{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:20:59Z","timestamp":1776842459700,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"267","license":[{"start":{"date-parts":[[2010,1,22]],"date-time":"2010-01-22T00:00:00Z","timestamp":1264118400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We present a convergence analysis for exponential splitting methods applied to linear evolution equations. Our main result states that the classical order of the splitting method is retained in a setting of unbounded operators, without requiring any additional order condition. This is achieved by basing the analysis on the abstract framework of (semi)groups. The convergence analysis also includes generalizations to splittings consisting of more than two operators, and to variable time steps. We conclude by illustrating that the abstract results are applicable in the context of the Schr\u00f6dinger equation with an external magnetic field or with an unbounded potential.<\/p>","DOI":"10.1090\/s0025-5718-09-02213-3","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1485-1496","source":"Crossref","is-referenced-by-count":59,"title":["Exponential splitting for unbounded operators"],"prefix":"10.1090","volume":"78","author":[{"given":"Eskil","family":"Hansen","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Ostermann","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,22]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.apnum.2004.10.005","article-title":"On the necessity of negative coefficients for operator splitting schemes of order higher than two","volume":"54","author":"Blanes, Sergio","year":"2005","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1007\/BF01396704","article-title":"The method of fractional steps for conservation laws","volume":"34","author":"Crandall, Michael","year":"1980","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"3","isbn-type":"print","volume-title":"Mathematical analysis and numerical methods for science and technology. Vol. 1","author":"Dautray, Robert","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540502076"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/S0021-7824(01)01216-8","article-title":"Strang\u2019s formula for holomorphic semi-groups","volume":"81","author":"Descombes, St\u00e9phane","year":"2002","journal-title":"J. Math. Pures Appl. (9)","ISSN":"https:\/\/id.crossref.org\/issn\/0021-7824","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1051\/m2an\/1996300303431","article-title":"Commutateurs de certains semi-groupes holomorphes et applications aux directions altern\u00e9es","volume":"30","author":"Dia, Boun Oumar","year":"1996","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"6","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","volume-title":"The Cauchy problem","volume":"18","author":"Fattorini, Hector O.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0201135175"},{"key":"7","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Geometric numerical integration","volume":"31","author":"Hairer, Ernst","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/3540306633","edition":"2"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1007\/s00211-007-0129-3","article-title":"Dimension splitting for evolution equations","volume":"108","author":"Hansen, Eskil","year":"2008","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"187","key":"9","doi-asserted-by":"publisher","first-page":"81","DOI":"10.2307\/2008350","article-title":"Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems","volume":"53","author":"Hundsdorfer, W. H.","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"10","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1023\/A:1022396519656","article-title":"Error bounds for exponential operator splittings","volume":"40","author":"Jahnke, Tobias","year":"2000","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/s002110050291","article-title":"An operator splitting method for nonlinear convection-diffusion equations","volume":"77","author":"Karlsen, Kenneth Hvistendahl","year":"1997","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"264","key":"12","doi-asserted-by":"publisher","first-page":"2141","DOI":"10.1090\/S0025-5718-08-02101-7","article-title":"On splitting methods for Schr\u00f6dinger-Poisson and cubic nonlinear Schr\u00f6dinger equations","volume":"77","author":"Lubich, Christian","year":"2008","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"13","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1017\/S0962492902000053","article-title":"Splitting methods","volume":"11","author":"McLachlan, Robert I.","year":"2002","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"key":"14","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5561-1","volume-title":"Semigroups of linear operators and applications to partial differential equations","volume":"44","author":"Pazy, A.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0387908455"},{"key":"15","volume-title":"Methods of modern mathematical physics. I. Functional analysis","author":"Reed, Michael","year":"1972"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1137\/0731003","article-title":"On the accuracy of fractional step methods for conservation laws in two dimensions","volume":"31","author":"Teng, Zhen Huan","year":"1994","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"17","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/b13355","volume-title":"Adiabatic perturbation theory in quantum dynamics","volume":"1821","author":"Teufel, Stefan","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/3540407235"},{"issue":"4","key":"18","doi-asserted-by":"publisher","first-page":"2022","DOI":"10.1137\/060674636","article-title":"High-order exponential operator splitting methods for time-dependent Schr\u00f6dinger equations","volume":"46","author":"Thalhammer, Mechthild","year":"2008","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02213-3\/S0025-5718-09-02213-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02213-3\/S0025-5718-09-02213-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:08:21Z","timestamp":1776787701000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02213-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,22]]},"references-count":18,"journal-issue":{"issue":"267","published-print":{"date-parts":[[2009,7]]}},"alternative-id":["S0025-5718-09-02213-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02213-3","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2009,1,22]]}}}