{"id":"https://openalex.org/W2074306248","doi":"https://doi.org/10.1007/s002110050406","title":"Numerical approximation of quadratic observables of Schr\u00f6dinger-type equations in the semi-classical limit","display_name":"Numerical approximation of quadratic observables of Schr\u00f6dinger-type equations in the semi-classical limit","publication_year":1999,"publication_date":"1999-02-01","ids":{"openalex":"https://openalex.org/W2074306248","doi":"https://doi.org/10.1007/s002110050406","mag":"2074306248"},"language":"en","primary_location":{"id":"doi:10.1007/s002110050406","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s002110050406","pdf_url":null,"source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110260067","display_name":"Peter A. Markowich","orcid":"https://orcid.org/0000-0002-3704-1821"},"institutions":[{"id":"https://openalex.org/I4577782","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40","country_code":"DE","type":"education","lineage":["https://openalex.org/I4577782"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Peter A. Markowich","raw_affiliation_strings":["Fachbereich Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, D-10623 Berlin, Germany; e-mail: markowic@math.tu-berlin.de\n, , , , , , DE","Fachbereich Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, D-10623 Berlin, Germany; e-mail: markowic@math.tu-berlin.de , DE"],"affiliations":[{"raw_affiliation_string":"Fachbereich Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, D-10623 Berlin, Germany; e-mail: markowic@math.tu-berlin.de\n, , , , , , DE","institution_ids":["https://openalex.org/I4577782"]},{"raw_affiliation_string":"Fachbereich Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, D-10623 Berlin, Germany; e-mail: markowic@math.tu-berlin.de , DE","institution_ids":["https://openalex.org/I4577782"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056631958","display_name":"Paola Pietra","orcid":"https://orcid.org/0000-0002-5967-6983"},"institutions":[{"id":"https://openalex.org/I4210129724","display_name":"Istituto di Genetica Molecolare","ror":"https://ror.org/03qpd8w66","country_code":"IT","type":"facility","lineage":["https://openalex.org/I4210129724","https://openalex.org/I4210155236"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Paola Pietra","raw_affiliation_strings":["Istituto di Analisi Numerica del C. N. R., Via Abbiategrasso 209, I-27100 Pavia, Italy\n, , , , , , IT","Istituto di Analisi Numerica del C. N. R., Via Abbiategrasso 209, I-27100 Pavia, Italy#N#, , IT"],"affiliations":[{"raw_affiliation_string":"Istituto di Analisi Numerica del C. N. R., Via Abbiategrasso 209, I-27100 Pavia, Italy\n, , , , , , IT","institution_ids":["https://openalex.org/I4210129724"]},{"raw_affiliation_string":"Istituto di Analisi Numerica del C. N. R., Via Abbiategrasso 209, I-27100 Pavia, Italy#N#, , IT","institution_ids":["https://openalex.org/I4210129724"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5005241309","display_name":"Carsten Pohl","orcid":null},"institutions":[{"id":"https://openalex.org/I4577782","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40","country_code":"DE","type":"education","lineage":["https://openalex.org/I4577782"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Carsten Pohl","raw_affiliation_strings":["Fachbereich Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, D-10623 Berlin, Germany; e-mail: markowic@math.tu-berlin.de\n, , , , , , DE","Fachbereich Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, D-10623 Berlin, Germany; e-mail: markowic@math.tu-berlin.de , DE"],"affiliations":[{"raw_affiliation_string":"Fachbereich Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, D-10623 Berlin, Germany; e-mail: markowic@math.tu-berlin.de\n, , , , , , DE","institution_ids":["https://openalex.org/I4577782"]},{"raw_affiliation_string":"Fachbereich Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, D-10623 Berlin, Germany; e-mail: markowic@math.tu-berlin.de , DE","institution_ids":["https://openalex.org/I4577782"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5110260067"],"corresponding_institution_ids":["https://openalex.org/I4577782"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":null,"fwci":4.5484,"has_fulltext":false,"cited_by_count":106,"citation_normalized_percentile":{"value":0.94952681,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"81","issue":"4","first_page":"595","last_page":"630"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.994700014591217,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10466","display_name":"Meteorological Phenomena and Simulations","score":0.9919000267982483,"subfield":{"id":"https://openalex.org/subfields/1902","display_name":"Atmospheric Science"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8294164538383484},{"id":"https://openalex.org/keywords/observable","display_name":"Observable","score":0.7561882734298706},{"id":"https://openalex.org/keywords/planck-constant","display_name":"Planck constant","score":0.6808215975761414},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.6277849674224854},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.5864007472991943},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5851675868034363},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5418710112571716},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5328667163848877},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4940069317817688},{"id":"https://openalex.org/keywords/planck","display_name":"Planck","score":0.440601110458374},{"id":"https://openalex.org/keywords/schr\u00f6dinger-equation","display_name":"Schr\u00f6dinger equation","score":0.42777764797210693},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38157734274864197},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.16569840908050537},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14620396494865417},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.1239486038684845},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12382158637046814}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8294164538383484},{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.7561882734298706},{"id":"https://openalex.org/C197111296","wikidata":"https://www.wikidata.org/wiki/Q122894","display_name":"Planck constant","level":3,"score":0.6808215975761414},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.6277849674224854},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5864007472991943},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5851675868034363},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5418710112571716},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5328667163848877},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4940069317817688},{"id":"https://openalex.org/C166126730","wikidata":"https://www.wikidata.org/wiki/Q725309","display_name":"Planck","level":2,"score":0.440601110458374},{"id":"https://openalex.org/C63036615","wikidata":"https://www.wikidata.org/wiki/Q165498","display_name":"Schr\u00f6dinger equation","level":2,"score":0.42777764797210693},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38157734274864197},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.16569840908050537},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14620396494865417},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.1239486038684845},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12382158637046814},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s002110050406","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s002110050406","pdf_url":null,"source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W1532154335","https://openalex.org/W1983797262","https://openalex.org/W1989121669","https://openalex.org/W1992147072","https://openalex.org/W2010414883","https://openalex.org/W2063727277","https://openalex.org/W2068731834","https://openalex.org/W2073339836","https://openalex.org/W2080711197","https://openalex.org/W2081907035","https://openalex.org/W2083916485","https://openalex.org/W2088552913","https://openalex.org/W2228489595","https://openalex.org/W2367355694","https://openalex.org/W2482762127","https://openalex.org/W3165898261"],"related_works":["https://openalex.org/W2800394832","https://openalex.org/W4377819710","https://openalex.org/W1960737133","https://openalex.org/W2005258390","https://openalex.org/W4385408225","https://openalex.org/W1676129229","https://openalex.org/W4299726473","https://openalex.org/W2560188568","https://openalex.org/W2027049858","https://openalex.org/W4233396376"],"abstract_inverted_index":null,"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":8},{"year":2013,"cited_by_count":11},{"year":2012,"cited_by_count":4}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}