{"id":"https://openalex.org/W2499023944","doi":"https://doi.org/10.1137/16m1088387","title":"On Spectral Properties of the Bloch--Torrey Operator in Two Dimensions","display_name":"On Spectral Properties of the Bloch--Torrey Operator in Two Dimensions","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2499023944","doi":"https://doi.org/10.1137/16m1088387","mag":"2499023944"},"language":"en","primary_location":{"id":"doi:10.1137/16m1088387","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1088387","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1608.01925","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5077899527","display_name":"Denis S. Grebenkov","orcid":"https://orcid.org/0000-0002-6273-9164"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Denis S. Grebenkov","raw_affiliation_strings":["Laboratoire de Physique de la Mati\u00e8re Condens\u00e9e, CNRS-Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire de Physique de la Mati\u00e8re Condens\u00e9e, CNRS-Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau, France","institution_ids":["https://openalex.org/I142476485","https://openalex.org/I1294671590"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5076342883","display_name":"Bernard Helffer","orcid":null},"institutions":[{"id":"https://openalex.org/I102197404","display_name":"Universit\u00e9 Paris-Sud","ror":"https://ror.org/028rypz17","country_code":"FR","type":"education","lineage":["https://openalex.org/I102197404"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I277688954","display_name":"Universit\u00e9 Paris-Saclay","ror":"https://ror.org/03xjwb503","country_code":"FR","type":"education","lineage":["https://openalex.org/I277688954"]},{"id":"https://openalex.org/I4210153365","display_name":"Laboratoire de Math\u00e9matiques Jean Leray","ror":"https://ror.org/04g1hjn96","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I4210141950","https://openalex.org/I4210153365","https://openalex.org/I97188460"]},{"id":"https://openalex.org/I4210155549","display_name":"Laboratoire de Math\u00e9matiques Blaise Pascal","ror":"https://ror.org/05sd5r855","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I198244214","https://openalex.org/I4210141950","https://openalex.org/I4210155549"]},{"id":"https://openalex.org/I97188460","display_name":"Nantes Universit\u00e9","ror":"https://ror.org/03gnr7b55","country_code":"FR","type":"education","lineage":["https://openalex.org/I97188460"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Bernard Helffer","raw_affiliation_strings":["Laboratoire de Math\u00e9matiques Jean Leray, Universit\u00e9 de Nantes 2 rue de la Houssini\u00e8re 44322 Nantes, France, and","Laboratoire de Math\u00e9matiques, Universit\u00e9 Paris-Sud, CNRS, Univ. Paris Saclay, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire de Math\u00e9matiques Jean Leray, Universit\u00e9 de Nantes 2 rue de la Houssini\u00e8re 44322 Nantes, France, and","institution_ids":["https://openalex.org/I4210153365","https://openalex.org/I97188460"]},{"raw_affiliation_string":"Laboratoire de Math\u00e9matiques, Universit\u00e9 Paris-Sud, CNRS, Univ. Paris Saclay, France","institution_ids":["https://openalex.org/I277688954","https://openalex.org/I102197404","https://openalex.org/I4210155549","https://openalex.org/I1294671590"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.1063,"has_fulltext":false,"cited_by_count":28,"citation_normalized_percentile":{"value":0.96263281,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"50","issue":"1","first_page":"622","last_page":"676"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9998000264167786,"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9988999962806702,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9987999796867371,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8743935823440552},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.6702367663383484},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5866588354110718},{"id":"https://openalex.org/keywords/spectrum","display_name":"Spectrum (functional analysis)","score":0.4750364124774933},{"id":"https://openalex.org/keywords/spectral-properties","display_name":"Spectral properties","score":0.4308076500892639},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3363913297653198},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10729333758354187},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08342918753623962}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8743935823440552},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.6702367663383484},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5866588354110718},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.4750364124774933},{"id":"https://openalex.org/C2985906921","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Spectral properties","level":2,"score":0.4308076500892639},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3363913297653198},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10729333758354187},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08342918753623962},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C44870925","wikidata":"https://www.wikidata.org/wiki/Q37547","display_name":"Astrophysics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/16m1088387","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1088387","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1608.01925","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1608.01925","pdf_url":"https://arxiv.org/pdf/1608.01925","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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:1608.01925","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1608.01925","pdf_url":"https://arxiv.org/pdf/1608.01925","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":[{"id":"https://metadata.un.org/sdg/11","display_name":"Sustainable cities and communities","score":0.5199999809265137}],"awards":[{"id":"https://openalex.org/G1674858504","display_name":null,"funder_award_id":"ANR-13-JSV5-0006-01","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G3912487373","display_name":"INferring Anomalous Diffusions in LIving Cells from single particle tracking experiments","funder_award_id":"ANR-13-JSV5-0006","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W147647273","https://openalex.org/W628388122","https://openalex.org/W1552223237","https://openalex.org/W1574334325","https://openalex.org/W1575147392","https://openalex.org/W1586623601","https://openalex.org/W1592958377","https://openalex.org/W1982451610","https://openalex.org/W1986269278","https://openalex.org/W1994808547","https://openalex.org/W2001992126","https://openalex.org/W2005485284","https://openalex.org/W2013289102","https://openalex.org/W2025794242","https://openalex.org/W2027803689","https://openalex.org/W2032958005","https://openalex.org/W2047319386","https://openalex.org/W2057357669","https://openalex.org/W2067028551","https://openalex.org/W2078563308","https://openalex.org/W2084844985","https://openalex.org/W2104816824","https://openalex.org/W2130630106","https://openalex.org/W2135485000","https://openalex.org/W2231753645","https://openalex.org/W2253638727","https://openalex.org/W2305344133","https://openalex.org/W2502855060","https://openalex.org/W2553415643","https://openalex.org/W2585382012","https://openalex.org/W2599747786","https://openalex.org/W2602458368","https://openalex.org/W2963645054","https://openalex.org/W2963749139","https://openalex.org/W2964219608","https://openalex.org/W3100853037","https://openalex.org/W4252176847","https://openalex.org/W4297215405","https://openalex.org/W4300657400"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1041577264","https://openalex.org/W2926402417","https://openalex.org/W2372100074","https://openalex.org/W2018402387","https://openalex.org/W1640766871","https://openalex.org/W2992850321","https://openalex.org/W2724424878","https://openalex.org/W2316101023","https://openalex.org/W2883715623"],"abstract_inverted_index":{"We":[0,38],"investigate":[1],"a":[2,10,40,107,119,131],"two-dimensional":[3],"Schr\u00f6dinger":[4],"operator,":[5,101],"$-h^2":[6],"\\Delta":[7],"+iV(x)$,":[8],"with":[9,29],"purely":[11],"complex":[12],"potential":[13],"$iV(x)$.":[14],"A":[15],"rigorous":[16],"definition":[17],"of":[18,98,122,127,138,152],"this":[19,128],"non-self-adjoint":[20],"operator":[21,129],"is":[22,57,110,116,145],"provided":[23],"for":[24,67,105,130],"bounded":[25,69],"and":[26,36,64,70,125,133],"unbounded":[27,71],"domains":[28],"common":[30],"boundary":[31,143],"conditions":[32],"(Dirichlet,":[33],"Neumann,":[34],"Robin,":[35],"transmission).":[37],"propose":[39],"general":[41,89],"perturbative":[42],"approach":[43],"to":[44,75,80],"construct":[45],"its":[46],"quasi":[47],"modes":[48],"in":[49,83,149],"the":[50,77,84,95,99,123,141,150],"semiclassical":[51],"limit.":[52,87],"An":[53],"alternative":[54],"WKB":[55],"construction":[56],"also":[58],"discussed.":[59,158],"These":[60],"approaches":[61],"are":[62,91,157],"local":[63],"thus":[65],"valid":[66],"both":[68],"domains,":[72],"allowing":[73],"one":[74],"compute":[76],"approximate":[78],"eigenvalues":[79,124],"any":[81],"order":[82],"small":[85],"$h$":[86],"The":[88,136],"results":[90],"further":[92],"illustrated":[93],"on":[94],"particular":[96],"case":[97],"Bloch--Torrey":[100],"$-h^2\\Delta":[102],"+":[103],"ix_1$,":[104],"which":[106],"four-term":[108],"asymptotics":[109],"explicitly":[111],"computed.":[112],"Its":[113],"high":[114],"accuracy":[115],"confirmed":[117],"by":[118],"numerical":[120],"computation":[121],"eigenfunctions":[126,139],"disk":[132],"circular":[134],"annuli.":[135],"localization":[137],"near":[140],"specific":[142],"points":[144],"revealed.":[146],"Some":[147],"applications":[148],"field":[151],"diffusion":[153],"nuclear":[154],"magnetic":[155],"resonance":[156]},"counts_by_year":[{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":4}],"updated_date":"2026-06-20T22:02:38.213706","created_date":"2025-10-10T00:00:00"}
