Limit behavior of Weyl coefficients

IF 0.7 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal Pub Date : 2022-08-24 DOI:10.1090/spmj/1729

R. Pruckner, H. Woracek

{"title":"Limit behavior of Weyl coefficients","authors":"R. Pruckner, H. Woracek","doi":"10.1090/spmj/1729","DOIUrl":null,"url":null,"abstract":"<p>The sets of radial or nontangential limit points towards <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i normal infinity\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>i</mml:mi>\n <mml:mi mathvariant=\"normal\">∞</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">i\\infty</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of a Nevanlinna function <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q\">\n <mml:semantics>\n <mml:mi>q</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are studied. Given a nonempty, closed, and connected subset <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper L\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">L</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">{\\mathcal {L}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C Subscript plus Baseline overbar\">\n <mml:semantics>\n <mml:mover>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo>+</mml:mo>\n </mml:msub>\n <mml:mo accent=\"false\">¯</mml:mo>\n </mml:mover>\n <mml:annotation encoding=\"application/x-tex\">\\overline {{\\mathbb {C}}_+}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, a Hamiltonian <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is constructed explicitly such that the radial and outer angular cluster sets towards <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i normal infinity\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>i</mml:mi>\n <mml:mi mathvariant=\"normal\">∞</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">i\\infty</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of the Weyl coefficient <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q Subscript upper H\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>q</mml:mi>\n <mml:mi>H</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">q_H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are both equal to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper L\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">L</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">{\\mathcal {L}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. The method is based on a study of the continuous group action of rescaling operators on the set of all Hamiltonians.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1729","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 1

Abstract

The sets of radial or nontangential limit points towards i ∞ i\infty of a Nevanlinna function q q are studied. Given a nonempty, closed, and connected subset L {\mathcal {L}} of C + ¯ \overline {{\mathbb {C}}_+} , a Hamiltonian H H is constructed explicitly such that the radial and outer angular cluster sets towards i ∞ i\infty of the Weyl coefficient q H q_H are both equal to L {\mathcal {L}} . The method is based on a study of the continuous group action of rescaling operators on the set of all Hamiltonians.

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Weyl系数的极限行为

研究了Nevanlinna函数q q向i∞i \infty方向的径向或非切向极限点集。给定一个非空的、封闭的、连通的子集L {\mathcal L{ (C +¯}}\overline{{\mathbb C＿{+)}}明确地构造了一个哈密顿量H H，使得Weyl系数q H q＿H向i∞i }\infty方向的径向和外角簇集都等于L {\mathcal L{。该方法是基于对所有哈密顿算子集合上的重标算子的连续群作用的研究。}}

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来源期刊

St Petersburg Mathematical Journal MATHEMATICS-

CiteScore

1.00

自引率

12.50%

发文量

审稿时长

>12 weeks

期刊介绍： This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.