[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-183601-105":59,"doc-detail-183601-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","equidistribution-of-cusp-forms-on-pslz-pslr","Equidistribution of Cusp Forms on PSL₂(Z) - PSL₂(R)","","The paper studies equidistribution properties of cusp forms on the non-compact quotient PSL₂(Z)\\PSL₂(R) under ergodic geodesic flow. It formulates and proves an analogue of earlier results on PSL₂(Z)\\PSL₂(R) by lifting measures to the unit tangent bundle and analyzing the Wigner distributions associated with zero-order pseudodifferential operators. The main results show decay of matrix elements against smooth test functions, with sharp exponents, and establish non-positivity for certain Wigner distributions. The analysis relies on Fourier expansions, approximation arguments, exponential sum estimates via the Petersson–Kuznetsov trace formula, and bounds for Fourier coefficients.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/equidistribution-of-cusp-forms-on-pslz-pslr/183601/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/equidistribution-of-cusp-forms-on-pslz-pslr/183601.png","ImageObject",300,407,{"name":92,"@type":93},"Lute","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-24","2026-09-02",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What equidistribution problem does the paper address?","Question",{"text":112,"@type":113},"It investigates how measures derived from cusp-form eigenfunctions become equidistributed in the phase space associated to PSL₂(Z)\\PSL₂(R), extending earlier compact-surface results to the non-compact setting.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the role of Wigner distributions in the main approach?",{"text":117,"@type":113},"Wigner distributions lift the cusp-form measures from the quotient X to the unit tangent bundle S*X (or its generalization), enabling the authors to state and prove the equidistribution theorem in that lifted setting.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the paper show about the positivity of Wigner distributions?",{"text":121,"@type":113},"It proves that, for certain test functions, the Wigner distribution is not a positive distribution, motivating the study of a related positive counterpart dv_j defined through Friedrichs symmetrization.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},183601,1788353315,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":56,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143,"preview_page_count":4},137454149569,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","ANNALES DE L'INSTITUT FOURIER  \nDMITRI JAKOBSONEquidistribution of cusp forms on PSL₂(Z)\\PSL₂(R)  \nAnnales de l'institut Fourier,tome 47,n3(1997),p.967-984\u003Chttp://www.numdam.org/item?id=AIF_1997__47_3_967_0>  \n◎ Annales de l'institut Fourier,1997,tous droits réservés.  \nL'accès aux archives de la revue《 Annales de I'institut Fourier》(http://annalif.ujf-grenoble.fr/)implique I'accord avec les conditions gé-nérales d'utilisation (http//www.numdam.org/conditions).Toute utilisa-tion commerciale ou impression systématique est constitutive d’une in-fracion pénale.Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.  \nNUMDAM  \nArticle numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/  \n# EQUIDISTRIBUTION OF CUSP FORMS ON\n\nPSL₂(Z)\\PSL₂(R)  \nby Dmitri JAKOBSON  \n## 1.Introduction.\n\nLet X be a compact Riemannian manifold with ergodic geodesic flow,  \n△ the corresponding Laplace-Beltrami operator,φj an orthonormal basisof eigenfunction with eigenvalues 0=λo\u003Cλ1≤λ2≤……Let dμj denotethe probability measures lφ;l²du where dv is the volume form on X.Shnirelman,Zelditch and Colin de Verdiere([Sn74],[Sn93],[Ze87],[CdV])have proved that there exists a subsequence入je of λ;of the full density(i.e.\\#{k:jk≤n}～n as n→x)such that for every f∈C∞(X),Jfdμjk→ʃf dv.Zelditch in [Ze92]extended that result to non-compactX=PSL₂(Z)\\H.He also proved that for every)such thatJxf dv=0  \n(1)  \n(here φjin the definition of dμj form a basis of the space of cusp forms-  \nL²-eigenfunctions of the hyperbolic Laplacian△on X).  \nIn [LS]Luo and Sarnak proved (assuming that φj-s are Heckeeigenforms)the following  \nTHEOREM 1.—Letand let.Then  \n(2)  \nfor any ε>0.  \n   \nDMITRY JAKOBSON  \nThis improves(1)(proved for more general surfaces).Luo and Sarnakalso remark that the exponent 1/2 in (2)is the best possible.  \nTheorem 1 is proved without considering S*X=PSL₂(Z)\\PSL₂(R)=T\\G(unlike(1)which is a corollary of a statement about S*X)so it is nat-ural to generalize the theorem to a statement about T\\G,which we do inthis paper.  \nTo formulate an analogue of Theorem 1 one has to “lift”the measuresdμj on Xto measures dwj on S*X.This is done as follows:given a pseudo-differential operator A of order zero with symbol σA,one defines the Wignerdistribution dwj by  \n(3)  \nIn the usual coordinates {(x,y,θ):y>0,0≤θ\u003C2π}of the Iwasawadecomposition(1)on SL₂(R)dwj is given (cf.[Ze91])by  \ndwj=φj(z)ū;(z,θ)dw,  \n(4)  \nwhere the Liouville measure dw on T\\G is given by(dxdydθ)/(2πy²),is anwhere φj,κ(z)are“shifted”Maass cusp forms of weight 2k.eigenfunctions of Casimir operator with the same eigenvaluefor every k.The Fourier expansion of φj and φj,k-s was computed in[Ja94].  \nThe Fourier expansion of φ;in(x,y,θ)coordinates is given by  \n(5)  \n,Wo,ir,is a Whittaker function,e(nx)denotes e2πinwhereandwhere λ;(n)-s are Hecke eigenvalues of 4j·  \nThe Fourier expansion of 4j,k for k>0(weight 2k)is given by  \nHere Wk,ir,is a Whittaker function and c;(n)-s are as before.  \nThe Fourier expansion of φj,-k,k>0(weight -2k)is given by  \nThe analogue of Theorem 1 can now be stated as follows:  \nTHEOREM 2.—Let f be a finite sum of functions of even weight andof compact support on r\\G,and let Jr\\cf dw=0.Then for every e>0  \n(8)  \nWigner distribution dw;is not a positive distribution.In some prob-lems it is necessary to consider its positive counterpart which we shall calldv;and which is defined by  \n(o,dvj〉=(F,dwj)  \n(9)  \nwhere σFis a Friedrichs symmetrization of o(cf.[Ze87],[Ze91]).One canshow([Ze91],Prop.3.8)that o-σFis a symbol of order-1+e for any  \nE>0,so  \nKf,dv;〉-\u003Cf,dw;>|《e |A;|-1/2+e.  \nThis implies that the estimate(8)also holds with dwj-s replaced by dv;-S.  \nTheorem 2 is proved by an approximation argument.It is necessaryto study the asymptotics of (f,dwj〉with f being an incomplete Eisensteinseries or Poincaré series on T\\G.Studying the asymptotics can be reducedto estimating exponential sums by using Petersson-Ku","cbCaiuikNvnKvi9O","https://ap.wps.com/l/cbCaiuikNvnKvi9O","pdf",1246933,"English","# 1. Introduction\n## 2. Incomplete Eisenstein series","[{\"question\":\"What equidistribution problem does the paper address?\",\"answer\":\"It investigates how measures derived from cusp-form eigenfunctions become equidistributed in the phase space associated to PSL₂(Z)\\\\PSL₂(R), extending earlier compact-surface results to the non-compact setting.\"},{\"question\":\"What is the role of Wigner distributions in the main approach?\",\"answer\":\"Wigner distributions lift the cusp-form measures from the quotient X to the unit tangent bundle S*X (or its generalization), enabling the authors to state and prove the equidistribution theorem in that lifted setting.\"},{\"question\":\"What does the paper show about the positivity of Wigner distributions?\",\"answer\":\"It proves that, for certain test functions, the Wigner distribution is not a positive distribution, motivating the study of a related positive counterpart dv_j defined through Friedrichs symmetrization.\"}]","Equidistribution of Cusp Forms on PSL₂(Z) - PSL₂(R) | PDF",48]