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Bögli S, Brown BM, Marletta M, Tretter C, Wagenhofer M. Guaranteed resonance enclosures and exclosures for atoms and molecules. Proc Math Phys Eng Sci 2014; 470:20140488. [PMID: 25383033 PMCID: PMC4197464 DOI: 10.1098/rspa.2014.0488] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2014] [Accepted: 09/01/2014] [Indexed: 11/15/2022] Open
Abstract
In this paper, we confirm, with absolute certainty, a conjecture on a certain oscillatory behaviour of higher auto-ionizing resonances of atoms and molecules beyond a threshold. These results not only definitely settle a more than 30 year old controversy in Rittby et al. (1981 Phys. Rev. A24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) and Korsch et al. (1982 Phys. Rev. A26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)), but also provide new and reliable information on the threshold. Our interval-arithmetic-based method allows one, for the first time, to enclose and to exclude resonances with guaranteed certainty. The efficiency of our approach is demonstrated by the fact that we are able to show that the approximations in Rittby et al. (1981 Phys. Rev. A24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) do lie near true resonances, whereas the approximations of higher resonances in Korsch et al. (1982 Phys. Rev. A26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)) do not, and further that there exist two new pairs of resonances as suggested in Abramov et al. (2001 J. Phys. A34, 57–72 (doi:10.1088/0305-4470/34/1/304)).
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Affiliation(s)
- Sabine Bögli
- Mathematisches Institut, Universität Bern , Alpeneggstrasse 22, 3012 Bern, Switzerland
| | - B Malcolm Brown
- School of Computer Science, Cardiff University , 5 The Parade, Cardiff CF24 3AA, UK
| | - Marco Marletta
- School of Mathematics, Cardiff University , 21-23 Senghennydd Road, Cardiff CF24 4AG, UK
| | - Christiane Tretter
- Mathematisches Institut, Universität Bern , Sidlerstrasse 5, 3012 Bern, Switzerland ; Matematiska institutionen, Stockholms universitet , 10691 Stockholm, Sweden
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Cerioni A, Genovese L, Duchemin I, Deutsch T. Accurate complex scaling of three dimensional numerical potentials. J Chem Phys 2013; 138:204111. [PMID: 23742458 DOI: 10.1063/1.4807495] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The complex scaling method, which consists in continuing spatial coordinates into the complex plane, is a well-established method that allows to compute resonant eigenfunctions of the time-independent Schrödinger operator. Whenever it is desirable to apply the complex scaling to investigate resonances in physical systems defined on numerical discrete grids, the most direct approach relies on the application of a similarity transformation to the original, unscaled Hamiltonian. We show that such an approach can be conveniently implemented in the Daubechies wavelet basis set, featuring a very promising level of generality, high accuracy, and no need for artificial convergence parameters. Complex scaling of three dimensional numerical potentials can be efficiently and accurately performed. By carrying out an illustrative resonant state computation in the case of a one-dimensional model potential, we then show that our wavelet-based approach may disclose new exciting opportunities in the field of computational non-Hermitian quantum mechanics.
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Affiliation(s)
- Alessandro Cerioni
- European Synchrotron Radiation Facility, 6 rue Horowitz, BP220 38043 Grenoble Cedex 9, France.
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Abramov DI, Mar’yasin VS. Central potential reconstruction from scattering data by the variable boundary method with the use of Bargmann potentials. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2012. [DOI: 10.1134/s1990793112020145] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Korsch HJ, Mohlenkamp R, Meyer HD. On the canonical product expansion of the S matrix. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0022-3700/17/15/010] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Korsch HJ, Mohlenkamp R, Thylwe KE. Semiclassical complex energy theory of orbiting resonances in curve crossing systems. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0022-3700/19/14/010] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Krylstedt P, Rittby M, Elander N, Brandas E. A complex rotated approach to resonant electron scattering on atoms in a static exchange plus polarisation formulation. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0022-3700/20/6/016] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Atabek O, Lefebvre R, Jacon M. Poles of the scattering amplitude for the repulsive exponential potential: analytic and complex rotation studies. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0022-3700/15/16/017] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Meyer HD. The analytically continued S matrix for potentials defined as a sum of exponentials. I. The single-channel problem. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0022-3700/16/13/002] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Lehr H, Chatzidimitriou-Dreismann CA. General properties of the spectrum of complex scaled Hamiltonians: Detachment point and localization threshold. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1995; 51:3005-3016. [PMID: 9911935 DOI: 10.1103/physreva.51.3005] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Lehr H, Chatzidimitriou-Dreismann CA. General properties of the spectrum of complex scaled Hamiltonians: Phenomenological description of pole string curves. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1994; 50:2347-2365. [PMID: 9911151 DOI: 10.1103/physreva.50.2347] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Germann TC, Kais S. Large order dimensional perturbation theory for complex energy eigenvalues. J Chem Phys 1993. [DOI: 10.1063/1.465703] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Scrinzi A, Elander N. A finite element implementation of exterior complex scaling for the accurate determination of resonance energies. J Chem Phys 1993. [DOI: 10.1063/1.464014] [Citation(s) in RCA: 75] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Lipkin N, Moiseyev N, Brändas E. Resonances by the exterior-scaling method within the framework of the finite-basis-set approximation. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 40:549-553. [PMID: 9902183 DOI: 10.1103/physreva.40.549] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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He X, Atabek O, Giusti-Suzor A. Laser-induced resonances in molecular dissociation in intense fields. PHYSICAL REVIEW. A, GENERAL PHYSICS 1988; 38:5586-5594. [PMID: 9900295 DOI: 10.1103/physreva.38.5586] [Citation(s) in RCA: 62] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Bandarage G, Thorson WR. Molecular-state close-coupling theory including continuum states. II. Packet states and couplings for the proton-hydrogen-atom system. PHYSICAL REVIEW. A, GENERAL PHYSICS 1988; 37:716-728. [PMID: 9899713 DOI: 10.1103/physreva.37.716] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Snitchler GL, Watson DK. Quantum defects for beryllium size-2I from the poles of the multichannel T matrix. PHYSICAL REVIEW. A, GENERAL PHYSICS 1987; 36:1533-1538. [PMID: 9899033 DOI: 10.1103/physreva.36.1533] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Möhlenkamp R, Korsch HJ. Semiclassical complex-energy quantization for coupled equations: Feshbach resonances and predissociation. PHYSICAL REVIEW. A, GENERAL PHYSICS 1986; 34:4716-4721. [PMID: 9897854 DOI: 10.1103/physreva.34.4716] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kundu B, Ray D, Mukherjee PK. Dynamic polarizabilities and Rydberg states of the sodium isoelectronic sequence. PHYSICAL REVIEW. A, GENERAL PHYSICS 1986; 34:62-70. [PMID: 9897226 DOI: 10.1103/physreva.34.62] [Citation(s) in RCA: 37] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Pajunen P, Luppi J. The Prüfer phase function calculation of complex energy eigenvalues for cubic anharmonic oscillator. J Chem Phys 1985. [DOI: 10.1063/1.448634] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Modern Aspects of Diatomic Interaction Theory. ADVANCES IN QUANTUM CHEMISTRY 1985. [DOI: 10.1016/s0065-3276(08)60302-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register]
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Luppi J, Pajunen P. Evaluation of complex higher‐order semiclassical phase integrals. J Chem Phys 1984. [DOI: 10.1063/1.447856] [Citation(s) in RCA: 31] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Connor JNL, Smith AD. Quantum complex rotation and uniform semiclassical calculations of complex energy eigenvalues. J Chem Phys 1983. [DOI: 10.1063/1.444579] [Citation(s) in RCA: 40] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Deguchi K, Kato Y, Nishikawa K, Aono S. On the localization of the resonance state wave function in the complex coordinate method. J Chem Phys 1983. [DOI: 10.1063/1.445354] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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