pp., ages ; arXiv:0810.2728 | DOI

[42] Nussenzveig H. M., Causality and dispersion relations, Mathematics in Science and Engineering, 95, Academic Press, New York – London, 1972 | MR

[43] Bohm A., Gadella M., Dirac kets, Gamow vectors and Gel'fand triplets. The rigged Hilbert space formulation of quantum mechanics, Lecture Notes in Physics, 348, Springer-Verlag, Berlin, 1989 | MR | Zbl

[44] Sternheim M. M., Walker J. F., “Non-Hermitian Hamiltonians, decaying states, and perturbation theory”, Phys. Rev. C, 6 (1972), 114–121 | DOI

[45] Bender C. M., Boettcher S.,, “Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry”, Phys. Rev. Lett., 80 (1998), 5243–5246 ; physics/9712001 | DOI | MR | Zbl

[46] Mostafazadeh A., “Pseudo-Hermiticity versus $\mathcal{PT}$ symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian”, J. Math. Phys., 43 (2002), 205–214 ; math-ph/0107001 | DOI | MR | Zbl

[47] Mostafazadeh A., Batal A., “Physical aspects of pseudo-Hermitian and $\mathcal{PT}$-symmetric quantum mechanics”, J. Phys. A: Math. Gen., 37 (2004), 11645–11679 ; quant-ph/0408132 | DOI | MR | Zbl

[48] Ahmed Z., Bender C. M., Berry M. V., “Reflectionless potentials and $\mathcal{PT}$ symmetry”, J. Phys. A: Math. Gen., 38 (2005), L627–L630 ; quant-ph/0508117 | DOI | MR | Zbl

[49] Znojil M., “Coupled-channel version of the $\mathcal{PT}$-symmetric square well”, J. Phys. A: Math. Gen., 39 (2006), 441–455 ; quant-ph/0511085 | DOI | MR | Zbl

[50] Andrianov A. A., Cannata F., Sokolov A. V., “Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians. I. General properties”, Nuclear Phys. B, 773 (2007), 107–136 ; math-ph/0610024 | DOI | MR | Zbl

[51] Bender C. M., Brody D. C., Jones H. F., Meister B. K., “Faster than Hermitian quantum mechanics”, Phys. Rev. Lett., 98 (2007), 040403, 4 pp., ages ; quant-ph/0609032 | DOI | MR

[52] Bender C. M., “Making sense of non-Hermitian Hamiltonians”, Rep. Progr. Phys., 70 (2007), 947–1018 ; hep-th/0703096 | DOI | MR