{"title":"Morse势扰动的Weyl渐近性及其与Riemann-zeta函数的联系","authors":"R. Rahm","doi":"10.1515/conop-2022-0139","DOIUrl":null,"url":null,"abstract":"Abstract Let N ( T ; V ) N\\left(T;\\hspace{0.33em}V) denote the number of eigenvalues of the Schrödinger operator − y ″ + V y -{y}^{^{\\prime\\prime} }+Vy with absolute value less than T T . This article studies the Weyl asymptotics of perturbations of the Schrödinger operator − y ″ + 1 4 e 2 t y -{y}^{^{\\prime\\prime} }+\\frac{1}{4}{e}^{2t}y on [ x 0 , ∞ ) \\left[{x}_{0},\\infty ) . In particular, we show that perturbations by functions ε ( t ) \\varepsilon \\left(t) that satisfy ∣ ε ( t ) ∣ ≲ e t | \\varepsilon \\left(t)| \\hspace{0.33em}\\lesssim \\hspace{0.33em}{e}^{t} do not change the Weyl asymptotics very much. Special emphasis is placed on connections to the asymptotics of the zeros of the Riemann zeta function.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"10 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2018-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Weyl asymptotics for perturbations of Morse potential and connections to the Riemann zeta function\",\"authors\":\"R. Rahm\",\"doi\":\"10.1515/conop-2022-0139\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let N ( T ; V ) N\\\\left(T;\\\\hspace{0.33em}V) denote the number of eigenvalues of the Schrödinger operator − y ″ + V y -{y}^{^{\\\\prime\\\\prime} }+Vy with absolute value less than T T . This article studies the Weyl asymptotics of perturbations of the Schrödinger operator − y ″ + 1 4 e 2 t y -{y}^{^{\\\\prime\\\\prime} }+\\\\frac{1}{4}{e}^{2t}y on [ x 0 , ∞ ) \\\\left[{x}_{0},\\\\infty ) . In particular, we show that perturbations by functions ε ( t ) \\\\varepsilon \\\\left(t) that satisfy ∣ ε ( t ) ∣ ≲ e t | \\\\varepsilon \\\\left(t)| \\\\hspace{0.33em}\\\\lesssim \\\\hspace{0.33em}{e}^{t} do not change the Weyl asymptotics very much. Special emphasis is placed on connections to the asymptotics of the zeros of the Riemann zeta function.\",\"PeriodicalId\":53800,\"journal\":{\"name\":\"Concrete Operators\",\"volume\":\"10 1\",\"pages\":\"\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2018-11-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Concrete Operators\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/conop-2022-0139\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Concrete Operators","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/conop-2022-0139","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Weyl asymptotics for perturbations of Morse potential and connections to the Riemann zeta function
Abstract Let N ( T ; V ) N\left(T;\hspace{0.33em}V) denote the number of eigenvalues of the Schrödinger operator − y ″ + V y -{y}^{^{\prime\prime} }+Vy with absolute value less than T T . This article studies the Weyl asymptotics of perturbations of the Schrödinger operator − y ″ + 1 4 e 2 t y -{y}^{^{\prime\prime} }+\frac{1}{4}{e}^{2t}y on [ x 0 , ∞ ) \left[{x}_{0},\infty ) . In particular, we show that perturbations by functions ε ( t ) \varepsilon \left(t) that satisfy ∣ ε ( t ) ∣ ≲ e t | \varepsilon \left(t)| \hspace{0.33em}\lesssim \hspace{0.33em}{e}^{t} do not change the Weyl asymptotics very much. Special emphasis is placed on connections to the asymptotics of the zeros of the Riemann zeta function.