{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T17:03:24Z","timestamp":1774458204119,"version":"3.50.1"},"reference-count":189,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2011,4,28]],"date-time":"2011-04-28T00:00:00Z","timestamp":1303948800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Acta Numerica"],"published-print":{"date-parts":[[2011,5]]},"abstract":"<jats:p>We consider time-dependent (linear and nonlinear) Schr\u00f6dinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schr\u00f6dinger equation in the semiclassical regime.<\/jats:p>","DOI":"10.1017\/s0962492911000031","type":"journal-article","created":{"date-parts":[[2011,4,28]],"date-time":"2011-04-28T13:35:26Z","timestamp":1303997726000},"page":"121-209","source":"Crossref","is-referenced-by-count":101,"title":["Mathematical and computational methods for semiclassical Schr\u00f6dinger equations"],"prefix":"10.1017","volume":"20","author":[{"given":"Shi","family":"Jin","sequence":"first","affiliation":[]},{"given":"Peter","family":"Markowich","sequence":"additional","affiliation":[]},{"given":"Christof","family":"Sparber","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2011,4,28]]},"reference":[{"key":"S0962492911000031_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/j.wavemoti.2006.05.005"},{"key":"S0962492911000031_ref100","doi-asserted-by":"publisher","DOI":"10.3934\/krm.2011.4.295"},{"key":"S0962492911000031_ref078","doi-asserted-by":"publisher","DOI":"10.1142\/S0129055X93000061"},{"key":"S0962492911000031_ref063","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C"},{"key":"S0962492911000031_ref159","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-2125(96)00021-2"},{"key":"S0962492911000031_ref184","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994270636"},{"key":"S0962492911000031_ref181","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.40.749"},{"key":"S0962492911000031_ref101","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L"},{"key":"S0962492911000031_ref175","doi-asserted-by":"publisher","DOI":"10.1063\/1.1675788"},{"key":"S0962492911000031_ref171","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2009.08.028"},{"key":"S0962492911000031_ref125","doi-asserted-by":"publisher","DOI":"10.1190\/1.2752136"},{"key":"S0962492911000031_ref169","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(84)90003-2"},{"key":"S0962492911000031_ref189","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.11.005"},{"key":"S0962492911000031_ref052","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(96)00023-4"},{"key":"S0962492911000031_ref161","doi-asserted-by":"publisher","DOI":"10.1063\/1.477416"},{"key":"S0962492911000031_ref158","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2000124"},{"key":"S0962492911000031_ref162","first-page":"153","article-title":"Wigner functions versus WKB-methods in multivalued geometrical optics","volume":"33","author":"Sparber","year":"2003","journal-title":"Asymptot. Anal."},{"key":"S0962492911000031_ref151","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-001-8181-3"},{"key":"S0962492911000031_ref144","doi-asserted-by":"publisher","DOI":"10.1080\/00411459508206018"},{"key":"S0962492911000031_ref143","doi-asserted-by":"publisher","DOI":"10.1016\/j.wavemoti.2010.02.001"},{"key":"S0962492911000031_ref142","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-7824(00)00158-6"},{"key":"S0962492911000031_ref140","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-8410-3"},{"key":"S0962492911000031_ref138","doi-asserted-by":"publisher","DOI":"10.1063\/1.530629"},{"key":"S0962492911000031_ref111","volume-title":"Semiclassical Soliton Ensembles for the Focusing Nonlinear Schr\u00f6dinger Equation, Vol. 154 of Annals of Mathematics Studies","author":"Kamvissis","year":"2003"},{"key":"S0962492911000031_ref185","unstructured":"Yin D. and Zheng C. (2011), Composite Gaussian beam approximation method for multi-phased wave functions. Submitted."},{"key":"S0962492911000031_ref139","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050406"},{"key":"S0962492911000031_ref134","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-08-02101-7"},{"key":"S0962492911000031_ref133","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2011.v9.n3.a2"},{"key":"S0962492911000031_ref131","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.12.018"},{"key":"S0962492911000031_ref124","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2009.01.007"},{"key":"S0962492911000031_ref123","volume-title":"Simulating Hamiltonian Dynamics, Vol. 14 of Cambridge Monographs on Applied and Computational Mathematics","author":"Leimkuhler","year":"2004"},{"key":"S0962492911000031_ref057","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160330605"},{"key":"S0962492911000031_ref056","doi-asserted-by":"publisher","DOI":"10.1063\/1.1527221"},{"key":"S0962492911000031_ref121","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2007.v5.n4.a3"},{"key":"S0962492911000031_ref115","first-page":"193","article-title":"On a construction of the fundamental solution for Schr\u00f6dinger equations","volume":"27","author":"Kitada","year":"1980","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math."},{"key":"S0962492911000031_ref188","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141001393407"},{"key":"S0962492911000031_ref112","doi-asserted-by":"publisher","DOI":"10.1063\/1.466320"},{"key":"S0962492911000031_ref109","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.091009.160310s"},{"key":"S0962492911000031_ref180","volume-title":"Linear and Nonlinear Waves","author":"Whitham","year":"1974"},{"key":"S0962492911000031_ref105","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.06.001"},{"key":"S0962492911000031_ref017","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2005.v3.n1.a5"},{"key":"S0962492911000031_ref102","doi-asserted-by":"publisher","DOI":"10.1137\/060676192"},{"key":"S0962492911000031_ref122","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-57-02471-7"},{"key":"S0962492911000031_ref098","first-page":"1106","article-title":"Computation of high frequency wave diffraction by a half plane via the Liouville equation and geometric theory of diffraction","volume":"4","author":"Jin","year":"2008","journal-title":"Comm. Comput. Phys."},{"key":"S0962492911000031_ref097","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-007-9137-9"},{"key":"S0962492911000031_ref096","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996314366"},{"key":"S0962492911000031_ref187","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1932.0165"},{"key":"S0962492911000031_ref095","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.10.012"},{"key":"S0962492911000031_ref091","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2010.v8.n1.a13"},{"key":"S0962492911000031_ref089","doi-asserted-by":"publisher","DOI":"10.1137\/060653214"},{"key":"S0962492911000031_ref088","doi-asserted-by":"publisher","DOI":"10.1142\/S0219891606000999"},{"key":"S0962492911000031_ref086","doi-asserted-by":"publisher","DOI":"10.1142\/9789812838766_0003"},{"key":"S0962492911000031_ref045","first-page":"339","article-title":"Large time, small coupling behaviour of a quantum particle in a random field","volume":"39","author":"Dell'Antonio","year":"1983","journal-title":"Ann. Inst. H. Poincar\u00e9 Sect. A"},{"key":"S0962492911000031_ref148","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-35657-6_22"},{"key":"S0962492911000031_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2006.10.003"},{"key":"S0962492911000031_ref141","first-page":"341","volume-title":"Acta Numerica","volume":"11","author":"McLachlan","year":"2002"},{"key":"S0962492911000031_ref083","doi-asserted-by":"publisher","DOI":"10.1016\/j.wavemoti.2008.07.001"},{"key":"S0962492911000031_ref082","doi-asserted-by":"publisher","DOI":"10.1137\/070699433"},{"key":"S0962492911000031_ref080","volume-title":"The Analysis of Linear Partial Differential Operators III","author":"H\u00f6rmander","year":"1985"},{"key":"S0962492911000031_ref079","doi-asserted-by":"publisher","DOI":"10.1063\/1.1522712"},{"key":"S0962492911000031_ref128","doi-asserted-by":"publisher","DOI":"10.4171\/RMI\/143"},{"key":"S0962492911000031_ref076","first-page":"2923","article-title":"A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations","volume":"91","author":"Herman","year":"1984","journal-title":"J. Chem. Phys."},{"key":"S0962492911000031_ref087","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(03)00124-6"},{"key":"S0962492911000031_ref072","doi-asserted-by":"publisher","DOI":"10.1007\/BF01229201"},{"key":"S0962492911000031_ref070","doi-asserted-by":"publisher","DOI":"10.1109\/5.757251"},{"key":"S0962492911000031_ref067","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.12.004"},{"key":"S0962492911000031_ref066","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100309"},{"key":"S0962492911000031_ref060","volume-title":"Wave Propagation and Time Reversal in Randomly Layered Media, Vol. 56 of Stochastic Modelling and Applied Probability","author":"Fouque","year":"2007"},{"key":"S0962492911000031_ref114","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-0436-2"},{"key":"S0962492911000031_ref059","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511626357"},{"key":"S0962492911000031_ref058","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.102476599"},{"key":"S0962492911000031_ref071","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-98-04164-1"},{"key":"S0962492911000031_ref106","doi-asserted-by":"publisher","DOI":"10.1137\/090774185"},{"key":"S0962492911000031_ref054","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5"},{"key":"S0962492911000031_ref152","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(82)90016-6"},{"key":"S0962492911000031_ref120","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20087"},{"key":"S0962492911000031_ref039","first-page":"347","article-title":"Reflection in a level set framework for geometric optics","volume":"5","author":"Cheng","year":"2004","journal-title":"CMES Comput. Model. Eng. Sci."},{"key":"S0962492911000031_ref001","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1995.1098"},{"key":"S0962492911000031_ref146","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90002-2"},{"key":"S0962492911000031_ref174","doi-asserted-by":"publisher","DOI":"10.1063\/1.459170"},{"key":"S0962492911000031_ref040","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2003.v1.n3.a10"},{"key":"S0962492911000031_ref167","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.59.14915"},{"key":"S0962492911000031_ref150","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1999.6345"},{"key":"S0962492911000031_ref081","doi-asserted-by":"publisher","DOI":"10.1137\/060652026"},{"key":"S0962492911000031_ref173","volume-title":"Adiabatic Perturbation Theory in Quantum Dynamics, Vol. 1821 of Lecture Notes in Mathematics","author":"Teufel","year":"2003"},{"key":"S0962492911000031_ref119","first-page":"46","article-title":"Zur Theorie der Energie\u00fcbertragung II","volume":"2","author":"Landau","year":"1932","journal-title":"Physics of the Soviet Union"},{"key":"S0962492911000031_ref145","doi-asserted-by":"publisher","DOI":"10.24033\/asens.1738"},{"key":"S0962492911000031_ref170","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2008.v6.n2.a9"},{"key":"S0962492911000031_ref022","first-page":"163","article-title":"Asymptotic analysis of quantum scattering under mesoscopic scaling","volume":"40","author":"Benedetto","year":"2004","journal-title":"Asymptot. Anal."},{"key":"S0962492911000031_ref030","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-006-0017-5"},{"key":"S0962492911000031_ref046","doi-asserted-by":"publisher","DOI":"10.1007\/BF01393835"},{"key":"S0962492911000031_ref132","doi-asserted-by":"crossref","unstructured":"Liu H. , Runborg O. and Tanushev N. (2011), Error estimates for Gaussian beam superpositions. Submitted.","DOI":"10.1090\/S0025-5718-2012-02656-1"},{"key":"S0962492911000031_ref036","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511529399"},{"key":"S0962492911000031_ref129","doi-asserted-by":"publisher","DOI":"10.1137\/090756909"},{"key":"S0962492911000031_ref075","doi-asserted-by":"publisher","DOI":"10.1021\/ar040196y"},{"key":"S0962492911000031_ref011","doi-asserted-by":"publisher","DOI":"10.3934\/krm.2010.3.529"},{"key":"S0962492911000031_ref153","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.06.043"},{"key":"S0962492911000031_ref044","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(81)90052-8"},{"key":"S0962492911000031_ref099","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2008.02.029"},{"key":"S0962492911000031_ref093","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2005.v3.n3.a2"},{"key":"S0962492911000031_ref149","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(90)90228-S"},{"key":"S0962492911000031_ref176","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.06.026"},{"key":"S0962492911000031_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-2125(99)00018-9"},{"key":"S0962492911000031_ref019","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0312(199911)52:11<1443::AID-CPA3>3.0.CO;2-Y"},{"key":"S0962492911000031_ref178","doi-asserted-by":"publisher","DOI":"10.1137\/090758295"},{"key":"S0962492911000031_ref029","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(01)01972-3"},{"key":"S0962492911000031_ref147","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7080"},{"key":"S0962492911000031_ref051","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-007-0119-5"},{"key":"S0962492911000031_ref053","first-page":"181","volume-title":"Acta Numerica","volume":"12","author":"Engquist","year":"2003"},{"key":"S0962492911000031_ref154","first-page":"206","volume-title":"Studies in Partial Differential Equations, Vol. 23 of MAA Stud. Math.","author":"Ralston","year":"1982"},{"key":"S0962492911000031_ref084","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.02.026"},{"key":"S0962492911000031_ref172","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500020606"},{"key":"S0962492911000031_ref038","doi-asserted-by":"publisher","DOI":"10.1137\/0723019"},{"key":"S0962492911000031_ref130","doi-asserted-by":"crossref","first-page":"517","DOI":"10.4310\/MAA.2002.v9.n4.a3","article-title":"Semi-classical limit of the nonlinear Schr\u00f6dinger\u2013 Poisson equation with sub-critical initial data","volume":"9","author":"Liu","year":"2002","journal-title":"Methods Appl. Anal."},{"key":"S0962492911000031_ref016","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827501393253"},{"key":"S0962492911000031_ref090","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2007.06.006"},{"key":"S0962492911000031_ref103","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.04.020"},{"key":"S0962492911000031_ref021","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(02)00907-X"},{"key":"S0962492911000031_ref047","first-page":"419","article-title":"A time- and space-adaptive algorithm for the linear time dependent Schr\u00f6dinger equation","volume":"73","author":"D\u00f6rfler","year":"1998","journal-title":"Numer. Math."},{"key":"S0962492911000031_ref031","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-006-0077-2"},{"key":"S0962492911000031_ref166","volume-title":"The Nonlinear Schr\u00f6dinger Equation: Self-Focusing and Wave Collapse, Vol. 139 of Applied Mathematical Sciences","author":"Sulem","year":"1999"},{"key":"S0962492911000031_ref163","doi-asserted-by":"publisher","DOI":"10.1007\/BF01014347"},{"key":"S0962492911000031_ref065","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/39\/33\/018"},{"key":"S0962492911000031_ref025","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901398040"},{"key":"S0962492911000031_ref164","doi-asserted-by":"publisher","DOI":"10.1007\/s002200100535"},{"key":"S0962492911000031_ref028","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.2000.49.1804"},{"key":"S0962492911000031_ref179","first-page":"45","article-title":"The l1-stability of a Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials","volume":"27","author":"Wen","year":"2009","journal-title":"J. Comput. Math."},{"key":"S0962492911000031_ref037","doi-asserted-by":"publisher","DOI":"10.1137\/0724025"},{"key":"S0962492911000031_ref048","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1999.6287"},{"key":"S0962492911000031_ref018","doi-asserted-by":"publisher","DOI":"10.1063\/1.1421635"},{"key":"S0962492911000031_ref116","first-page":"116","article-title":"Fourth order time-stepping for low dispersion Korteweg\u2013de Vries and nonlinear Schr\u00f6dinger equations","volume":"29","author":"Klein","year":"2008","journal-title":"Electron. Trans. Numer. Anal."},{"key":"S0962492911000031_ref085","doi-asserted-by":"publisher","DOI":"10.1137\/S106482759631041X"},{"key":"S0962492911000031_ref006","volume-title":"Solid State Physics","author":"Ashcroft","year":"1976"},{"key":"S0962492911000031_ref068","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.05.020"},{"key":"S0962492911000031_ref074","doi-asserted-by":"publisher","DOI":"10.1063\/1.442382"},{"key":"S0962492911000031_ref073","first-page":"536","article-title":"Molecular propagation through electron energy level crossings","volume":"111","author":"Hagedorn","year":"1994","journal-title":"Mem. Amer. Math. Soc."},{"key":"S0962492911000031_ref032","doi-asserted-by":"publisher","DOI":"10.1142\/6753"},{"key":"S0962492911000031_ref104","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.11.008"},{"key":"S0962492911000031_ref117","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112086002732"},{"key":"S0962492911000031_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-004-0367-2"},{"key":"S0962492911000031_ref009","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004598015978"},{"key":"S0962492911000031_ref014","doi-asserted-by":"publisher","DOI":"10.1088\/0953-4075\/37\/2\/003"},{"key":"S0962492911000031_ref182","doi-asserted-by":"publisher","DOI":"10.1007\/BF02790171"},{"key":"S0962492911000031_ref062","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-010-9063-3"},{"key":"S0962492911000031_ref126","doi-asserted-by":"publisher","DOI":"10.1007\/BF02179552"},{"key":"S0962492911000031_ref135","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(94)90198-8"},{"key":"S0962492911000031_ref155","volume-title":"Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness","author":"Reed","year":"1975"},{"key":"S0962492911000031_ref077","doi-asserted-by":"publisher","DOI":"10.1190\/1.1442788"},{"key":"S0962492911000031_ref041","volume-title":"Methods of Mathematical Physics","volume":"2","author":"Courant","year":"1962"},{"key":"S0962492911000031_ref118","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2008.11.016"},{"key":"S0962492911000031_ref023","volume-title":"Asymptotic Analysis for Periodic Structures","volume":"5","author":"Bensoussan","year":"1978"},{"key":"S0962492911000031_ref027","doi-asserted-by":"publisher","DOI":"10.1016\/S0294-1449(97)89298-0"},{"key":"S0962492911000031_ref061","doi-asserted-by":"publisher","DOI":"10.1007\/BF01209475"},{"key":"S0962492911000031_ref033","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202507002376"},{"key":"S0962492911000031_ref186","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.05.008"},{"key":"S0962492911000031_ref043","doi-asserted-by":"publisher","DOI":"10.1137\/070700711"},{"key":"S0962492911000031_ref024","doi-asserted-by":"publisher","DOI":"10.1007\/BF01339455"},{"key":"S0962492911000031_ref049","volume-title":"Fourier Integral Operators, Vol. 130 of Progress in Mathematics","author":"Duistermaat","year":"1996"},{"key":"S0962492911000031_ref157","doi-asserted-by":"publisher","DOI":"10.1142\/S0129055X1000417X"},{"key":"S0962492911000031_ref035","volume-title":"Semilinear Schr\u00f6dinger Equations, Vol. 10 of Courant Lecture Notes in Mathematics","author":"Cazenave","year":"2003"},{"key":"S0962492911000031_ref005","doi-asserted-by":"publisher","DOI":"10.1137\/S1540345902419616"},{"key":"S0962492911000031_ref092","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2003.v1.n3.a9"},{"key":"S0962492911000031_ref069","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202503003082"},{"key":"S0962492911000031_ref020","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6530"},{"key":"S0962492911000031_ref160","volume-title":"Geometric Phases in Physics, Vol. 5 of Advanced Series in Mathematical Physics","author":"Shapere","year":"1989"},{"key":"S0962492911000031_ref094","doi-asserted-by":"publisher","DOI":"10.1137\/050631343"},{"key":"S0962492911000031_ref004","doi-asserted-by":"crossref","unstructured":"Ariel G. , Engquist B. , Tanushev N. and Tsai R. (2011), Gaussian beam decomposition of high frequency wave fields using expectation-maximization. Submitted.","DOI":"10.1016\/j.jcp.2010.12.018"},{"key":"S0962492911000031_ref177","doi-asserted-by":"publisher","DOI":"10.4208\/jcm.2009.27.4.014"},{"key":"S0962492911000031_ref012","doi-asserted-by":"publisher","DOI":"10.1137\/030601211"},{"key":"S0962492911000031_ref055","unstructured":"Faou E. and Grebert B. (2010), Resonances in long time integration of semi linear Hamiltonian PDEs. Preprint available at: www.irisa.fr\/ipso\/perso\/faou\/."},{"key":"S0962492911000031_ref136","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202593000072"},{"key":"S0962492911000031_ref008","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2004.v2.n3.a9"},{"key":"S0962492911000031_ref137","doi-asserted-by":"publisher","DOI":"10.1007\/s100920050029"},{"key":"S0962492911000031_ref026","doi-asserted-by":"crossref","first-page":"973","DOI":"10.4310\/CMS.2009.v7.n4.a9","article-title":"Gaussian beams summation for the wave equation in a convex domain","volume":"7","author":"Bougacha","year":"2009","journal-title":"Comm. Math. Sci."},{"key":"S0962492911000031_ref183","doi-asserted-by":"publisher","DOI":"10.1063\/1.1851990"},{"key":"S0962492911000031_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9991(03)00102-5"},{"key":"S0962492911000031_ref156","volume-title":"Methods of Modern Mathematical Physics IV: Analysis of Operators","author":"Reed","year":"1976"},{"key":"S0962492911000031_ref042","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1983-0690039-8"},{"key":"S0962492911000031_ref050","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2006.11.024"},{"key":"S0962492911000031_ref165","volume-title":"The Wadsworth and Brooks\/Cole Mathematics Series","author":"Strikwerda","year":"1989"},{"key":"S0962492911000031_ref113","first-page":"3","article-title":"The Herman\u2013Kluk approximation: Derivation and semiclassical corrections","volume":"322","author":"Kay","year":"2006","journal-title":"J. Chem. Phys."},{"key":"S0962492911000031_ref015","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2001.6956"},{"key":"S0962492911000031_ref064","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7085"},{"key":"S0962492911000031_ref034","doi-asserted-by":"publisher","DOI":"10.1023\/B:JOSS.0000044070.34410.17"},{"key":"S0962492911000031_ref107","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2008.v6.n4.a10"},{"key":"S0962492911000031_ref127","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.70.016502"},{"key":"S0962492911000031_ref168","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-008-0681-4"},{"key":"S0962492911000031_ref110","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.01.025"},{"key":"S0962492911000031_ref108","doi-asserted-by":"crossref","first-page":"261","DOI":"10.4208\/jcm.2009.10-m1005","article-title":"A numerical study of the Gaussian beam methods for Schr\u00f6dinger\u2013Poisson equations","volume":"28","author":"Jin","year":"2010","journal-title":"J. Comput. Math."}],"container-title":["Acta Numerica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0962492911000031","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T14:20:49Z","timestamp":1686061249000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0962492911000031\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,28]]},"references-count":189,"alternative-id":["S0962492911000031"],"URL":"https:\/\/doi.org\/10.1017\/s0962492911000031","relation":{},"ISSN":["0962-4929","1474-0508"],"issn-type":[{"value":"0962-4929","type":"print"},{"value":"1474-0508","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,4,28]]}}}