[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81893-en":3,"doc-seo-81893-105":31,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81893,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Guaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods with Application to Schrödinger Operators","Spectral Galerkin methods provide highly accurate eigenvalue approximations, but direct rigorous lower bounds from the spectral discretisation had been missing because classical Kato and Weinstein–Temple enclosures depend on prior information on a neighbouring eigenvalue. This work extends a projection-based guaranteed lower-bound framework, originally developed for finite elements, to conforming spectral Galerkin spaces. For trial spaces spanned by exact eigenfunctions, the optimal projection constant is given in closed form, while for −Δ + V with 0 ≤ V ∈ L∞ explicit constants follow, and composite discretisation removes ∥V ∥L∞ dependence.","arXiv :2607 .04247v1 [math .NA] 5 Jul 2026  \nGuaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods with Application to Schr¨odinger Operators ∗  \nXuefeng Liu†  \nAbstract  \nSpectral Galerkin methods are renowned for high-precision eigenvalue approximation, yet a rigorous lower bound obtained directly from a spectral discretisation has remained unavailable: the classical Kato and Weinstein–Temple enclosures do apply, but require a priori information on a neighbouring eigenvalue. This paper resolves the issue by extending the author’s projection-based framework for guaranteed lower eigenvalue bounds—so far realised only through finite element methods—to conforming spectral Galerkin methods. For trial spaces of exact eigenfunctions the required projection constant is the closed-form optimal value CN = λ−M121, the inverse square root of the first omitted eigenvalue. For −∆ + V with 0 ≤ V ∈ L∞ , a projection-gap estimate yields an explicit constant for the standard Galerkin matrix (exact at V = 0), and a composite discretisation removes the ∥V ∥L∞ -dependence for large potentials. With Neumann domain truncation these give certified two-sided bounds on Rd ; for two benchmark potentials on R2 the spectral enclosures match or surpass certified finite element ones at two orders of magnitude fewer degrees of freedom. The same auxiliaryprojector mechanism extends to singular potentials with an unbounded L ∞ norm—in particular to attractive Coulomb singularities in three dimensions, via a localised Hardy inequality—which we develop in a companion paper.  \nKeywords: eigenvalue bounds, spectral Galerkin method, verified computation, projection error constant, Schr¨odinger operator, domain truncation  \nMSC codes: 65N25, 65N35, 65G20, 35P15, 81Q05  \n1 Introduction  \nThe eigenvalues of self-adjoint differential operators govern vibration frequencies, stability thresholds, spectral gaps of quantum systems, and the a posteriori  \n∗ Submitted to the editors July 7, 2026 .  \n†Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan ([xfliu@lab.twcu.ac.jp](xfliu@lab.twcu.ac.jp)).  \nerror constants used in computer-assisted proofs for partial differential equations. The Rayleigh–Ritz principle makes certified upper bounds easy: every conforming Galerkin eigenvalue lies above the corresponding exact one. Certified lower bounds are substantially harder. Classical approaches—Temple– Kato bounds [25, 11], the Lehmann–Goerisch method [13, 2], Weinberger’sand Kuttler–Sigillito’s constructions [26, 12], and homotopy-based enclosures [22, 21]—require a priori spectral information (e.g. a rough lower bound for the next eigenvalue) or problem-specific auxiliary constructions (e.g. a base problem with a closed-form eigensystem) .  \nA different route was opened by the early ideas of Birkhoff et al. [3], Kikuchi– Liu [9, 14], Liu–Oishi [15], Kobayashi [10], and Carstensen et al. [5, 6], all seeking eigenvalue bounds without a priori spectral information. This approach is formalised in an abstract Hilbert-space framework by Liu [16, 17]: if the Ritz projection Ph onto the trial space satisfies an explicit projection-error estimate ∥ (I − Ph)v∥b ≤ Ch ∥ (I − Ph)v∥a , then every Galerkin eigenvalue λk,h yields, unconditionally, the bound  \nλk ≥ 1~~ ~~+~~ ~~λkC2h,hλk,h , k = 1 , 2 ,..., dim Vh. (1.1)  \nNo a priori information on the spectrum is needed. In existing realisations of (1.1) the constant Ch arises from the error estimation of finite element methods (FEM) on meshes: it is obtained either from global error estimation via the hypercircle method or interpolation error constants for conforming FEMs or from local interpolation estimates for non-conforming FEMs. Examples include the Crouzeix–Raviart and enriched Crouzeix–Raviart elements for the Laplacian and Schr¨odinger operators [16, 17 , 18], Crouzeix–Raviart pairs for the Stokes problem [27], trace-type constants for the Steklov problem [28], and the Fujino– Morley el","cbCaick7SxtNqV8c","https://ap.wps.com/l/cbCaick7SxtNqV8c","pdf",648600,4,1,28,"English","en",105,"# Introduction\n# Contributions\n## Spectral projection constant\n## Projection-gap mechanism for −Δ + V\n## Removing ∥V∥L∞ dependence\n## Extension to singular potentials\n# Guaranteed lower bounds framework","[{\"question\":\"Why were guaranteed lower eigenvalue bounds difficult for spectral Galerkin methods before this work?\",\"answer\":\"Classical guaranteed lower-bound tools require a priori spectral information about nearby eigenvalues, and existing projection-based constants were tied to finite element mesh/interpolation structure. No explicit spectral Galerkin projection constant with the needed form had been published, leaving a gap for guaranteed lower bounds derived directly from spectral discretisations.\"},{\"question\":\"What is the optimal projection constant when the trial space is spanned by the first M exact eigenfunctions?\",\"answer\":\"For such trial spaces, the required projection constant is given in closed form as CN = λ−M+1, where λM+1 denotes the first omitted eigenvalue.\"},{\"question\":\"How does the paper handle potentials and prevent dependence on ∥V∥L∞ for large potentials?\",\"answer\":\"For −Δ + V with 0 ≤ V ∈ L∞, it first uses a projection-gap estimate to obtain explicit constants for the standard Galerkin matrix. Then a composite discretisation (a bandlimited under-approximation with a slaved weighted projection) removes the ∥V∥L∞ dependence for large potentials, enabling certified two-sided bounds under Neumann domain truncation.\"}]","Guaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods with Application to Schrödinger Operators | PDF",1784176911,71,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"guaranteed-lower-eigenvalue-bounds-for-spectral-galerkin-methods-with-application-to-schrodinger-operators","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":20},"https://docshare.wps.com/document/guaranteed-lower-eigenvalue-bounds-for-spectral-galerkin-methods-with-application-to-schrodinger-operators/81893/",{"url":53,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why were guaranteed lower eigenvalue bounds difficult for spectral Galerkin methods before this work?","Question",{"text":76,"@type":77},"Classical guaranteed lower-bound tools require a priori spectral information about nearby eigenvalues, and existing projection-based constants were tied to finite element mesh/interpolation structure. No explicit spectral Galerkin projection constant with the needed form had been published, leaving a gap for guaranteed lower bounds derived directly from spectral discretisations.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the optimal projection constant when the trial space is spanned by the first M exact eigenfunctions?",{"text":81,"@type":77},"For such trial spaces, the required projection constant is given in closed form as CN = λ−M+1, where λM+1 denotes the first omitted eigenvalue.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the paper handle potentials and prevent dependence on ∥V∥L∞ for large potentials?",{"text":85,"@type":77},"For −Δ + V with 0 ≤ V ∈ L∞, it first uses a projection-gap estimate to obtain explicit constants for the standard Galerkin matrix. Then a composite discretisation (a bandlimited under-approximation with a slaved weighted projection) removes the ∥V∥L∞ dependence for large potentials, enabling certified two-sided bounds under Neumann domain truncation.","https://schema.org",{"og:url":53,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]