{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:40:49Z","timestamp":1776836449090,"version":"3.51.2"},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"339","license":[{"start":{"date-parts":[[2023,9,23]],"date-time":"2023-09-23T00:00:00Z","timestamp":1695427200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2017-06588"],"award-info":[{"award-number":["RGPIN-2017-06588"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this article, we propose a new numerical method and its analysis to solve eigenvalue problems for self-adjoint Schr\u00f6dinger operators, by combining the Feshbach\u2013Schur perturbation theory with the spectral Fourier discretization. In order to analyze the method, we establish an abstract framework of Feshbach\u2013Schur perturbation theory with minimal regularity assumptions on the potential that is then applied to the setting of the new spectral Fourier discretization method. Finally, we present some numerical results that underline the theoretical findings.<\/p>","DOI":"10.1090\/mcom\/3774","type":"journal-article","created":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T16:16:41Z","timestamp":1663949801000},"page":"217-249","source":"Crossref","is-referenced-by-count":1,"title":["Analysis of the Feshbach\u2013Schur method for the Fourier spectral discretizations of Schr\u00f6dinger operators"],"prefix":"10.1090","volume":"92","author":[{"given":"Genevi\u00e8ve","family":"Dusson","sequence":"first","affiliation":[]},{"given":"Israel","family":"Sigal","sequence":"additional","affiliation":[]},{"given":"Benjamin","family":"Stamm","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,9,23]]},"reference":[{"key":"1","isbn-type":"print","first-page":"641","article-title":"Eigenvalue problems","author":"Babu\u0161ka, I.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0444703659"},{"issue":"186","key":"2","doi-asserted-by":"publisher","first-page":"275","DOI":"10.2307\/2008468","article-title":"Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems","volume":"52","author":"Babu\u0161ka, I.","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492910000012","article-title":"Finite element approximation of eigenvalue problems","volume":"19","author":"Boffi, Daniele","year":"2010","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"D. Brust, Electronic spectra of crystalline germanium and silicon, Phys. Rev. 134 (1964). no. 5A, A1337\u2013A1353.","DOI":"10.1103\/PhysRev.134.A1337"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1093\/imanum\/drw053","article-title":"Two-grid methods for a class of nonlinear elliptic eigenvalue problems","volume":"38","author":"Canc\u00e8s, Eric","year":"2018","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"1-3","key":"6","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1007\/s10915-010-9358-1","article-title":"Numerical analysis of nonlinear eigenvalue problems","volume":"45","author":"Canc\u00e8s, Eric","year":"2010","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"7","unstructured":"E. Canc\u00e8s, R. Coyaud, and L. R. Scott. Van der Waals interactions between two hydrogen atoms: The next orders. arXiv preprint arXiv:2007.04227, 2020."},{"issue":"11","key":"8","doi-asserted-by":"publisher","first-page":"941","DOI":"10.1016\/j.crma.2014.09.014","article-title":"A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schr\u00f6dinger equations","volume":"352","author":"Canc\u00e8s, \u00c9ric","year":"2014","journal-title":"C. R. Math. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/1631-073X","issn-type":"print"},{"key":"9","doi-asserted-by":"publisher","first-page":"446","DOI":"10.1016\/j.jcp.2015.12.012","article-title":"A perturbation-method-based post-processing for the planewave discretization of Kohn-Sham models","volume":"307","author":"Canc\u00e8s, Eric","year":"2016","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"10","doi-asserted-by":"crossref","unstructured":"E. Canc\u00e8s, G. Dusson, Y. Maday, B. Stamm, and M. Vohral\u00edk. Post-processing of the planewave approximation of Schr\u00f6dinger equations. Part I: linear operators. IMA Journal of Numerical Analysis, 09 2020.","DOI":"10.1093\/imanum\/draa044"},{"key":"11","doi-asserted-by":"crossref","unstructured":"E. Canc\u00e8s and L. R. Scott, Van der Waals Interactions Between Two Hydrogen Atoms: The Slater\u2013Kirkwood Method Revisited, SIAM Journal on Mathematical Analysis, 50(1):381\u2013410, 2018.","DOI":"10.1137\/15M1021878"},{"key":"12","series-title":"Classics in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970678.ch1","volume-title":"Spectral approximation of linear operators","volume":"65","author":"Chatelin, Fran\u00e7oise","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9780898719994"},{"key":"13","series-title":"Texts and Monographs in Physics","isbn-type":"print","volume-title":"Schr\\\"{o}dinger operators with application to quantum mechanics and global geometry","author":"Cycon, H. L.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/3540167587"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1137\/06067780X","article-title":"Three-scale finite element discretizations for quantum eigenvalue problems","volume":"46","author":"Dai, Xiaoying","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"2456","DOI":"10.1093\/imanum\/draa052","article-title":"Post-processing of the plane-wave approximation of Schr\u00f6dinger equations. Part II: Kohn-Sham models","volume":"41","author":"Dusson, Genevi\u00e8ve","year":"2021","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"9","key":"16","doi-asserted-by":"publisher","first-page":"2329","DOI":"10.1016\/j.jfa.2008.01.015","article-title":"On the smooth Feshbach-Schur map","volume":"254","author":"Griesemer, M.","year":"2008","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"key":"17","series-title":"Monographs and Studies in Mathematics","isbn-type":"print","volume-title":"Elliptic problems in nonsmooth domains","volume":"24","author":"Grisvard, P.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0273086472"},{"key":"18","series-title":"Universitext","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-55729-3","volume-title":"Mathematical concepts of quantum mechanics","author":"Gustafson, Stephen J.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/3540441603"},{"key":"19","doi-asserted-by":"crossref","unstructured":"M. F. Herbst, A. Levitt, and E. Canc\u00e8s, DFTK: a Julian approach for simulating electrons in solids, JuliaCon Proc. 3 (2021), no. 26, 69.","DOI":"10.21105\/jcon.00069"},{"key":"20","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0741-2","volume-title":"Introduction to spectral theory","volume":"113","author":"Hislop, P. D.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0387945016"},{"key":"21","series-title":"Grundlehren der Mathematischen Wissenschaften, Band 132","volume-title":"Perturbation theory for linear operators","author":"Kato, Tosio","year":"1976","edition":"2"},{"key":"22","doi-asserted-by":"crossref","unstructured":"C. M\u00f8ller and M. S. Plesset, Note on an approximation treatment for many-electron systems, Phys. Rev. 46 (1934), no. 7, 618\u2013622.","DOI":"10.1103\/PhysRev.46.618"},{"issue":"6","key":"23","doi-asserted-by":"publisher","first-page":"4356","DOI":"10.1137\/090756545","article-title":"Convergence analysis of planewave expansion methods for 2D Schr\u00f6dinger operators with discontinuous periodic potentials","volume":"47","author":"Norton, Richard","year":"2010","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"24","unstructured":"M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. 2: Fourier Analysis, Self-Adjointness. Elsevier, 1975."},{"key":"25","unstructured":"M. Reed and B. Simon. Methods of Modern Mathematical Physics. Vol. 4. Operator Analysis. Academic Press, New York, 1979."},{"key":"26","volume-title":"Perturbation theory of eigenvalue problems","author":"Rellich, Franz","year":"1969"},{"issue":"233","key":"27","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1090\/S0025-5718-99-01180-1","article-title":"A two-grid discretization scheme for eigenvalue problems","volume":"70","author":"Xu, Jinchao","year":"2001","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03774-1\/mcom3774_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03774-1\/S0025-5718-2022-03774-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:46:03Z","timestamp":1776833163000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03774-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,23]]},"references-count":27,"journal-issue":{"issue":"339","published-print":{"date-parts":[[2023,1]]}},"alternative-id":["S0025-5718-2022-03774-1"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3774","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":[[2022,9,23]]}}}