{"id":"https://openalex.org/W2159001728","doi":"https://doi.org/10.1007/s11080-006-8222-0","title":"Energy Evolution in Time-Dependent Harmonic Oscillator","display_name":"Energy Evolution in Time-Dependent Harmonic Oscillator","publication_year":2006,"publication_date":"2006-05-19","ids":{"openalex":"https://openalex.org/W2159001728","doi":"https://doi.org/10.1007/s11080-006-8222-0","mag":"2159001728"},"language":"en","primary_location":{"id":"doi:10.1007/s11080-006-8222-0","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s11080-006-8222-0","pdf_url":null,"source":{"id":"https://openalex.org/S27332288","display_name":"Open Systems & Information Dynamics","issn_l":"1230-1612","issn":["1230-1612","1573-1324","1793-7191"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Open Systems & Information Dynamics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/nlin/0608004","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Marko Robnik","orcid":null},"institutions":[{"id":"https://openalex.org/I37696226","display_name":"University of Maribor","ror":"https://ror.org/01d5jce07","country_code":"SI","type":"education","lineage":["https://openalex.org/I37696226"]},{"id":"https://openalex.org/I4210104143","display_name":"Advanced Telecommunications Research Institute International","ror":"https://ror.org/01pe1d703","country_code":"JP","type":"facility","lineage":["https://openalex.org/I4210104143"]}],"countries":["JP","SI"],"is_corresponding":false,"raw_author_name":"Marko Robnik","raw_affiliation_strings":["ATR Advanced Telecommunications Research Institute International, 2-2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0288, Japan","CAMTP \u2014 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ATR Advanced Telecommunications Research Institute International, 2-2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0288, Japan","institution_ids":["https://openalex.org/I4210104143"]},{"raw_affiliation_string":"CAMTP \u2014 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia","institution_ids":["https://openalex.org/I37696226"]}]},{"author_position":"last","author":{"id":null,"display_name":"Valery G. Romanovski","orcid":null},"institutions":[{"id":"https://openalex.org/I37696226","display_name":"University of Maribor","ror":"https://ror.org/01d5jce07","country_code":"SI","type":"education","lineage":["https://openalex.org/I37696226"]}],"countries":["SI"],"is_corresponding":false,"raw_author_name":"Valery G. Romanovski","raw_affiliation_strings":["CAMTP \u2014 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CAMTP \u2014 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia","institution_ids":["https://openalex.org/I37696226"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.7757,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.74007855,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"13","issue":"02","first_page":"197","last_page":"222"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11262","display_name":"Quantum Mechanics and Non-Hermitian Physics","score":0.9700999855995178,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11262","display_name":"Quantum Mechanics and Non-Hermitian Physics","score":0.9700999855995178,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.009499999694526196,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11191","display_name":"Mathematical functions and polynomials","score":0.002300000051036477,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/adiabatic-process","display_name":"Adiabatic process","score":0.6481999754905701},{"id":"https://openalex.org/keywords/harmonic-oscillator","display_name":"Harmonic oscillator","score":0.583899974822998},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5407000184059143},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5149000287055969},{"id":"https://openalex.org/keywords/moment","display_name":"Moment (physics)","score":0.5070000290870667},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5067999958992004},{"id":"https://openalex.org/keywords/ideal","display_name":"Ideal (ethics)","score":0.4959999918937683},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.48969998955726624},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.48190000653266907}],"concepts":[{"id":"https://openalex.org/C109663097","wikidata":"https://www.wikidata.org/wiki/Q182453","display_name":"Adiabatic process","level":2,"score":0.6481999754905701},{"id":"https://openalex.org/C40934718","wikidata":"https://www.wikidata.org/wiki/Q190070","display_name":"Harmonic oscillator","level":2,"score":0.583899974822998},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5407000184059143},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5307999849319458},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5149000287055969},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.5070000290870667},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5067999958992004},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.4959999918937683},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.48969998955726624},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.48190000653266907},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.42399999499320984},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4115999937057495},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.4016999900341034},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3840000033378601},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.37619999051094055},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3425999879837036},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.33649998903274536},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.33379998803138733},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.3334999978542328},{"id":"https://openalex.org/C106553664","wikidata":"https://www.wikidata.org/wiki/Q1156744","display_name":"Time evolution","level":2,"score":0.326200008392334},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.3147999942302704},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.31220000982284546},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.3095000088214874},{"id":"https://openalex.org/C186603090","wikidata":"https://www.wikidata.org/wiki/Q55505926","display_name":"Distribution function","level":2,"score":0.3059999942779541},{"id":"https://openalex.org/C61062188","wikidata":"https://www.wikidata.org/wiki/Q835065","display_name":"Second moment of area","level":2,"score":0.2989000082015991},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.29760000109672546},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.2856000065803528},{"id":"https://openalex.org/C4285052","wikidata":"https://www.wikidata.org/wiki/Q3406268","display_name":"Moment problem","level":3,"score":0.2816999852657318},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.2599000036716461},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C101683677","wikidata":"https://www.wikidata.org/wiki/Q1077769","display_name":"Microcanonical ensemble","level":4,"score":0.2549999952316284},{"id":"https://openalex.org/C40560622","wikidata":"https://www.wikidata.org/wiki/Q356895","display_name":"Adiabatic theorem","level":3,"score":0.2535000145435333}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s11080-006-8222-0","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s11080-006-8222-0","pdf_url":null,"source":{"id":"https://openalex.org/S27332288","display_name":"Open Systems & Information Dynamics","issn_l":"1230-1612","issn":["1230-1612","1573-1324","1793-7191"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Open Systems & Information Dynamics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:nlin/0608004","is_oa":true,"landing_page_url":"http://arxiv.org/abs/nlin/0608004","pdf_url":"https://arxiv.org/pdf/nlin/0608004","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:nlin/0608004","is_oa":true,"landing_page_url":"http://arxiv.org/abs/nlin/0608004","pdf_url":"https://arxiv.org/pdf/nlin/0608004","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1545237387","https://openalex.org/W1965240898","https://openalex.org/W1979762763","https://openalex.org/W1984307133","https://openalex.org/W1990658140","https://openalex.org/W2005712362","https://openalex.org/W2019792577","https://openalex.org/W2044385102","https://openalex.org/W2045077849","https://openalex.org/W2048641938","https://openalex.org/W2058798349","https://openalex.org/W2063989121","https://openalex.org/W2069268781","https://openalex.org/W2077118202","https://openalex.org/W2093016738","https://openalex.org/W2094863020","https://openalex.org/W2133292567","https://openalex.org/W2158287655","https://openalex.org/W2185979903"],"related_works":[],"abstract_inverted_index":{"The":[0],"theory":[1],"of":[2,19,55,59,78,118,138,189],"adiabatic":[3],"invariants":[4],"has":[5],"a":[6],"long":[7],"history,":[8],"and":[9,13,22,73,111,142,154,170,198],"very":[10],"important":[11],"implications":[12],"applications":[14],"in":[15,49,144],"many":[16],"different":[17],"branches":[18],"physics,":[20],"classically":[21],"quantally,":[23],"but":[24],"is":[25,88,143,159,188,218],"rarely":[26],"founded":[27],"on":[28,134],"rigorous":[29],"results.":[30],"Here":[31],"we":[32,165],"treat":[33],"the":[34,53,76,93,100,123,139,155,174,200,212],"general":[35,175],"time-dependent":[36],"one-dimensional":[37],"harmonic":[38],"oscillator,":[39],"whose":[40],"Newton":[41],"equation":[42],"[Formula:":[43,103,108,151,167,191,214],"see":[44,104,109,152,168,192,215],"text]":[45,193,216],"cannot":[46],"be":[47,221],"solved":[48],"general.":[50],"We":[51,183],"follow":[52],"time-evolution":[54],"an":[56],"initial":[57],"ensemble":[58],"phase":[60],"points":[61],"with":[62],"sharply":[63],"defined":[64],"energy":[65,79],"E":[66,80],"0":[67,72],"at":[68],"time":[69,83],"t":[70,84],"=":[71,85],"calculate":[74,166],"rigorously":[75],"distribution":[77,129],"1":[81],"after":[82],"T,":[86],"which":[87],"fully":[89],"(all":[90,194],"moments,":[91],"including":[92,199],"variance":[94,156],"\u03bc":[95,119,157,171,205,223],"2":[96,120,158,172],")":[97],"determined":[98],"by":[99],"first":[101],"moment":[102],"text].":[105],"For":[106],"example,":[107],"text],":[110,153,169],"all":[112,181],"higher":[113],"even":[114],"moments":[115],"are":[116,203],"powers":[117],",":[121,209],"whilst":[122,161,210],"odd":[124],"ones":[125],"vanish":[126],"identically.":[127],"This":[128],"function":[130,140],"does":[131],"not":[132],"depend":[133],"any":[135],"further":[136],"details":[137],"\u03c9(t)":[141,187],"this":[145],"sense":[146],"universal.":[147],"In":[148],"ideal":[149],"adiabaticity":[150],"zero,":[160],"for":[162,173,211],"finite":[163],"T":[164,207],"case":[176],"using":[177],"exact":[178],"WKB-theory":[179],"to":[180,197,220],"orders.":[182],"prove":[184],"that":[185],"if":[186],"class":[190,213],"derivatives":[195],"up":[196],"order":[201],"m":[202],"continuous)":[204],"\u221d":[206,224],"-(m+1)":[208],"it":[217],"known":[219],"exponential":[222],"exp":[225],"(-\u03b1T).":[226]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2016-06-24T00:00:00"}