{"id":"https://openalex.org/W7212768108","doi":"https://doi.org/10.48550/arxiv.2609.12369","title":"A spectral-element method for computing many eigenvalues and their asymptotics of the Schr\u00f6dinger operator with Robin boundary condition","display_name":"A spectral-element method for computing many eigenvalues and their asymptotics of the Schr\u00f6dinger operator with Robin boundary condition","publication_year":2026,"publication_date":"2026-09-11","ids":{"openalex":"https://openalex.org/W7212768108","doi":"https://doi.org/10.48550/arxiv.2609.12369"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.12369","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.12369","pdf_url":"https://arxiv.org/pdf/2609.12369","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.12369","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100759028","display_name":"Weizhu Bao","orcid":"https://orcid.org/0000-0003-3418-9625","observed_orcids":["https://orcid.org/0000-0003-3418-9625"]},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Bao, Weizhu","raw_affiliation_strings":["Department of Mathematics , National University of Singapore , Singapore 119076 , Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , National University of Singapore , Singapore 119076 , Singapore","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025261220","display_name":"Fukeng Huang","orcid":"https://orcid.org/0000-0003-3644-268X","observed_orcids":["https://orcid.org/0000-0003-3644-268X"]},"institutions":[{"id":"https://openalex.org/I4394709157","display_name":"Eastern Institute of Technology, Ningbo","ror":"https://ror.org/036mbz113","country_code":"CN","type":"education","lineage":["https://openalex.org/I4394709157"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Huang, Fukeng","raw_affiliation_strings":["School of Mathematical Sciences , Eastern Institute of Technology , Ningbo , Zhejiang 315200 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Eastern Institute of Technology , Ningbo , Zhejiang 315200 , P. R. China","institution_ids":["https://openalex.org/I4394709157"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9965999722480774,"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":0.9965999722480774,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.000699999975040555,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12727","display_name":"Singular Perturbation, Asymptotics and Difference Schemes","score":0.0003000000142492354,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/spectral-element-method","display_name":"spectral element method","score":0.8349999785423279},{"id":"https://openalex.org/keywords/schrodinger-operators","display_name":"Schr\u00f6dinger operators","score":0.9020000100135803},{"id":"https://openalex.org/keywords/robin-boundary-conditions","display_name":"Robin boundary conditions","score":0.9750000238418579},{"id":"https://openalex.org/keywords/eigenvalue-computation","display_name":"eigenvalue computation","score":0.640999972820282},{"id":"https://openalex.org/keywords/eigenvalue-asymptotics","display_name":"eigenvalue asymptotics","score":0.8159999847412109},{"id":"https://openalex.org/keywords/laplacian","display_name":"Laplacian","score":0.6629999876022339},{"id":"https://openalex.org/keywords/complex-geometries","display_name":"complex geometries","score":0.675000011920929},{"id":"https://openalex.org/keywords/convergence-rate","display_name":"convergence rate","score":0.6639999747276306}],"concepts":[{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.7809000015258789},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.7297999858856201},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.7114999890327454},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7110999822616577},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.6335999965667725},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5074999928474426},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4796999990940094},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.45329999923706055},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.44209998846054077},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.44029998779296875},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.423799991607666},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.3644999861717224},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.3643999993801117},{"id":"https://openalex.org/C62611344","wikidata":"https://www.wikidata.org/wiki/Q1062658","display_name":"Node (physics)","level":2,"score":0.32589998841285706},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.32330000400543213},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.3151000142097473},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.29490000009536743},{"id":"https://openalex.org/C203269682","wikidata":"https://www.wikidata.org/wiki/Q7116898","display_name":"p-Laplacian","level":3,"score":0.2721000015735626},{"id":"https://openalex.org/C2987355758","wikidata":"https://www.wikidata.org/wiki/Q25037251","display_name":"Operator splitting","level":2,"score":0.26980000734329224},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.26420000195503235},{"id":"https://openalex.org/C138268822","wikidata":"https://www.wikidata.org/wiki/Q1051925","display_name":"Resolution (logic)","level":2,"score":0.26330000162124634}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.12369","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.12369","pdf_url":"https://arxiv.org/pdf/2609.12369","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},{"id":"doi:10.48550/arxiv.2609.12369","doi":"https://doi.org/10.48550/arxiv.2609.12369","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.12369","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.12369","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.12369","pdf_url":"https://arxiv.org/pdf/2609.12369","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212768108.pdf","grobid_xml":"https://content.openalex.org/works/W7212768108.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4281642811","https://openalex.org/W3043338902","https://openalex.org/W2117430212","https://openalex.org/W2772259509","https://openalex.org/W2963080509","https://openalex.org/W2981763651","https://openalex.org/W2066209299","https://openalex.org/W7092367447","https://openalex.org/W6980068901","https://openalex.org/W1537822161"],"abstract_inverted_index":{"We":[0],"propose":[1],"spectral":[2],"and":[3,21,55,156],"spectral-element":[4],"methods":[5],"for":[6,102],"accurately":[7,54],"computing":[8,50],"many":[9],"eigenvalues":[10,53,65,96],"of":[11,52,64,74,76,86,126,152,175,179],"the":[12,43,58,62,72,122,127,149,153,172,176,180],"Schr\u00f6dinger":[13,154,181],"operator":[14,129,155],"with":[15,57,66,83,97],"Robin":[16],"boundary":[17],"condition":[18],"on":[19,111,121,161,171],"simple":[20],"complex":[22,104],"geometries,":[23],"respectively.":[24],"Due":[25],"to":[26,35,71],"their":[27,157],"spectral-type":[28],"accuracy":[29,68],"in":[30,49,106,130],"approximating":[31],"those":[32],"eigenfunctions":[33],"corresponding":[34],"high-index":[36],"eigenvalues,":[37,114],"which":[38,132],"are":[39],"usually":[40],"highly":[41],"oscillatory,":[42],"proposed":[44],"approaches":[45],"have":[46],"excellent":[47],"resolution":[48,59],"thousands":[51],"efficiently":[56],"property":[60],"that":[61],"number":[63,73],"reasonable":[67],"is":[69],"proportional":[70],"degrees":[75],"freedom.":[77],"Using":[78],"a":[79,168],"standard":[80],"HPC":[81],"node":[82],"128":[84],"GB":[85],"memory,":[87],"we":[88,115,146,166],"can":[89],"obtain":[90],"numerically":[91],"more":[92],"than":[93],"5,000":[94],"reliable":[95],"relative":[98],"errors":[99],"below":[100],"$10^{-8}$":[101],"different":[103],"domains":[105],"two":[107],"dimensions":[108],"(2D).":[109],"Based":[110,160],"these":[112],"computed":[113],"first":[116],"confirm":[117],"some":[118],"theoretical":[119],"results":[120],"Robin-to-Neumann":[123],"(RtN)":[124],"gaps":[125,151,178],"Laplacian":[128],"2D,":[131],"were":[133],"recently":[134],"studied":[135],"by":[136],"Rudnick":[137],"\\textit{et":[138],"al.}":[139],"[\\textit{Comm.":[140],"Math.":[141],"Phys.}":[142],"388":[143],"(2021)].":[144],"Then":[145],"systematically":[147],"study":[148],"RtN":[150,177],"convergence":[158],"rates.":[159],"our":[162],"extensive":[163],"numerical":[164],"results,":[165],"formulate":[167],"unified":[169],"conjecture":[170],"cumulative":[173],"averages":[174],"operator.":[182]},"counts_by_year":[],"updated_date":"2026-11-05T12:25:35.769452","created_date":"2026-09-15T00:00:00"}