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Volumn 20, Issue , 2011, Pages 121-209

Mathematical and computational methods for semiclassical Schrödinger equations

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EID: 79956066352     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492911000031     Document Type: Article
Times cited : (160)

References (189)
  • 1
    • 28144437204 scopus 로고
    • A fast level set method for propagating interfaces
    • D. Adalsteinsson and J. A. Sethian (1995), 'A fast level set method for propagating interfaces', J. Comput. Phys. 118, 269-277.
    • (1995) J. Comput. Phys. , vol.118 , pp. 269-277
    • Adalsteinsson, D.1    Sethian, J.A.2
  • 2
    • 37849187598 scopus 로고    scopus 로고
    • Semi-classical limit of Schrödinger-Poisson equations in space dimension n ≥ 3
    • DOI 10.1016/j.jde.2006.10.003, PII S0022039606004050
    • T. Alazard and R. Carles (2007), 'Semi-classical limit of Schrödinger-Poisson equations in space dimension n ≥ 3', J. Diff. Equations 233, 241-275. (Pubitemid 44824765)
    • (2007) Journal of Differential Equations , vol.233 , Issue.1 , pp. 241-275
    • Alazard, T.1    Carles, R.2
  • 3
    • 7544243351 scopus 로고    scopus 로고
    • Transport equation and Cauchy problem for BV vector fields
    • DOI 10.1007/s00222-004-0367-2
    • L. Ambrosio (2004), 'Transport equation and Cauchy problem for BV vector fields', Invent. Math. 158, 227-260. (Pubitemid 39453950)
    • (2004) Inventiones Mathematicae , vol.158 , Issue.2 , pp. 227-260
    • Ambrosio, L.1
  • 7
    • 33746881461 scopus 로고    scopus 로고
    • Accuracy of transport models for waves in random media
    • DOI 10.1016/j.wavemoti.2006.05.005, PII S0165212506000357
    • G. Bal and O. Pinaud (2006), 'Accuracy of transport models for waves in random media', Wave Motion 43, 561-578. (Pubitemid 44184752)
    • (2006) Wave Motion , vol.43 , Issue.7 , pp. 561-578
    • Bal, G.1    Pinaud, O.2
  • 8
    • 79956117081 scopus 로고    scopus 로고
    • Time splitting for the Liouville equation in a random medium
    • G. Bal and L. Ryzhik (2004), 'Time splitting for the Liouville equation in a random medium', Comm. Math. Sci. 2, 515-534.
    • (2004) Comm. Math. Sci. , vol.2 , pp. 515-534
    • Bal, G.1    Ryzhik, L.2
  • 10
    • 0033363402 scopus 로고    scopus 로고
    • Transport theory for acoustic waves with reflection and transmission at interfaces
    • DOI 10.1016/S0165-2125(99)00018-9
    • G. Bal, J. B. Keller, G. Papanicolaou and L. Ryzhik (1999b), 'Transport theory for acoustic waves with reflection and transmission at interfaces', Wave Motion 30, 303-327. (Pubitemid 30499229)
    • (1999) Wave Motion , vol.30 , Issue.4 , pp. 303-327
    • Bal, G.1    Keller, J.B.2    Papanicolaou, G.3    Ryzhik, L.4
  • 12
    • 18144389899 scopus 로고    scopus 로고
    • A fourth-order time-splitting laguerre-hermite pseudospectral method for bose-einstein condensates
    • DOI 10.1137/030601211
    • W. Bao and J. Shen (2005), 'A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates', SIAM J. Sci. Comput. 26, 2010-2028. (Pubitemid 41641360)
    • (2005) SIAM Journal of Scientific Computing , vol.26 , Issue.6 , pp. 2010-2028
    • Bao, W.1    Shen, J.2
  • 13
    • 0037408314 scopus 로고    scopus 로고
    • Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation
    • DOI 10.1016/S0021-9991(03)00102-5
    • W. Bao, D. Jaksch and P. A. Markowich (2003a), 'Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation', J. Comput. Phys. 187, 318-342. (Pubitemid 36515583)
    • (2003) Journal of Computational Physics , vol.187 , Issue.1 , pp. 318-342
    • Bao, W.1    Jaksch, D.2    Markowich, P.A.3
  • 14
    • 0442279560 scopus 로고    scopus 로고
    • Three dimensional simulation of jet formation in collapsing condensates
    • W. Bao, D. Jaksch and P. A. Markowich (2004), 'Three dimensional simulation of jet formation in collapsing condensates', J. Phys. B: At. Mol. Opt. Phys. 37, 329-343.
    • (2004) J. Phys. B: At. Mol. Opt. Phys. , vol.37 , pp. 329-343
    • Bao, W.1    Jaksch, D.2    Markowich, P.A.3
  • 15
    • 0037138007 scopus 로고    scopus 로고
    • On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
    • W. Bao, S. Jin and P. A. Markowich (2002), 'On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime', J. Comput. Phys. 175, 487-524.
    • (2002) J. Comput. Phys. , vol.175 , pp. 487-524
    • Bao, W.1    Jin, S.2    Markowich, P.A.3
  • 16
    • 1842531889 scopus 로고    scopus 로고
    • Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes
    • W. Bao, S. Jin and P. A. Markowich (2003b), 'Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes', SIAM J. Sci. Comput. 25, 27-64.
    • (2003) SIAM J. Sci. Comput. , vol.25 , pp. 27-64
    • Bao, W.1    Jin, S.2    Markowich, P.A.3
  • 17
    • 47349106037 scopus 로고    scopus 로고
    • Ground, symmetric and central vortex states in rotating Bose-Einstein condensates
    • W. Bao, H. Wang and P. A. Markowich (2005), 'Ground, symmetric and central vortex states in rotating Bose-Einstein condensates', Comm. Math. Sci. 3, 57-88.
    • (2005) Comm. Math. Sci. , vol.3 , pp. 57-88
    • Bao, W.1    Wang, H.2    Markowich, P.A.3
  • 18
    • 0036033113 scopus 로고    scopus 로고
    • Coupling one-dimensional time-dependent classical and quantum transport models
    • N. Ben Abdallah, P. Degond and I. M. Gamba (2002), 'Coupling one-dimensional time-dependent classical and quantum transport models', J. Math. Phys. 43, 1-24.
    • (2002) J. Math. Phys. , vol.43 , pp. 1-24
    • Ben Abdallah, N.1    Degond, P.2    Gamba, I.M.3
  • 19
    • 0033421722 scopus 로고    scopus 로고
    • Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations
    • J.-D. Benamou (1999), 'Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations', Comm. Pure Appl. Math. 52, 1443-1475.
    • (1999) Comm. Pure Appl. Math. , vol.52 , pp. 1443-1475
    • Benamou, J.-D.1
  • 20
    • 0001415492 scopus 로고    scopus 로고
    • An Eulerian method for capturing caustics
    • J.-D. Benamou and I. Solliec (2000), 'An Eulerian method for capturing caustics', J. Comput. Phys. 162, 132-163.
    • (2000) J. Comput. Phys. , vol.162 , pp. 132-163
    • Benamou, J.-D.1    Solliec, I.2
  • 21
    • 0038547744 scopus 로고    scopus 로고
    • A geometrical opticsbased numerical method for high frequency electromagnetic fields computations near fold caustics I
    • J.-D. Benamou, O. Lafitte, R. Sentis and I. Solliec (2003), 'A geometrical opticsbased numerical method for high frequency electromagnetic fields computations near fold caustics I', J. Comput. Appl. Math. 156, 93-125.
    • (2003) J. Comput. Appl. Math. , vol.156 , pp. 93-125
    • Benamou, J.-D.1    Lafitte, O.2    Sentis, R.3    Solliec, I.4
  • 22
    • 23744508288 scopus 로고    scopus 로고
    • Asymptotic analysis of quantum scattering under mesoscopic scaling
    • D. Benedetto, R. Esposito and M. Pulvirenti (2004), 'Asymptotic analysis of quantum scattering under mesoscopic scaling', Asymptot. Anal. 40, 163-187.
    • (2004) Asymptot. Anal. , vol.40 , pp. 163-187
    • Benedetto, D.1    Esposito, R.2    Pulvirenti, M.3
  • 24
    • 34250940025 scopus 로고
    • Über die Quantenmechanik der Elektronen in Kristallgittern
    • F. Bloch (1928), 'Über die Quantenmechanik der Elektronen in Kristallgittern', Z. Phys. 52, 555-600.
    • (1928) Z. Phys. , vol.52 , pp. 555-600
    • Bloch, F.1
  • 25
    • 0442331533 scopus 로고    scopus 로고
    • Numerical approximations of pressureless and isothermal gas dynamics
    • F. Bouchut, S. Jin and X. Li (2003), 'Numerical approximations of pressureless and isothermal gas dynamics', SIAM J. Numer. Anal. 41, 135-158.
    • (2003) SIAM J. Numer. Anal. , vol.41 , pp. 135-158
    • Bouchut, F.1    Jin, S.2    Li, X.3
  • 26
    • 77953933814 scopus 로고    scopus 로고
    • Gaussian beams summation for the wave equation in a convex domain
    • S. Bougacha, J.-L. Akian and R. Alexandre (2009), 'Gaussian beams summation for the wave equation in a convex domain', Comm. Math. Sci. 7, 973-1008.
    • (2009) Comm. Math. Sci. , vol.7 , pp. 973-1008
    • Bougacha, S.1    Akian, J.-L.2    Alexandre, R.3
  • 28
    • 0001605822 scopus 로고    scopus 로고
    • Geometric optics with caustic crossing for some nonlinear Schrödinger equations
    • R. Carles (2000), 'Geometric optics with caustic crossing for some nonlinear Schrödinger equations', Indiana Univ. Math. J. 49, 475-551.
    • (2000) Indiana Univ. Math. J. , vol.49 , pp. 475-551
    • Carles, R.1
  • 30
    • 33846837356 scopus 로고    scopus 로고
    • Geometric optics and instability for semi-classical Schrödinger equations
    • R. Carles (2007a), 'Geometric optics and instability for semi-classical Schrödinger equations', Arch. Ration. Mech. Anal. 183, 525-553.
    • (2007) Arch. Ration. Mech. Anal. , vol.183 , pp. 525-553
    • Carles, R.1
  • 31
    • 33751119261 scopus 로고    scopus 로고
    • WKB analysis for nonlinear Schrödinger equations with potential
    • DOI 10.1007/s00220-006-0077-2
    • R. Carles (2007b), 'WKB analysis for nonlinear Schrödinger equations with potential', Comm. Math. Phys. 269, 195-221. (Pubitemid 44770798)
    • (2007) Communications in Mathematical Physics , vol.269 , Issue.1 , pp. 195-221
    • Carles, R.1
  • 33
    • 34548616678 scopus 로고    scopus 로고
    • Numerical aspects of nonlinear Schrödinger equations in the presence of caustics
    • R. Carles and L. Gosse (2007), 'Numerical aspects of nonlinear Schrödinger equations in the presence of caustics', Math. Models Methods Appl. Sci. 17, 1531-1553.
    • (2007) Math. Models Methods Appl. Sci. , vol.17 , pp. 1531-1553
    • Carles, R.1    Gosse, L.2
  • 34
    • 5444243818 scopus 로고    scopus 로고
    • Semiclassical asymptotics for weakly nonlinear Bloch waves
    • DOI 10.1023/B:JOSS.0000044070.34410.17
    • R. Carles, P. A. Markowich and C. Sparber (2004), 'Semiclassical asymptotics for weakly nonlinear Bloch waves', J. Statist. Phys. 117, 343-375. (Pubitemid 39363800)
    • (2004) Journal of Statistical Physics , vol.117 , Issue.1-2 , pp. 343-375
    • Carles, R.1    Markowich, P.A.2    Sparber, C.3
  • 35
    • 79956065331 scopus 로고    scopus 로고
    • Courant Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, New York University
    • T. Cazenave (2003), Semilinear Schrödinger Equations, Vol. 10 of Courant Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, New York University.
    • (2003) Semilinear Schrödinger Equations , vol.10
    • Cazenave, T.1
  • 37
    • 0023331015 scopus 로고
    • Stability analysis of difference scheme for variable coefficient Schrödinger type equations
    • T. F. Chan and L. Shen (1987), 'Stability analysis of difference scheme for variable coefficient Schrödinger type equations', SIAM J. Numer. Anal. 24, 336-349. (Pubitemid 17550509)
    • (1987) SIAM Journal on Numerical Analysis , vol.24 , Issue.2 , pp. 336-349
    • Chan, T.F.1    Shen, L.2
  • 38
    • 0022695120 scopus 로고
    • Stable explicit schemes for equations of the Schrödinger type
    • T. Chan, D. Lee and L. Shen (1986), 'Stable explicit schemes for equations of the Schrödinger type', SIAM J. Numer. Anal. 23, 274-281. (Pubitemid 16517410)
    • (1986) SIAM Journal on Numerical Analysis , vol.23 , Issue.2 , pp. 274-281
    • Chan, T.F.1    Lee, D.2    Shen, L.3
  • 40
    • 3242697504 scopus 로고    scopus 로고
    • Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations
    • L.-T. Cheng, H. Liu and S. Osher (2003), 'Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations', Comm. Math. Sci. 1, 593-621.
    • (2003) Comm. Math. Sci. , vol.1 , pp. 593-621
    • Cheng, L.-T.1    Liu, H.2    Osher, S.3
  • 42
    • 84967758647 scopus 로고
    • Viscosity solutions of Hamilton-Jacobi equations
    • M. G. Crandall and P.-L. Lions (1983), 'Viscosity solutions of Hamilton-Jacobi equations', Trans. Amer. Math. Soc. 277, 1-42.
    • (1983) Trans. Amer. Math. Soc. , vol.277 , pp. 1-42
    • Crandall, M.G.1    Lions, P.-L.2
  • 43
    • 55349096773 scopus 로고    scopus 로고
    • On the time splitting spectral method for the complex Ginzburg-Landau equation in the large time and space scale limit
    • P. Degond, S. Jin and M. Tang (2008), 'On the time splitting spectral method for the complex Ginzburg-Landau equation in the large time and space scale limit', SIAM J. Sci. Comput. 30, 2466-2487.
    • (2008) SIAM J. Sci. Comput. , vol.30 , pp. 2466-2487
    • Degond, P.1    Jin, S.2    Tang, M.3
  • 44
    • 49149137309 scopus 로고
    • Finite-difference solutions of a nonlinear Schrödinger equation
    • M. Delfour, M. Fortin and G. Payre (1981), 'Finite-difference solutions of a nonlinear Schrödinger equation', J. Comput. Phys. 44, 277-288.
    • (1981) J. Comput. Phys. , vol.44 , pp. 277-288
    • Delfour, M.1    Fortin, M.2    Payre, G.3
  • 45
    • 0009293975 scopus 로고
    • Large time, small coupling behaviour of a quantum particle in a random field
    • G. F. Dell'Antonio (1983), 'Large time, small coupling behaviour of a quantum particle in a random field', Ann. Inst. H. Poincaŕe Sect. A (NS) 39, 339-384.
    • (1983) Ann. Inst. H. Poincaŕe Sect. A (NS) , vol.39 , pp. 339-384
    • Dell'Antonio, G.F.1
  • 46
    • 0002306354 scopus 로고
    • Ordinary differential equations, transport theory and Sobolev spaces
    • R. J. DiPerna and P.-L. Lions (1989), 'Ordinary differential equations, transport theory and Sobolev spaces', Invent. Math. 98, 511-547.
    • (1989) Invent. Math. , vol.98 , pp. 511-547
    • Diperna, R.J.1    Lions, P.-L.2
  • 47
    • 0039270375 scopus 로고    scopus 로고
    • A time- and space-adaptive algorithm for the linear time dependent Schrödinger equation
    • W. Dörfler (1998), 'A time- and space-adaptive algorithm for the linear time dependent Schrödinger equation', Numer. Math. 73, 419-448.
    • (1998) Numer. Math. , vol.73 , pp. 419-448
    • Dörfler, W.1
  • 48
    • 0000341962 scopus 로고    scopus 로고
    • Basics of surface hopping in mixed quantum/classical simulations
    • K. Drukker (1999), 'Basics of surface hopping in mixed quantum/classical simulations', J. Comput. Phys. 153, 225-272.
    • (1999) J. Comput. Phys. , vol.153 , pp. 225-272
    • Drukker, K.1
  • 50
    • 33845984363 scopus 로고    scopus 로고
    • Long time behavior of splitting methods applied to the linear Schrödinger equation
    • G. Dujardin and E. Faou (2007a), 'Long time behavior of splitting methods applied to the linear Schrödinger equation', CR Math. Acad. Sci. Paris 344, 89-92.
    • (2007) CR Math. Acad. Sci. Paris , vol.344 , pp. 89-92
    • Dujardin, G.1    Faou, E.2
  • 51
    • 36649004371 scopus 로고    scopus 로고
    • Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential
    • G. Dujardin and E. Faou (2007b), 'Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential', Numer. Math. 108, 223-262.
    • (2007) Numer. Math. , vol.108 , pp. 223-262
    • Dujardin, G.1    Faou, E.2
  • 52
    • 0030571228 scopus 로고    scopus 로고
    • Multi-phase computations in geometrical optics
    • PII S0377042796000234
    • B. Engquist and O. Runborg (1996), 'Multi-phase computations in geometrical optics', J. Comput. Appl. Math. 74, 175-192. TICAM Symposium (Austin, TX, 1995). (Pubitemid 126381013)
    • (1996) Journal of Computational and Applied Mathematics , vol.74 , Issue.1-2 , pp. 175-192
    • Engquist, B.1    Runborg, O.2
  • 53
    • 24644487123 scopus 로고    scopus 로고
    • Computational high frequency wave propagation
    • Cambridge University Press
    • B. Engquist and O. Runborg (2003), Computational high frequency wave propagation. In Acta Numerica, Vol. 12, Cambridge University Press, pp. 181-266.
    • (2003) Acta Numerica , vol.12 , pp. 181-266
    • Engquist, B.1    Runborg, O.2
  • 54
    • 0034418177 scopus 로고    scopus 로고
    • Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation
    • L. Erdös and H.-T. Yau (2000), 'Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation', Comm. Pure Appl. Math. 53, 667-735.
    • (2000) Comm. Pure Appl. Math. , vol.53 , pp. 667-735
    • Erdös, L.1    Yau, H.-T.2
  • 56
    • 0037324071 scopus 로고    scopus 로고
    • Wigner measures and codimension two crossings
    • DOI 10.1063/1.1527221
    • C. Fermanian Kammerer and C. Lasser (2003), 'Wigner measures and codimension two crossings', J. Math. Phys. 44, 507-527. (Pubitemid 36229106)
    • (2003) Journal of Mathematical Physics , vol.44 , Issue.2 , pp. 507-527
    • Kammerer, C.F.1    Lasser, C.2
  • 57
    • 84980178052 scopus 로고
    • Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation
    • H. Flaschka, M. G. Forest and D. W. McLaughlin (1980), 'Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation', Comm. Pure Appl. Math. 33, 739-784.
    • (1980) Comm. Pure Appl. Math. , vol.33 , pp. 739-784
    • Flaschka, H.1    Forest, M.G.2    McLaughlin, D.W.3
  • 59
    • 0003583110 scopus 로고    scopus 로고
    • Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press
    • B. Fornberg (1996), A Practical Guide to Pseudospectral Methods, Vol. 1 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.
    • (1996) A Practical Guide to Pseudospectral Methods , vol.1
    • Fornberg, B.1
  • 61
    • 0000284780 scopus 로고
    • Absence of diffusion in the Anderson tight binding model for large disorder or low energy
    • J. Fröhlich and T. Spencer (1983), 'Absence of diffusion in the Anderson tight binding model for large disorder or low energy', Comm. Math. Phys. 88, 151-184.
    • (1983) Comm. Math. Phys. , vol.88 , pp. 151-184
    • Fröhlich, J.1    Spencer, T.2
  • 62
    • 77953323050 scopus 로고    scopus 로고
    • Splitting integrators for nonlinear Schrödinger equations over long times
    • L. Gauckler and C. Lubich (2010), 'Splitting integrators for nonlinear Schrödinger equations over long times', Found. Comput. Math. 10, 275-302.
    • (2010) Found. Comput. Math. , vol.10 , pp. 275-302
    • Gauckler, L.1    Lubich, C.2
  • 64
    • 0037143007 scopus 로고    scopus 로고
    • Using K-branch entropy solutions for multivalued geometric optics computations
    • L. Gosse (2002), 'Using K-branch entropy solutions for multivalued geometric optics computations', J. Comput. Phys. 180, 155-182.
    • (2002) J. Comput. Phys. , vol.180 , pp. 155-182
    • Gosse, L.1
  • 65
    • 33746895844 scopus 로고    scopus 로고
    • Multiphase semiclassical approximation of the one-dimensional harmonic crystal: I. The periodic case
    • DOI 10.1088/0305-4470/39/33/018, PII S0305447006145035, 018
    • L. Gosse (2006), 'Multiphase semiclassical approximation of the one-dimensional harmonic crystal I: The periodic case', J. Phys. A 39, 10509-10521. (Pubitemid 44183569)
    • (2006) Journal of Physics A: Mathematical and General , vol.39 , Issue.33 , pp. 10509-10521
    • Gosse, L.1
  • 66
    • 0036003237 scopus 로고    scopus 로고
    • Convergence results for an inhomogeneous system arising in various high frequency approximations
    • DOI 10.1007/s002110100309
    • L. Gosse and F. James (2002), 'Convergence results for an inhomogeneous system arising in various high frequency approximations', Numer. Math. 90, 721-753. (Pubitemid 36161559)
    • (2002) Numerische Mathematik , vol.90 , Issue.4 , pp. 721-753
    • Gosse, L.1    James, F.2
  • 67
    • 3242737651 scopus 로고    scopus 로고
    • Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice I. Homogeneous problems
    • DOI 10.1016/j.jcp.2003.12.004, PII S0021999103006454
    • L. Gosse and P. A. Markowich (2004), 'Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice I: Homogeneous problems', J. Comput. Phys. 197, 387-417. (Pubitemid 38963380)
    • (2004) Journal of Computational Physics , vol.197 , Issue.2 , pp. 387-417
    • Gosse, L.1    Markowich, P.A.2
  • 68
    • 25444466220 scopus 로고    scopus 로고
    • Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice - III. From ab initio models to WKB for Schrödinger-Poisson
    • DOI 10.1016/j.jcp.2005.05.020, PII S0021999105002779
    • L. Gosse and N. J. Mauser (2006), 'Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice III: From ab initio models to WKB for Schrödinger-Poisson', J. Comput. Phys. 211, 326-346. (Pubitemid 41363988)
    • (2006) Journal of Computational Physics , vol.211 , Issue.1 , pp. 326-346
    • Gosse, L.1    Mauser, N.J.2
  • 69
    • 0442275913 scopus 로고    scopus 로고
    • Two moment systems for computing multiphase semiclassical limits of the schrödinger equation
    • DOI 10.1142/S0218202503003082
    • L. Gosse, S. Jin and X. Li (2003), 'Two moment systems for computing multiphase semiclassical limits of the Schrödinger equation', Math. Models Methods Appl. Sci. 13, 1689-1723. (Pubitemid 38191701)
    • (2003) Mathematical Models and Methods in Applied Sciences , vol.13 , Issue.12 , pp. 1689-1723
    • Gosse, L.1    Jin, S.2    Li, X.3
  • 70
    • 0032661414 scopus 로고    scopus 로고
    • Vacuum electronics at the dawn of the twenty-first century
    • V. L. Granastein, R. K. Parker and C. Armstrong (1999), 'Vacuum electronics at the dawn of the twenty-first century', Proc. IEEE, 87, 702-716.
    • (1999) Proc. IEEE , vol.87 , pp. 702-716
    • Granastein, V.L.1    Parker, R.K.2    Armstrong, C.3
  • 71
    • 21944449213 scopus 로고    scopus 로고
    • Semiclassical limit of the nonlinear Schrödinger equation in small time
    • E. Grenier (1998), 'Semiclassical limit of the nonlinear Schrödinger equation in small time', Proc. Amer. Math. Soc. 126, 523-530.
    • (1998) Proc. Amer. Math. Soc. , vol.126 , pp. 523-530
    • Grenier, E.1
  • 72
    • 0001742215 scopus 로고
    • Semiclassical asymptotics in solid-state physics
    • J.-C. Guillot, J. Ralston and E. Trubowitz (1988), 'Semiclassical asymptotics in solid-state physics', Comm. Math. Phys. 116, 401-415.
    • (1988) Comm. Math. Phys. , vol.116 , pp. 401-415
    • Guillot, J.-C.1    Ralston, J.2    Trubowitz, E.3
  • 73
    • 0001118605 scopus 로고
    • Molecular propagation through electron energy level crossings
    • # 536
    • G. A. Hagedorn (1994), 'Molecular propagation through electron energy level crossings', Mem. Amer. Math. Soc. 111, #536.
    • (1994) Mem. Amer. Math. Soc. , pp. 111
    • Hagedorn, G.A.1
  • 74
    • 36749113388 scopus 로고
    • Frozen Gaussians: A very simple semiclassical approximation
    • E. J. Heller (1981), 'Frozen Gaussians: A very simple semiclassical approximation', J. Chem. Phys. 75, 2923-2931.
    • (1981) J. Chem. Phys. , vol.75 , pp. 2923-2931
    • Heller, E.J.1
  • 75
    • 33644973307 scopus 로고    scopus 로고
    • Guided Gaussian wave packets
    • E. J. Heller (2006), 'Guided Gaussian wave packets', Acc. Chem. Res. 39, 127-134.
    • (2006) Acc. Chem. Res. , vol.39 , pp. 127-134
    • Heller, E.J.1
  • 76
    • 79956146900 scopus 로고
    • A semiclassical justification for the use of nonspreading wavepackets in dynamics calculations
    • M. Herman and E. Kluk (1984), 'A semiclassical justification for the use of nonspreading wavepackets in dynamics calculations', J. Chem. Phys. 91, 2923-2931.
    • (1984) J. Chem. Phys. , vol.91 , pp. 2923-2931
    • Herman, M.1    Kluk, E.2
  • 77
    • 0025626799 scopus 로고
    • Gaussian beam migration
    • N. Hill (1990), 'Gaussian beam migration', Geophysics 55, 1416-1428.
    • (1990) Geophysics , vol.55 , pp. 1416-1428
    • Hill, N.1
  • 78
    • 0000111115 scopus 로고
    • On the weak coupling limit for a Fermi gas in a random potential
    • T. G. Ho, L. J. Landau and A. J. Wilkins (1993), 'On the weak coupling limit for a Fermi gas in a random potential', Rev. Math. Phys. 5, 209-298.
    • (1993) Rev. Math. Phys. , vol.5 , pp. 209-298
    • Ho, T.G.1    Landau, L.J.2    Wilkins, A.J.3
  • 79
    • 0037461490 scopus 로고    scopus 로고
    • Quantum-classical Liouville approach to molecular dynamics: Surface hopping Gaussian phasespace packets
    • I. Horenko, C. Salzmann, B. Schmidt and C. Schütte (2002), 'Quantum-classical Liouville approach to molecular dynamics: Surface hopping Gaussian phasespace packets', J. Chem. Phys. 117, 11075.
    • (2002) J. Chem. Phys. , vol.117 , pp. 11075
    • Horenko, I.1    Salzmann, C.2    Schmidt, B.3    Schütte, C.4
  • 81
    • 41449101890 scopus 로고    scopus 로고
    • A bloch decomposition-based split-step pseudospectral method for quantum dynamics with periodic potentials
    • DOI 10.1137/060652026
    • Z. Huang, S. Jin, P. A. Markowich and C. Sparber (2007), 'A Bloch decompositionbased split-step pseudospectral method for quantum dynamics with periodic potentials', SIAM J. Sci. Comput. 29, 515-538. (Pubitemid 351456403)
    • (2007) SIAM Journal of Scientific Computing , vol.29 , Issue.2 , pp. 515-538
    • Huang, Z.1    Jin, S.2    Markowich, P.A.3    Sparber, C.4
  • 82
    • 55149094759 scopus 로고    scopus 로고
    • Numerical simulation of the nonlinear Schrödinger equation with multidimensional periodic potentials
    • Z. Huang, S. Jin, P. A. Markowich and C. Sparber (2008), 'Numerical simulation of the nonlinear Schrödinger equation with multidimensional periodic potentials', Multiscale Model. Simul. 7, 539-564.
    • (2008) Multiscale Model. Simul. , vol.7 , pp. 539-564
    • Huang, Z.1    Jin, S.2    Markowich, P.A.3    Sparber, C.4
  • 83
    • 56549089394 scopus 로고    scopus 로고
    • On the Bloch decomposition based spectral method for wave propagation in periodic media
    • Z. Huang, S. Jin, P. A. Markowich and C. Sparber (2009), 'On the Bloch decomposition based spectral method for wave propagation in periodic media', Wave Motion 46, 15-28.
    • (2009) Wave Motion , vol.46 , pp. 15-28
    • Huang, Z.1    Jin, S.2    Markowich, P.A.3    Sparber, C.4
  • 84
    • 19744363102 scopus 로고    scopus 로고
    • A time-splitting spectral scheme for the Maxwell-Dirac system
    • DOI 10.1016/j.jcp.2005.02.026, PII S0021999105001300
    • Z. Huang, S. Jin, P. A. Markowich, C. Sparber and C. Zheng (2005), 'A timesplitting spectral scheme for the Maxwell-Dirac system', J. Comput. Phys. 208, 761-789. (Pubitemid 40746723)
    • (2005) Journal of Computational Physics , vol.208 , Issue.2 , pp. 761-789
    • Huang, Z.1    Jin, S.2    Markowich, P.A.3    Sparber, C.4    Zheng, C.5
  • 85
    • 0032206542 scopus 로고    scopus 로고
    • Nonoscillatory central schemes for multidimensional hyperbolic conservation laws
    • PII S106482759631041X
    • G. Jiang and E. Tadmor (1998), 'Nonoscillatory central schemes for multidimensional hyperbolic conservation laws', SIAM J. Sci. Comput. 19, 1892-1917. (Pubitemid 128614424)
    • (1998) SIAM Journal of Scientific Computing , vol.19 , Issue.6 , pp. 1892-1917
    • Jiang, G.-S.1    Tadmor, E.2
  • 86
    • 85116523702 scopus 로고    scopus 로고
    • Recent computational methods for high frequency waves in heterogeneous media
    • Ser. Contemp. Appl. Math. CAM, Higher Education Press, Beijing
    • S. Jin (2009), Recent computational methods for high frequency waves in heterogeneous media. In Industrial and Applied Mathematics in China, Vol. 10 of Ser. Contemp. Appl. Math. CAM, Higher Education Press, Beijing, pp. 49-64.
    • (2009) Industrial and Applied Mathematics in China , vol.10 , pp. 49-64
    • Jin, S.1
  • 87
    • 0442331531 scopus 로고    scopus 로고
    • Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs. Wigner
    • S. Jin and X. Li (2003), 'Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs. Wigner', Phys. D 182, 46-85.
    • (2003) Phys. D , vol.182 , pp. 46-85
    • Jin, S.1    Li, X.2
  • 88
    • 34548683732 scopus 로고    scopus 로고
    • A Hamiltonian-preserving scheme for high frequency elastic waves in heterogeneous media
    • S. Jin and X. Liao (2006), 'A Hamiltonian-preserving scheme for high frequency elastic waves in heterogeneous media', J. Hyperbolic Diff. Eqn. 3, 741-777.
    • (2006) J. Hyperbolic Diff. Eqn. , vol.3 , pp. 741-777
    • Jin, S.1    Liao, X.2
  • 89
    • 34548686622 scopus 로고    scopus 로고
    • A semiclassical transport model for thin quantum barriers
    • DOI 10.1137/060653214
    • S. Jin and K. A. Novak (2006), 'A semiclassical transport model for thin quantum barriers', Multiscale Model. Simul. 5, 1063-1086. (Pubitemid 350285615)
    • (2006) Multiscale Modeling and Simulation , vol.5 , Issue.4 , pp. 1063-1086
    • Jin, S.1    Novak, K.A.2
  • 90
    • 34548689607 scopus 로고    scopus 로고
    • A semiclassical transport model for twodimensional thin quantum barriers
    • S. Jin and K. A. Novak (2007), 'A semiclassical transport model for twodimensional thin quantum barriers', J. Comput. Phys. 226, 1623-1644.
    • (2007) J. Comput. Phys. , vol.226 , pp. 1623-1644
    • Jin, S.1    Novak, K.A.2
  • 91
    • 77954748545 scopus 로고    scopus 로고
    • A coherent semiclassical transport model for purestate quantum scattering
    • S. Jin and K. A. Novak (2010), 'A coherent semiclassical transport model for purestate quantum scattering', Comm. Math. Sci. 8, 253-275.
    • (2010) Comm. Math. Sci. , vol.8 , pp. 253-275
    • Jin, S.1    Novak, K.A.2
  • 92
    • 3242673880 scopus 로고    scopus 로고
    • A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations
    • S. Jin and S. Osher (2003), 'A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations', Comm. Math. Sci. 1, 575-591.
    • (2003) Comm. Math. Sci. , vol.1 , pp. 575-591
    • Jin, S.1    Osher, S.2
  • 93
    • 33644957151 scopus 로고    scopus 로고
    • Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials
    • S. Jin and X. Wen (2005), 'Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials', Comm. Math. Sci. 3, 285-315.
    • (2005) Comm. Math. Sci. , vol.3 , pp. 285-315
    • Jin, S.1    Wen, X.2
  • 94
    • 34548672178 scopus 로고    scopus 로고
    • A Hamiltonian-preserving scheme for the Liouville equation of geometrical optics with partial transmissions and reflections
    • DOI 10.1137/050631343
    • S. Jin and X. Wen (2006a), 'A Hamiltonian-preserving scheme for the Liouville equation of geometrical optics with partial transmissions and reflections', SIAM J. Numer. Anal. 44, 1801-1828. (Pubitemid 350012041)
    • (2006) SIAM Journal on Numerical Analysis , vol.44 , Issue.5 , pp. 1801-1828
    • Jin, S.1    Wen, X.2
  • 95
    • 33644952544 scopus 로고    scopus 로고
    • Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds
    • DOI 10.1016/j.jcp.2005.10.012, PII S0021999105004687
    • S. Jin and X.Wen (2006b), 'Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds', J. Comput. Phys. 214, 672-697. (Pubitemid 43407349)
    • (2006) Journal of Computational Physics , vol.214 , Issue.2 , pp. 672-697
    • Jin, S.1    Wen, X.2
  • 96
    • 0000997075 scopus 로고    scopus 로고
    • Numerical passage from systems of conservation laws to hamilton-jacobi equations, and relaxation schemes
    • S. Jin and Z. Xin (1998), 'Numerical passage from systems of conservation laws to Hamilton-Jacobi equations, relaxation schemes', SIAM J. Numer. Anal. 35, 2385-2404. (Pubitemid 128464113)
    • (1998) SIAM Journal on Numerical Analysis , vol.35 , Issue.6 , pp. 2385-2404
    • Jin, S.1    Xin, Z.2
  • 97
    • 49749131952 scopus 로고    scopus 로고
    • Computation of the semiclassical limit of the Schrödinger equation with phase shift by a level set method
    • S. Jin and X. Yang (2008), 'Computation of the semiclassical limit of the Schrödinger equation with phase shift by a level set method', J. Sci. Comput. 35, 144-169.
    • (2008) J. Sci. Comput. , vol.35 , pp. 144-169
    • Jin, S.1    Yang, X.2
  • 98
    • 57849152968 scopus 로고    scopus 로고
    • Computation of high frequency wave diffraction by a half plane via the Liouville equation and geometric theory of diffraction
    • S. Jin and D. Yin (2008a), 'Computation of high frequency wave diffraction by a half plane via the Liouville equation and geometric theory of diffraction', Comm. Comput. Phys. 4, 1106-1128.
    • (2008) Comm. Comput. Phys. , vol.4 , pp. 1106-1128
    • Jin, S.1    Yin, D.2
  • 99
    • 43049091907 scopus 로고    scopus 로고
    • Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction
    • S. Jin and D. Yin (2008b), 'Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction', J. Comput. Phys. 227, 6106-6139.
    • (2008) J. Comput. Phys. , vol.227 , pp. 6106-6139
    • Jin, S.1    Yin, D.2
  • 100
    • 79956152646 scopus 로고    scopus 로고
    • Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction
    • S. Jin and D. Yin (2011), 'Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction', Kinetic and Related Models 4, 295-316.
    • (2011) Kinetic and Related Models , vol.4 , pp. 295-316
    • Jin, S.1    Yin, D.2
  • 101
    • 0033438093 scopus 로고    scopus 로고
    • The semiclassical limit of the defocusing NLS hierarchy
    • S. Jin, C. D. Levermore and D. W. McLaughlin (1999), 'The semiclassical limit of the defocusing NLS hierarchy', Comm. Pure Appl. Math. 52, 613-654.
    • (1999) Comm. Pure Appl. Math. , vol.52 , pp. 613-654
    • Jin, S.1    Levermore, C.D.2    McLaughlin, D.W.3
  • 102
    • 55549106473 scopus 로고    scopus 로고
    • Computation of interface reflection and regular or diffuse transmission of the planar symmetric radiative transfer equation with isotropic scattering and its diffusion limit
    • S. Jin, X. Liao and X. Yang (2008a), 'Computation of interface reflection and regular or diffuse transmission of the planar symmetric radiative transfer equation with isotropic scattering and its diffusion limit', SIAM J. Sci. Comput. 30, 1992-2017.
    • (2008) SIAM J. Sci. Comput. , vol.30 , pp. 1992-2017
    • Jin, S.1    Liao, X.2    Yang, X.3
  • 103
    • 23944519965 scopus 로고    scopus 로고
    • Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems
    • DOI 10.1016/j.jcp.2005.04.020, PII S0021999105002378
    • S. Jin, H. Liu, S. Osher and R. Tsai (2005a), 'Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems', J. Comput. Phys. 210, 497-518. (Pubitemid 41183686)
    • (2005) Journal of Computational Physics , vol.210 , Issue.2 , pp. 497-518
    • Jin, S.1    Liu, H.2    Osher, S.3    Tsai, R.4
  • 104
    • 16844373042 scopus 로고    scopus 로고
    • Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
    • S. Jin, H. Liu, S. Osher and Y.-H. R. Tsai (2005b), 'Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation', J. Comput. Phys. 205, 222-241.
    • (2005) J. Comput. Phys. , vol.205 , pp. 222-241
    • Jin, S.1    Liu, H.2    Osher, S.3    Tsai, Y.-H.R.4
  • 105
    • 13844262830 scopus 로고    scopus 로고
    • Numerical simulation of a generalized Zakharov system
    • DOI 10.1016/j.jcp.2004.06.001, PII S0021999104002347
    • S. Jin, P. A. Markowich and C. Zheng (2004), 'Numerical simulation of a generalized Zakharov system', J. Comput. Phys. 201, 376-395. (Pubitemid 40256599)
    • (2004) Journal of Computational Physics , vol.201 , Issue.1 , pp. 376-395
    • Jin, S.1    Markowich, P.A.2    Zheng, C.3
  • 106
    • 79955918897 scopus 로고    scopus 로고
    • An Eulerian surface hopping method for the Schrödinger equation with conical crossings
    • S. Jin, P. Qi and Z. Zhang (2011), 'An Eulerian surface hopping method for the Schrödinger equation with conical crossings', Multiscale Model. Simul. 9, 258-281.
    • (2011) Multiscale Model. Simul. , vol.9 , pp. 258-281
    • Jin, S.1    Qi, P.2    Zhang, Z.3
  • 107
    • 77950364040 scopus 로고    scopus 로고
    • Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations
    • S. Jin, H. Wu and X. Yang (2008b), 'Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations', Comm. Math. Sci. 6, 995-1020.
    • (2008) Comm. Math. Sci. , vol.6 , pp. 995-1020
    • Jin, S.1    Wu, H.2    Yang, X.3
  • 108
    • 77950359784 scopus 로고    scopus 로고
    • A numerical study of the Gaussian beam methods for Schrödinger- Poisson equations
    • S. Jin, H. Wu and X. Yang (2010a), 'A numerical study of the Gaussian beam methods for Schrödinger-Poisson equations', J. Comput. Math. 28, 261-272.
    • (2010) J. Comput. Math. , vol.28 , pp. 261-272
    • Jin, S.1    Wu, H.2    Yang, X.3
  • 109
    • 79551510529 scopus 로고    scopus 로고
    • Semi-Eulerian and high order Gaussian beam methods for the Schrödinger equation in the semiclassical regime
    • S. Jin, H. Wu and X. Yang (2011), 'Semi-Eulerian and high order Gaussian beam methods for the Schrödinger equation in the semiclassical regime', Comm. Comput. Phys. 9, 668-687.
    • (2011) Comm. Comput. Phys. , vol.9 , pp. 668-687
    • Jin, S.1    Wu, H.2    Yang, X.3
  • 110
    • 77952425757 scopus 로고    scopus 로고
    • Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials
    • S. Jin, H. Wu, X. Yang and Z. Huang (2010b), 'Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials', J. Comput. Phys. 229, 4869-4883.
    • (2010) J. Comput. Phys. , vol.229 , pp. 4869-4883
    • Jin, S.1    Wu, H.2    Yang, X.3    Huang, Z.4
  • 112
    • 36449004965 scopus 로고
    • Integral expressions for the semi-classical time-dependent propagator
    • K. Kay (1994), 'Integral expressions for the semi-classical time-dependent propagator', J. Chem. Phys. 100, 4437-4392.
    • (1994) J. Chem. Phys. , vol.100 , pp. 4437-4392
    • Kay, K.1
  • 113
    • 33244471525 scopus 로고    scopus 로고
    • The Herman-Kluk approximation: Derivation and semiclassical corrections
    • K. Kay (2006), 'The Herman-Kluk approximation: Derivation and semiclassical corrections', J. Chem. Phys. 322, 3-12.
    • (2006) J. Chem. Phys. , vol.322 , pp. 3-12
    • Kay, K.1
  • 114
    • 0001786794 scopus 로고
    • Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell's equations
    • Plenum
    • J. B. Keller and R. M. Lewis (1995), Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell's equations. In Surveys in Applied Mathematics, Vol. 1, Plenum, pp. 1-82.
    • (1995) Surveys in Applied Mathematics , vol.1 , pp. 1-82
    • Keller, J.B.1    Lewis, R.M.2
  • 115
    • 0000175040 scopus 로고
    • On a construction of the fundamental solution for Schrödinger equations
    • H. Kitada (1980), 'On a construction of the fundamental solution for Schrödinger equations', J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 193-226.
    • (1980) J. Fac. Sci. Univ. Tokyo Sect. IA Math. , vol.27 , pp. 193-226
    • Kitada, H.1
  • 116
    • 62749134561 scopus 로고    scopus 로고
    • Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations
    • C. Klein (2008), 'Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations', Electron. Trans. Numer. Anal. 29, 116-135.
    • (2008) Electron. Trans. Numer. Anal. , vol.29 , pp. 116-135
    • Klein, C.1
  • 117
    • 0022735928 scopus 로고
    • A study of singularity formation in a vortex sheet by the pointvortex approximation
    • R. Krasny (1986), 'A study of singularity formation in a vortex sheet by the pointvortex approximation', J. Fluid Mech. 167, 65-93. (Pubitemid 16603670)
    • (1986) Journal of Fluid Mechanics , vol.167 , pp. 65-93
    • Krasny, R.1
  • 118
    • 59149086364 scopus 로고    scopus 로고
    • Monte Carlo sampling of Wigner functions and surface hopping quantum dynamics
    • S. Kube, C. Lasser and M. Weber (2009), 'Monte Carlo sampling of Wigner functions and surface hopping quantum dynamics', J. Comput. Phys. 228, 1947-1962.
    • (2009) J. Comput. Phys. , vol.228 , pp. 1947-1962
    • Kube, S.1    Lasser, C.2    Weber, M.3
  • 119
    • 0002636472 scopus 로고
    • Zur Theorie der EnergieÜbertragung II
    • L. Landau (1932), 'Zur Theorie der EnergieÜbertragung II', Physics of the Soviet Union 2, 46-51.
    • (1932) Physics of the Soviet Union , vol.2 , pp. 46-51
    • Landau, L.1
  • 120
    • 25444450804 scopus 로고    scopus 로고
    • Propagation through conical crossings: An asymptotic semigroup
    • DOI 10.1002/cpa.20087
    • C. Lasser and S. Teufel (2005), 'Propagation through conical crossings: An asymptotic semigroup', Comm. Pure Appl. Math. 58, 1188-1230. (Pubitemid 41366832)
    • (2005) Communications on Pure and Applied Mathematics , vol.58 , Issue.9 , pp. 1188-1230
    • Lasser, C.1    Teufel, S.2
  • 121
    • 39049111633 scopus 로고    scopus 로고
    • Construction and validation of a rigorous surface hopping algorithm for conical crossings
    • C. Lasser, T. Swart and S. Teufel (2007), 'Construction and validation of a rigorous surface hopping algorithm for conical crossings', Comm. Math. Sci. 5, 789-814.
    • (2007) Comm. Math. Sci. , vol.5 , pp. 789-814
    • Lasser, C.1    Swart, T.2    Teufel, S.3
  • 122
    • 84972538741 scopus 로고
    • Asymptotic solutions of oscillatory initial value problems
    • P. D. Lax (1957), 'Asymptotic solutions of oscillatory initial value problems', Duke Math. J. 24, 627-646.
    • (1957) Duke Math. J. , vol.24 , pp. 627-646
    • Lax, P.D.1
  • 123
    • 28844494719 scopus 로고    scopus 로고
    • Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press
    • B. Leimkuhler and S. Reich (2004), Simulating Hamiltonian Dynamics, Vol. 14 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.
    • (2004) Simulating Hamiltonian Dynamics , vol.14
    • Leimkuhler, B.1    Reich, S.2
  • 124
    • 61349126449 scopus 로고    scopus 로고
    • Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime
    • S. Leung and J. Qian (2009), 'Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime', J. Comput. Phys. 228, 2951-2977.
    • (2009) J. Comput. Phys. , vol.228 , pp. 2951-2977
    • Leung, S.1    Qian, J.2
  • 125
    • 34548534954 scopus 로고    scopus 로고
    • Eulerian Gaussian beams for high frequency wave propagation
    • S. Leung, J. Qian and R. Burridge (2007), 'Eulerian Gaussian beams for high frequency wave propagation', Geophysics 72, 61-76.
    • (2007) Geophysics , vol.72 , pp. 61-76
    • Leung, S.1    Qian, J.2    Burridge, R.3
  • 126
    • 0030534703 scopus 로고    scopus 로고
    • Moment closure hierarchies for kinetic theories
    • C. Levermore (1996), 'Moment closure hierarchies for kinetic theories', J. Statist. Phys. 83, 1021-1065. (Pubitemid 126326619)
    • (1996) Journal of Statistical Physics , vol.83 , Issue.5-6 , pp. 1021-1065
    • Levermore, C.D.1
  • 127
    • 37649029921 scopus 로고    scopus 로고
    • An Eulerian method for computing multi-valued solutions to the Euler-Poisson equations and applications to wave breaking in klystrons
    • X. Li, J. G. Wöhlbier, S. Jin and J. Booske (2004), 'An Eulerian method for computing multi-valued solutions to the Euler-Poisson equations and applications to wave breaking in klystrons', Phys. Rev. E 70, 016502.
    • (2004) Phys. Rev. E , vol.70 , pp. 016502
    • Li, X.1    Wöhlbier, J.G.2    Jin, S.3    Booske, J.4
  • 129
    • 77955846856 scopus 로고    scopus 로고
    • Recovery of high frequency wave fields from phase space-based measurements
    • H. Liu and J. Ralston (2010), 'Recovery of high frequency wave fields from phase space-based measurements', Multiscale Model. Simul. 8, 622-644.
    • (2010) Multiscale Model. Simul. , vol.8 , pp. 622-644
    • Liu, H.1    Ralston, J.2
  • 130
    • 13844272991 scopus 로고    scopus 로고
    • Semiclassical Limit of the Nonlinear Schrodinger-Poisson Equation with Subcritical Initial Data
    • H. Liu and E. Tadmor (2002), 'Semi-classical limit of the nonlinear Schrödinger- Poisson equation with sub-critical initial data', Methods Appl. Anal. 9, 517-532. (Pubitemid 37316795)
    • (2002) Methods Appl. Anal. , vol.9 , Issue.4 , pp. 517-532
    • Liu, H.1    Tadmor, E.2
  • 131
    • 34447264074 scopus 로고    scopus 로고
    • A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations
    • H. Liu and Z. Wang (2007), 'A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations', J. Comput. Phys. 225, 591-614.
    • (2007) J. Comput. Phys. , vol.225 , pp. 591-614
    • Liu, H.1    Wang, Z.2
  • 133
    • 79953052469 scopus 로고    scopus 로고
    • Frozen Gaussian approximation for high frequency wave propagation
    • J. Lu and X. Yang (2011), 'Frozen Gaussian approximation for high frequency wave propagation', Commun. Math. Sci. 9, 663-683.
    • (2011) Commun. Math. Sci. , vol.9 , pp. 663-683
    • Lu, J.1    Yang, X.2
  • 134
    • 57049171510 scopus 로고    scopus 로고
    • On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations
    • C. Lubich (2008), 'On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations', Math. Comp. 77, 2141-2153.
    • (2008) Math. Comp. , vol.77 , pp. 2141-2153
    • Lubich, C.1
  • 135
    • 0001599323 scopus 로고
    • Concentrations in the onedimensional Vlasov-Poisson equations I: Temporal development and nonunique weak solutions in the single component case
    • A. J. Majda, G. Majda and Y. X. Zheng (1994), 'Concentrations in the onedimensional Vlasov-Poisson equations I: Temporal development and nonunique weak solutions in the single component case', Phys. D 74, 268-300.
    • (1994) Phys. D , vol.74 , pp. 268-300
    • Majda, A.J.1    Majda, G.2    Zheng, Y.X.3
  • 136
    • 0000269855 scopus 로고
    • The classical limit of a self-consistent quantum-Vlasov equation in 3D
    • P. A. Markowich and N. J. Mauser (1993), 'The classical limit of a self-consistent quantum-Vlasov equation in 3D', Math. Models Methods Appl. Sci. 3, 109-124.
    • (1993) Math. Models Methods Appl. Sci. , vol.3 , pp. 109-124
    • Markowich, P.A.1    Mauser, N.J.2
  • 137
    • 0011569744 scopus 로고    scopus 로고
    • The pseudo-differential approach to finite differences revisited
    • P. A. Markowich and F. Poupaud (1999), 'The pseudo-differential approach to finite differences revisited', Calcolo 36, 161-186.
    • (1999) Calcolo , vol.36 , pp. 161-186
    • Markowich, P.A.1    Poupaud, F.2
  • 138
    • 36449006348 scopus 로고
    • A Wigner-function approach to (semi)classical limits: Electrons in a periodic potential
    • P. A. Markowich, N. J. Mauser and F. Poupaud (1994), 'A Wigner-function approach to (semi)classical limits: Electrons in a periodic potential', J. Math. Phys. 35, 1066-1094.
    • (1994) J. Math. Phys. , vol.35 , pp. 1066-1094
    • Markowich, P.A.1    Mauser, N.J.2    Poupaud, F.3
  • 139
    • 0033480955 scopus 로고    scopus 로고
    • Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit
    • P. A. Markowich, P. Pietra and C. Pohl (1999), 'Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit', Numer. Math. 81, 595-630.
    • (1999) Numer. Math. , vol.81 , pp. 595-630
    • Markowich, P.A.1    Pietra, P.2    Pohl, C.3
  • 141
    • 85095839561 scopus 로고    scopus 로고
    • Splitting methods
    • Cambridge University Press
    • R. I. McLachlan and G. R. W. Quispel (2002), Splitting methods. In Acta Numerica, Vol. 11, Cambridge University Press, pp. 341-434.
    • (2002) Acta Numerica , vol.11 , pp. 341-434
    • McLachlan, R.I.1    Quispel, G.R.W.2
  • 142
    • 0034148496 scopus 로고    scopus 로고
    • Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary
    • L. Miller (2000), 'Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary', J. Math. Pures Appl. (9) 79, 227-269.
    • (2000) J. Math. Pures Appl. (9) , vol.79 , pp. 227-269
    • Miller, L.1
  • 143
    • 77955843316 scopus 로고    scopus 로고
    • Taylor expansion and discretization errors in Gaussian beam superposition
    • M. Motamed and O. Runborg (2010), 'Taylor expansion and discretization errors in Gaussian beam superposition', Wave Motion 47, 421-439.
    • (2010) Wave Motion , vol.47 , pp. 421-439
    • Motamed, M.1    Runborg, O.2
  • 144
    • 21844502570 scopus 로고
    • Asymptotic analysis of a scaled Wigner equation and quantum scattering
    • F. Nier (1995), 'Asymptotic analysis of a scaled Wigner equation and quantum scattering', Transport Theory Statist. Phys. 24, 591-628.
    • (1995) Transport Theory Statist. Phys. , vol.24 , pp. 591-628
    • Nier, F.1
  • 145
    • 0001485147 scopus 로고    scopus 로고
    • A semi-classical picture of quantum scattering
    • F. Nier (1996), 'A semi-classical picture of quantum scattering', Ann. Sci. ́ Ecole Norm. Sup. (4) 29, 149-183.
    • (1996) Ann. Sci. ́ Ecole Norm. Sup. (4) , vol.29 , pp. 149-183
    • Nier, F.1
  • 146
    • 44749084234 scopus 로고
    • Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
    • S. Osher and J. A. Sethian (1988), 'Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations', J. Comput. Phys. 79, 12-49.
    • (1988) J. Comput. Phys. , vol.79 , pp. 12-49
    • Osher, S.1    Sethian, J.A.2
  • 147
    • 0036647932 scopus 로고    scopus 로고
    • Geometric optics in a phase-space-based level set and Eulerian framework
    • S. Osher, L.-T. Cheng, M. Kang, H. Shim and Y.-H. Tsai (2002), 'Geometric optics in a phase-space-based level set and Eulerian framework', J. Comput. Phys. 179, 622-648.
    • (2002) J. Comput. Phys. , vol.179 , pp. 622-648
    • Osher, S.1    Cheng, L.-T.2    Kang, M.3    Shim, H.4    Tsai, Y.-H.5
  • 149
    • 0003172958 scopus 로고
    • Pseudo-spectral solution of nonlinear Schrödinger equations
    • D. Pathria and J. Morris (1990), 'Pseudo-spectral solution of nonlinear Schrödinger equations', J. Comput. Phys. 87, 108-125.
    • (1990) J. Comput. Phys. , vol.87 , pp. 108-125
    • Pathria, D.1    Morris, J.2
  • 151
    • 0012588539 scopus 로고    scopus 로고
    • A kinetic scheme for the Saint-Venant system with a source term
    • B. Perthame and C. Simeoni (2001), 'A kinetic scheme for the Saint-Venant system with a source term', Calcolo 38, 201-231. (Pubitemid 33743526)
    • (2001) Calcolo , vol.38 , Issue.4 , pp. 201-231
    • Perthame, B.1    Simeoni, C.2
  • 152
    • 0019899451 scopus 로고
    • A new method of computation of wave fields using Gaussian beams
    • DOI 10.1016/0165-2125(82)90016-6
    • M. M. Popov (1982), 'A new method of computation of wave fields using Gaussian beams', Wave Motion 4, 85-97. (Pubitemid 12514951)
    • (1982) Wave Motion , vol.4 , Issue.1 , pp. 85-97
    • Popov, M.M.1
  • 153
    • 77955512346 scopus 로고    scopus 로고
    • Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation
    • J. Qian and L. Ying (2010), 'Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation', J. Comput. Phys. 229, 7848-7873.
    • (2010) J. Comput. Phys. , vol.229 , pp. 7848-7873
    • Qian, J.1    Ying, L.2
  • 154
    • 0000952462 scopus 로고
    • Gaussian beams and the propagation of singularities
    • MAA Stud. Math., Mathematical Association of America
    • J. Ralston (1982), Gaussian beams and the propagation of singularities. In Studies in Partial Differential Equations, Vol. 23 of MAA Stud. Math., Mathematical Association of America, pp. 206-248.
    • (1982) Studies in Partial Differential Equations , vol.23 , pp. 206-248
    • Ralston, J.1
  • 157
    • 78649525352 scopus 로고    scopus 로고
    • On the Herman-Kluk semiclassical approximation
    • D. Robert (2010), 'On the Herman-Kluk semiclassical approximation', Rev. Math. Phys. 22, 1123-1145.
    • (2010) Rev. Math. Phys. , vol.22 , pp. 1123-1145
    • Robert, D.1
  • 158
    • 0034551460 scopus 로고    scopus 로고
    • Some new results in multiphase geometrical optics
    • O. Runborg (2000), 'Some new results in multiphase geometrical optics', M2AN Math. Model. Numer. Anal. 34, 1203-1231.
    • (2000) M2AN Math. Model. Numer. Anal. , vol.34 , pp. 1203-1231
    • Runborg, O.1
  • 159
    • 0030388835 scopus 로고    scopus 로고
    • Transport equations for elastic and other waves in random media
    • DOI 10.1016/S0165-2125(96)00021-2, PII S0165212596000212
    • L. Ryzhik, G. Papanicolaou and J. B. Keller (1996), 'Transport equations for elastic and other waves in random media', Wave Motion 24, 327-370. (Pubitemid 126324199)
    • (1996) Wave Motion , vol.24 , Issue.4 , pp. 327-370
    • Ryzhik, L.1    Papanicolaou, G.2    Keller, J.B.3
  • 160
    • 0003468178 scopus 로고
    • Advanced Series in Mathematical Physics, World Scientific
    • A. Shapere and F. Wilczek, eds (1989), Geometric Phases in Physics, Vol. 5 of Advanced Series in Mathematical Physics, World Scientific.
    • (1989) Geometric Phases in Physics , vol.5
    • Shapere, A.1    Wilczek, F.2
  • 161
    • 0000689893 scopus 로고    scopus 로고
    • A generalized surface hopping method
    • D. Sholla and J. Tully (1998), 'A generalized surface hopping method', J. Chem. Phys. 109, 7702.
    • (1998) J. Chem. Phys. , vol.109 , pp. 7702
    • Sholla, D.1    Tully, J.2
  • 162
    • 84887251437 scopus 로고    scopus 로고
    • Wigner functions versus WKBmethods in multivalued geometrical optics
    • C. Sparber, P. Markowich and N. Mauser (2003), 'Wigner functions versus WKBmethods in multivalued geometrical optics', Asymptot. Anal. 33, 153-187.
    • (2003) Asymptot. Anal. , vol.33 , pp. 153-187
    • Sparber, C.1    Markowich, P.2    Mauser, N.3
  • 163
    • 0001200202 scopus 로고
    • Derivation of the transport equation for electrons moving through random impurities
    • H. Spohn (1977), 'Derivation of the transport equation for electrons moving through random impurities', J. Statist. Phys. 17, 385-412.
    • (1977) J. Statist. Phys. , vol.17 , pp. 385-412
    • Spohn, H.1
  • 164
    • 0035539929 scopus 로고    scopus 로고
    • Adiabatic decoupling and time-dependent Born- Oppenheimer theory
    • H. Spohn and S. Teufel (2001), 'Adiabatic decoupling and time-dependent Born- Oppenheimer theory', Comm. Math. Phys. 224, 113-132.
    • (2001) Comm. Math. Phys. , vol.224 , pp. 113-132
    • Spohn, H.1    Teufel, S.2
  • 167
    • 0000952833 scopus 로고    scopus 로고
    • Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects
    • G. Sundaram and Q. Niu (1999), 'Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects', Phys. Rev. B 59, 14915-14925.
    • (1999) Phys. Rev. B , vol.59 , pp. 14915-14925
    • Sundaram, G.1    Niu, Q.2
  • 168
    • 58849104349 scopus 로고    scopus 로고
    • A mathematical justification for the Herman-Kluk propagator
    • T. Swart and V. Rousse (2009), 'A mathematical justification for the Herman-Kluk propagator', Comm. Math. Phys. 286, 725-750.
    • (2009) Comm. Math. Phys. , vol.286 , pp. 725-750
    • Swart, T.1    Rousse, V.2
  • 169
    • 48549114390 scopus 로고
    • Analytical and numerical aspects of certain nonlinear evolution equations II: Numerical, nonlinear Schrödinger equations
    • T. Taha and M. J. Ablowitz (1984), 'Analytical and numerical aspects of certain nonlinear evolution equations II: Numerical, nonlinear Schrödinger equations', J. Comput. Phys. 55, 203-230.
    • (1984) J. Comput. Phys. , vol.55 , pp. 203-230
    • Taha, T.1    Ablowitz, M.J.2
  • 170
    • 49349099865 scopus 로고    scopus 로고
    • Superpositions and higher order Gaussian beams
    • N. M. Tanushev (2008), 'Superpositions and higher order Gaussian beams', Comm. Math. Sci. 6, 449-475.
    • (2008) Comm. Math. Sci. , vol.6 , pp. 449-475
    • Tanushev, N.M.1
  • 171
    • 70349733388 scopus 로고    scopus 로고
    • Gaussian beam decomposition of high frequency wave fields
    • N. M. Tanushev, B. Engquist and R. Tsai (2009), 'Gaussian beam decomposition of high frequency wave fields', J. Comput. Phys. 228, 8856-8871.
    • (2009) J. Comput. Phys. , vol.228 , pp. 8856-8871
    • Tanushev, N.M.1    Engquist, B.2    Tsai, R.3
  • 172
    • 84974050752 scopus 로고
    • H-measures: A new approach for studying homogenisation, oscillations and concentration effects in partial differential equations
    • L. Tartar (1990), 'H-measures: A new approach for studying homogenisation, oscillations and concentration effects in partial differential equations', Proc. Roy. Soc. Edinburgh Sect. A 115, 193-230.
    • (1990) Proc. Roy. Soc. Edinburgh Sect. A , vol.115 , pp. 193-230
    • Tartar, L.1
  • 174
    • 13044304431 scopus 로고
    • Molecular dynamics with electronic transitions
    • J. Tully (1990), 'Molecular dynamics with electronic transitions', J. Chem. Phys. 93, 1061-1071.
    • (1990) J. Chem. Phys. , vol.93 , pp. 1061-1071
    • Tully, J.1
  • 175
    • 2242440384 scopus 로고
    • Trajectory surface hopping approach to nonadiabatic molecular collisions: The reaction of h+ with d2
    • J. Tully and R. Preston (1971), 'Trajectory surface hopping approach to nonadiabatic molecular collisions: The reaction of h+ with d2', J. Chem. Phys. 55, 562-572.
    • (1971) J. Chem. Phys. , vol.55 , pp. 562-572
    • Tully, J.1    Preston, R.2
  • 176
    • 77955272630 scopus 로고    scopus 로고
    • A level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials
    • D. Wei, S. Jin, R. Tsai and X. Yang (2010), 'A level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials', Comm. Comput. Phys. 229, 7440-7455.
    • (2010) Comm. Comput. Phys. , vol.229 , pp. 7440-7455
    • Wei, D.1    Jin, S.2    Tsai, R.3    Yang, X.4
  • 177
    • 68049110342 scopus 로고    scopus 로고
    • Convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients II: Some related binomial coefficient inequalities
    • X. Wen (2009), 'Convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients II: Some related binomial coefficient inequalities', J. Comput. Math. 27, 474-483.
    • (2009) J. Comput. Math. , vol.27 , pp. 474-483
    • Wen, X.1
  • 178
    • 77953827201 scopus 로고    scopus 로고
    • High order numerical methods to three dimensional delta function integrals in level set methods
    • X. Wen (2010), 'High order numerical methods to three dimensional delta function integrals in level set methods', SIAM J. Sci. Comput. 32, 1288-1309.
    • (2010) SIAM J. Sci. Comput. , vol.32 , pp. 1288-1309
    • Wen, X.1
  • 179
    • 61349152299 scopus 로고    scopus 로고
    • The l1-stability of a Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials
    • X. Wen and S. Jin (2009), 'The l1-stability of a Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials', J. Comput. Math. 27, 45-67.
    • (2009) J. Comput. Math. , vol.27 , pp. 45-67
    • Wen, X.1    Jin, S.2
  • 181
    • 33745014742 scopus 로고
    • On the quantum correction for thermodynamic equilibrium
    • E. P. Wigner (1932), 'On the quantum correction for thermodynamic equilibrium', Phys. Rev. 40, 749-759.
    • (1932) Phys. Rev. , vol.40 , pp. 749-759
    • Wigner, E.P.1
  • 182
    • 51249187120 scopus 로고
    • Theory of Bloch waves
    • C. H. Wilcox (1978), 'Theory of Bloch waves', J. Anal. Math. 33, 146-167.
    • (1978) J. Anal. Math. , vol.33 , pp. 146-167
    • Wilcox, C.H.1
  • 183
    • 21444449256 scopus 로고    scopus 로고
    • Eulerian calculations of wave breaking and multivalued solutions in a traveling wave tube
    • J. G. Wöhlbier, S. Jin and S. Sengele (2005), 'Eulerian calculations of wave breaking and multivalued solutions in a traveling wave tube', Physics of Plasmas 12, 023106-023113.
    • (2005) Physics of Plasmas , vol.12 , pp. 023106-023113
    • Wöhlbier, J.G.1    Jin, S.2    Sengele, S.3
  • 184
    • 0011495032 scopus 로고    scopus 로고
    • Dufort-frankel-type methods for linear and nonlinear Schrödinger equations
    • L. Wu (1996), 'Dufort-Frankel-type methods for linear and nonlinear Schrödinger equations', SIAM J. Numer. Anal. 33, 1526-1533. (Pubitemid 126437204)
    • (1996) SIAM Journal on Numerical Analysis , vol.33 , Issue.4 , pp. 1526-1533
    • Wu, L.1
  • 186
    • 33751202687 scopus 로고    scopus 로고
    • The phase flow method
    • DOI 10.1016/j.jcp.2006.05.008, PII S0021999106002282
    • L. Ying and E. J. Cand̀es (2006), 'The phase flow method', J. Comput. Phys. 220, 184-215. (Pubitemid 44780567)
    • (2006) Journal of Computational Physics , vol.220 , Issue.1 , pp. 184-215
    • Ying, L.1    Candes, E.J.2
  • 187
    • 0001332554 scopus 로고
    • Non-adiabatic crossing of energy levels
    • C. Zener (1932), 'Non-adiabatic crossing of energy levels', Proc. Royal Soc. London, Ser. A 137, 692-702.
    • (1932) Proc. Royal Soc. London, Ser. A , vol.137 , pp. 692-702
    • Zener, C.1
  • 188
    • 0038237443 scopus 로고    scopus 로고
    • Wigner measure and the semiclassical limit of Schrödinger- Poisson equations
    • P. Zhang (2002), 'Wigner measure and the semiclassical limit of Schrödinger- Poisson equations', SIAM J. Math. Anal. 34, 700-718.
    • (2002) SIAM J. Math. Anal. , vol.34 , pp. 700-718
    • Zhang, P.1
  • 189
    • 33645915478 scopus 로고    scopus 로고
    • Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations
    • DOI 10.1016/j.jcp.2005.11.005, PII S0021999105005048
    • C. Zheng (2006), 'Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations', J. Comput. Phys. 215, 552-565. (Pubitemid 43585672)
    • (2006) Journal of Computational Physics , vol.215 , Issue.2 , pp. 552-565
    • Zheng, C.1


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