Calculation of Expectation Values of Operators in the Complex Scaling Method

Papadimitriou, G.

doi:10.1007/s00601-016-1126-9

Title: Calculation of Expectation Values of Operators in the Complex Scaling Method

Journal Article · Tue Jun 14 00:00:00 EDT 2016 · Few-Body Systems

DOI:https://doi.org/10.1007/s00601-016-1126-9· OSTI ID:1258519

Papadimitriou, G. ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

In the complex scaling method (CSM) provides with a way to obtain resonance parameters of particle unstable states by rotating the coordinates and momenta of the original Hamiltonian. It is convenient to use an L² integrable basis to resolve the complex rotated or complex scaled Hamiltonian H_θ , with θ being the angle of rotation in the complex energy plane. Within the CSM, resonance and scattering solutions have fall-off asymptotics. Furthermore, one of the consequences is that, expectation values of operators in a resonance or scattering complex scaled solution are calculated by complex rotating the operators. In this work we are exploring applications of the CSM on calculations of expectation values of quantum mechanical operators by using the regularized backrotation technique and calculating hence the expectation value using the unrotated operator. Moreover, the test cases involve a schematic two-body Gaussian model and also applications using realistic interactions.

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Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1258519

Report Number(s):: LLNL-JRNL-680078

Journal Information:: Few-Body Systems, Journal Name: Few-Body Systems; ISSN 0177-7963

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 2 works

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References (83)

R-Matrix Theory of Nuclear Reactions Lane, A. M.; Thomas, R. G. Reviews of Modern Physics, Vol. 30, Issue 2 https://doi.org/10.1103/RevModPhys.30.257	journal	April 1958
The R -matrix theory Descouvemont, P.; Baye, D. Reports on Progress in Physics, Vol. 73, Issue 3 https://doi.org/10.1088/0034-4885/73/3/036301	journal	February 2010
Coupled reaction channels calculations in nuclear physics Thompson, Ian J. Computer Physics Reports, Vol. 7, Issue 4 https://doi.org/10.1016/0167-7977(88)90005-6	journal	April 1988
Ab initio many-body calculations of nucleon-nucleus scattering Quaglioni, Sofia; Navrátil, Petr Physical Review C, Vol. 79, Issue 4 https://doi.org/10.1103/PhysRevC.79.044606	journal	April 2009
Continuum-discretized coupled-channels calculations for three-body models of deuteron-nucleus reactions Austern, N.; Iseri, Y.; Kamimura, M. Physics Reports, Vol. 154, Issue 3 https://doi.org/10.1016/0370-1573(87)90094-9	journal	October 1987
Nuclear Reactions for Astrophysics: Principles, Calculation and Applications of Low-Energy Reactions Thompson, Ian J.; Nunes, Filomena M. https://doi.org/10.1017/CBO9781139152150	book	January 2009
Shell model in the complex energy plane Michel, N.; Nazarewicz, W.; Płoszajczak, M. Journal of Physics G: Nuclear and Particle Physics, Vol. 36, Issue 1 https://doi.org/10.1088/0954-3899/36/1/013101	journal	October 2008
Shell model in the complex energy plane and two-particle resonances Id Betan, R.; Liotta, R. J.; Sandulescu, N. Physical Review C, Vol. 67, Issue 1 https://doi.org/10.1103/PhysRevC.67.014322	journal	January 2003
Ab initio no-core Gamow shell model calculations with realistic interactions Papadimitriou, G.; Rotureau, J.; Michel, N. Physical Review C, Vol. 88, Issue 4 https://doi.org/10.1103/PhysRevC.88.044318	journal	October 2013
Elastic proton scattering of medium mass nuclei from coupled-cluster theory Hagen, G.; Michel, N. Physical Review C, Vol. 86, Issue 2 https://doi.org/10.1103/PhysRevC.86.021602	journal	August 2012
Tunneling theory for tunable open quantum systems of ultracold atoms in one-dimensional traps Lundmark, R.; Forssén, C.; Rotureau, J. Physical Review A, Vol. 91, Issue 4 https://doi.org/10.1103/PhysRevA.91.041601	journal	April 2015
Gamow shell model description of proton scattering on ${}^{18}{Ne}$ Jaganathen, Y.; Michel, N.; Płoszajczak, M. Physical Review C, Vol. 89, Issue 3 https://doi.org/10.1103/PhysRevC.89.034624	journal	March 2014
On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes Berggren, Tore Nuclear Physics A, Vol. 109, Issue 2 https://doi.org/10.1016/0375-9474(68)90593-9	journal	February 1968
Dynamics of quantum systems embedded in a continuum Okołowicz, J.; Płoszajczak, M.; Rotter, I. Physics Reports, Vol. 374, Issue 4-5 https://doi.org/10.1016/S0370-1573(02)00366-6	journal	February 2003
Potential scattering for complex energy and angular momentum Bottino, A.; Longoni, A. M.; Regge, T. Il Nuovo Cimento, Vol. 23, Issue 6 https://doi.org/10.1007/BF02731254	journal	March 1962
Quadratic form techniques and the Balslev-Combes theorem Simon, Barry Communications in Mathematical Physics, Vol. 27, Issue 1 https://doi.org/10.1007/BF01649654	journal	March 1972
The rigged Hilbert space of the algebra of the one-dimensional rectangular barrier potential Madrid, Rafael de la Journal of Physics A: Mathematical and General, Vol. 37, Issue 33 https://doi.org/10.1088/0305-4470/37/33/011	journal	August 2004
Physical and mathematical aspects of Gamow states Civitarese, O.; Gadella, M. Physics Reports, Vol. 396, Issue 2 https://doi.org/10.1016/j.physrep.2004.03.001	journal	June 2004
Complex resonance eigenvalues by the lanczos recursion method Milfeld, Kent F.; Moiseyev, Nimrod Chemical Physics Letters, Vol. 130, Issue 1-2 https://doi.org/10.1016/0009-2614(86)80442-0	journal	January 1986
Non-Hermitian electronic theory and applications to clusters Santra, Robin; Cederbaum, Lorenz S. Physics Reports, Vol. 368, Issue 1 https://doi.org/10.1016/S0370-1573(02)00143-6	journal	September 2002
A new approach for many-body resonance spectroscopy with the complex scaling method Mizusaki, T.; Myo, T.; Kat , K. Progress of Theoretical and Experimental Physics, Vol. 2014, Issue 9 https://doi.org/10.1093/ptep/ptu124	journal	September 2014
$S$ -matrix and $R$ -matrix determination of the low-energy $^{5}$ He and $^{5}$ Li resonance parameters Csótó, Attila; Hale, G. M. Physical Review C, Vol. 55, Issue 1 https://doi.org/10.1103/PhysRevC.55.536	journal	January 1997
Complex Energy Method for Scattering Processes Kamada, H.; Koike, Y.; Glockle, W. Progress of Theoretical Physics, Vol. 109, Issue 5 https://doi.org/10.1143/PTP.109.869	journal	May 2003
Complex energy method in four-body Faddeev-Yakubovsky equations Uzu, E.; Kamada, H.; Koike, Y. Physical Review C, Vol. 68, Issue 6 https://doi.org/10.1103/PhysRevC.68.061001	journal	December 2003
Calculation of neutron- ${}^{3}{He}$ scattering up to 30 MeV Deltuva, A.; Fonseca, A. C. Physical Review C, Vol. 90, Issue 4 https://doi.org/10.1103/PhysRevC.90.044002	journal	October 2014
Calculation of Multichannel Reactions in the Four-Nucleon System above Breakup Threshold Deltuva, A.; Fonseca, A. C. Physical Review Letters, Vol. 113, Issue 10 https://doi.org/10.1103/PhysRevLett.113.102502	journal	September 2014
Radiative Capture Reactions in Lattice Effective Field Theory Rupak, Gautam; Lee, Dean Physical Review Letters, Vol. 111, Issue 3 https://doi.org/10.1103/PhysRevLett.111.032502	journal	July 2013
Method of Complex Coordinates for Three-Body Calculations above the Breakup Threshold Nuttall, J.; Cohen, H. L. Physical Review, Vol. 188, Issue 4 https://doi.org/10.1103/PhysRev.188.1542	journal	December 1969
Scattering amplitude without an explicit enforcement of boundary conditions Kruppa, A. T.; Suzuki, R.; Katō, K. Physical Review C, Vol. 75, Issue 4 https://doi.org/10.1103/PhysRevC.75.044602	journal	April 2007
Three-body model analysis of $α + d$ elastic scattering and the $^{2}$ H( $α, γ$ ) $^{6}$ Li reaction in complex-scaled solutions of the Lippmann-Schwinger equation Kikuchi, Yuma; Kurihara, Nozomi; Wano, Atsushi Physical Review C, Vol. 84, Issue 6 https://doi.org/10.1103/PhysRevC.84.064610	journal	December 2011
Relativistic extension of the complex scaling method for resonant states in deformed nuclei Shi, Min; Liu, Quan; Niu, Zhong-Ming Physical Review C, Vol. 90, Issue 3 https://doi.org/10.1103/PhysRevC.90.034319	journal	September 2014
Application of the complex-scaling method to four-nucleon scattering above break-up threshold Lazauskas, Rimantas Physical Review C, Vol. 86, Issue 4 https://doi.org/10.1103/PhysRevC.86.044002	journal	October 2012
Modern nuclear force predictions for $n - {}^{3}H$ scattering above the three- and four-nucleon breakup thresholds Lazauskas, Rimantas Physical Review C, Vol. 91, Issue 4 https://doi.org/10.1103/PhysRevC.91.041001	journal	April 2015
Nucleon-nucleon scattering with the complex scaling method and realistic interactions Papadimitriou, G.; Vary, J. P. Physical Review C, Vol. 91, Issue 2 https://doi.org/10.1103/PhysRevC.91.021001	journal	February 2015
Nucleon–nucleon resonances at intermediate energies using a complex energy formalism Papadimitriou, G.; Vary, J. P. Physics Letters B, Vol. 746 https://doi.org/10.1016/j.physletb.2015.04.058	journal	June 2015
Representation of several complex coordinate methods by similarity transformation operators Moiseyev, N.; Hirschfelder, J. O. The Journal of Chemical Physics, Vol. 88, Issue 2 https://doi.org/10.1063/1.454275	journal	January 1988
Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling Moiseyev, N. Physics Reports, Vol. 302, Issue 5-6 https://doi.org/10.1016/S0370-1573(98)00002-7	journal	September 1998
The method of complex coordinate rotation and its applications to atomic collision processes Ho, Y. K. Physics Reports, Vol. 99, Issue 1 https://doi.org/10.1016/0370-1573(83)90112-6	journal	October 1983
Response functions from integral transforms with a Lorentz kernel Efros, V. Physics Letters B, Vol. 338, Issue 2-3 https://doi.org/10.1016/0370-2693(94)91355-2	journal	October 1994
Electromagnetic and neutral-weak response functions of ${}^{4}{He}$ and ${}^{12}C$ Lovato, A.; Gandolfi, S.; Carlson, J. Physical Review C, Vol. 91, Issue 6 https://doi.org/10.1103/PhysRevC.91.062501	journal	June 2015
Finite-Temperature Pairing Gap of a Unitary Fermi Gas by Quantum Monte Carlo Calculations Magierski, Piotr; Wlazłowski, Gabriel; Bulgac, Aurel Physical Review Letters, Vol. 103, Issue 21 https://doi.org/10.1103/PhysRevLett.103.210403	journal	November 2009
LINPRO: Linear inverse problem library for data contaminated by statistical noise Magierski, Piotr; Wlazłowski, Gabriel Computer Physics Communications, Vol. 183, Issue 10 https://doi.org/10.1016/j.cpc.2012.05.005	journal	October 2012
Nuclear three-body problem in the complex energy plane: Complex-scaling Slater method Kruppa, A. T.; Papadimitriou, G.; Nazarewicz, W. Physical Review C, Vol. 89, Issue 1 https://doi.org/10.1103/PhysRevC.89.014330	journal	January 2014
A Modified Tikhonov Regularization for Stable Analytic Continuation Fu, Chu-Li; Deng, Zhi-Liang; Feng, Xiao-Li SIAM Journal on Numerical Analysis, Vol. 47, Issue 4 https://doi.org/10.1137/080730196	journal	January 2009
Wavefunctions of atomic resonances Buchleitner, A.; Gremaud, B.; Delande, D. Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 27, Issue 13 https://doi.org/10.1088/0953-4075/27/13/001	journal	July 1994
Back-rotation of the wave function in the complex scaling method Csótó, A.; Gyarmati, B.; Kruppa, A. T. Physical Review A, Vol. 41, Issue 7 https://doi.org/10.1103/PhysRevA.41.3469	journal	April 1990
Recent development of complex scaling method for many-body resonances and continua in light nuclei Myo, Takayuki; Kikuchi, Yuma; Masui, Hiroshi Progress in Particle and Nuclear Physics, Vol. 79 https://doi.org/10.1016/j.ppnp.2014.08.001	journal	November 2014
The Lagrange-mesh method Baye, Daniel Physics Reports, Vol. 565 https://doi.org/10.1016/j.physrep.2014.11.006	journal	March 2015
Low-energy $^{6} H e + p$ reactions in a microscopic multicluster model Arai, K.; Descouvemont, P.; Baye, D. Physical Review C, Vol. 63, Issue 4 https://doi.org/10.1103/PhysRevC.63.044611	journal	March 2001
The discrete variable–finite basis approach to quantum scattering Lill, J. V.; Parker, Gregory A.; Light, John C. The Journal of Chemical Physics, Vol. 85, Issue 2 https://doi.org/10.1063/1.451245	journal	July 1986
Matrix Elements of Physical Quantities Associated with Resonance States Homma, M.; Myo, T.; Kato, K. Progress of Theoretical Physics, Vol. 97, Issue 4 https://doi.org/10.1143/PTP.97.561	journal	April 1997
Resonance and Continuum Components of the Strength Function Myo, T.; Ohnishi, A.; Kato, K. Progress of Theoretical Physics, Vol. 99, Issue 5 https://doi.org/10.1143/PTP.99.801	journal	May 1998
Decomposition of scattering phase shifts and reaction cross sections using the complex scaling method Odsuren, Myagmarjav; Katō, Kiyoshi; Aikawa, Masayuki Physical Review C, Vol. 89, Issue 3 https://doi.org/10.1103/PhysRevC.89.034322	journal	March 2014
The contour deformation method in momentum space, applied to subatomic physics Hagen, G.; Vaagen, J. S.; Hjorth-Jensen, M. Journal of Physics A: Mathematical and General, Vol. 37, Issue 38 https://doi.org/10.1088/0305-4470/37/38/006	journal	September 2004
Gamow, a program for calculating the resonant state solution of the radial Schrödinger equation in an arbitrary optical potential Vertse, T.; Pál, K. F.; Balogh, Z. Computer Physics Communications, Vol. 27, Issue 3 https://doi.org/10.1016/0010-4655(82)90178-3	journal	September 1982
Inner product for resonant states and shell-model applications Romo, William J. Nuclear Physics A, Vol. 116, Issue 3 https://doi.org/10.1016/0375-9474(68)90395-3	journal	August 1968
On the normalization of Gamow functions Gyarmati, B.; Vertse, T. Nuclear Physics A, Vol. 160, Issue 3 https://doi.org/10.1016/0375-9474(71)90095-9	journal	January 1971
Rotational bands in the continuum illustrated by $^{8}$ Be results Garrido, E.; Jensen, A. S.; Fedorov, D. V. Physical Review C, Vol. 88, Issue 2 https://doi.org/10.1103/PhysRevC.88.024001	journal	August 2013
Expectation value of an operator in a resonant state Berggren, Tore Physics Letters B, Vol. 373, Issue 1-3 https://doi.org/10.1016/0370-2693(96)00132-3	journal	April 1996
Ab initio study of the photoabsorption of $^{4}$ He Horiuchi, W.; Suzuki, Y.; Arai, K. Physical Review C, Vol. 85, Issue 5 https://doi.org/10.1103/PhysRevC.85.054002	journal	May 2012
Precise comparison of the Gaussian expansion method and the Gamow shell model Masui, H.; Katō, K.; Michel, N. Physical Review C, Vol. 89, Issue 4 https://doi.org/10.1103/PhysRevC.89.044317	journal	April 2014
Complex-scaled spectrum completeness for pedestrians Giraud, B. G.; Katō, K. Annals of Physics, Vol. 308, Issue 1 https://doi.org/10.1016/S0003-4916(03)00134-9	journal	November 2003
Microscopic Study of Nucleon-4He Scattering and Effective Nuclear Potentials Kanada, H.; Kaneko, T.; Nagata, S. Progress of Theoretical Physics, Vol. 61, Issue 5 https://doi.org/10.1143/PTP.61.1327	journal	May 1979
Level Density in the Complex Scaling Method Suzuki, R.; Myo, T.; Kato, K. Progress of Theoretical Physics, Vol. 113, Issue 6 https://doi.org/10.1143/PTP.113.1273	journal	June 2005
Calculation of the continuum level density Kruppa, A. T. Physics Letters B, Vol. 431, Issue 3-4 https://doi.org/10.1016/S0370-2693(98)00573-5	journal	July 1998
Continuum level density in a microscopic cluster model: Parameters of resonances Arai, K.; Kruppa, A. T. Physical Review C, Vol. 60, Issue 6 https://doi.org/10.1103/PhysRevC.60.064315	journal	November 1999
Resonances and the continuum level density Kruppa, A. T.; Arai, K. Physical Review A, Vol. 59, Issue 5 https://doi.org/10.1103/PhysRevA.59.3556	journal	May 1999
Quasiparticle continuum and resonances in the Hartree-Fock-Bogoliubov theory Pei, J. C.; Kruppa, A. T.; Nazarewicz, W. Physical Review C, Vol. 84, Issue 2 https://doi.org/10.1103/PhysRevC.84.024311	journal	August 2011
Continuum Level Density of a Coupled-Channel System in the Complex Scaling Method Suzuki, R.; Kruppa, A. T.; Giraud, B. G. Progress of Theoretical Physics, Vol. 119, Issue 6 https://doi.org/10.1143/PTP.119.949	journal	June 2008
An extended Levinson’s theorem Osborn, T. A.; Bollé, D. Journal of Mathematical Physics, Vol. 18, Issue 3 https://doi.org/10.1063/1.523287	journal	March 1977
Levinson's theorems in classical scattering Osborn, T. A.; Froese, R. G.; Howes, S. F. Physical Review A, Vol. 22, Issue 1 https://doi.org/10.1103/PhysRevA.22.101	journal	July 1980
Resonances and time-delay in three-body scattering Svenne, J. P.; Osborn, T. A.; Pisent, G. Physical Review C, Vol. 40, Issue 3 https://doi.org/10.1103/PhysRevC.40.1136	journal	September 1989
Realistic nuclear Hamiltonian: Ab exitu approach Shirokov, A. M.; Vary, J. P.; Mazur, A. I. Physics Letters B, Vol. 644, Issue 1 https://doi.org/10.1016/j.physletb.2006.10.066	journal	January 2007
Improved L2-stabilization theory to compute resonances under multichannel conditions Salzgeber, R. F.; Manthe, U.; Weiss, Th. Chemical Physics Letters, Vol. 249, Issue 3-4 https://doi.org/10.1016/0009-2614(95)01390-3	journal	February 1996
Nuclear physics from lattice QCD Beane, S. R.; Detmold, W.; Orginos, K. Progress in Particle and Nuclear Physics, Vol. 66, Issue 1 https://doi.org/10.1016/j.ppnp.2010.08.002	journal	January 2011
Ab initio alpha–alpha scattering Elhatisari, Serdar; Lee, Dean; Rupak, Gautam Nature, Vol. 528, Issue 7580 https://doi.org/10.1038/nature16067	journal	December 2015
Volume dependence of the energy spectrum in massive quantum field theories: II. Scattering states Lüscher, M. Communications in Mathematical Physics, Vol. 105, Issue 2 https://doi.org/10.1007/BF01211097	journal	June 1986
An effective field theory approach to two trapped particles Stetcu, I.; Rotureau, J.; Barrett, B. R. Annals of Physics, Vol. 325, Issue 8 https://doi.org/10.1016/j.aop.2010.02.008	journal	August 2010
Nucleon-nucleon scattering in a harmonic potential Luu, Thomas; Savage, Martin J.; Schwenk, Achim Physical Review C, Vol. 82, Issue 3 https://doi.org/10.1103/PhysRevC.82.034003	journal	September 2010
Weyl's theory and the complex-rotation method applied to phenomena associated with a continuous spectrum Rittby, Magnus; Elander, Nils; Brändas, Erkki Physical Review A, Vol. 24, Issue 3 https://doi.org/10.1103/PhysRevA.24.1636	journal	September 1981
Reply to "Comment on `Weyl's theory and the complex-rotation method applied to phenomena associated with a continuous spectrum"' Rittby, Magnus; Elander, Nils; Brändas, Erkki Physical Review A, Vol. 26, Issue 3 https://doi.org/10.1103/PhysRevA.26.1804	journal	September 1982
Energy resolution with the Lorentz integral transform Leidemann, Winfried Physical Review C, Vol. 91, Issue 5 https://doi.org/10.1103/PhysRevC.91.054001	journal	May 2015
Accurate complex scaling of three dimensional numerical potentials Cerioni, Alessandro; Genovese, Luigi; Duchemin, Ivan The Journal of Chemical Physics, Vol. 138, Issue 20 https://doi.org/10.1063/1.4807495	journal	May 2013

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