{"title":"双光子不对称量子拉比模型的大特征值行为","authors":"A. Boutet de Monvel, M. Charif, L. Zielinski","doi":"10.1090/spmj/1793","DOIUrl":null,"url":null,"abstract":"<p>The asymptotic behavior of large eigenvalues is studied for the two-photon quantum Rabi model with a finite bias. It is proved that the spectrum of this Hamiltonian model consists of two eigenvalue sequences <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace upper E Subscript n Superscript plus Baseline right-brace Subscript n equals 0 Superscript normal infinity\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:msubsup> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\lbrace E_n^+\\rbrace _{n=0}^{\\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace upper E Subscript n Superscript minus Baseline right-brace Subscript n equals 0 Superscript normal infinity\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:msubsup> <mml:msubsup> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\lbrace E_n^-\\rbrace _{n=0}^{\\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and their large <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> asymptotic behavior with error term <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper O left-parenthesis n Superscript negative 1 slash 2 Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">O</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\operatorname {O}(n^{-1/2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is described. The principal tool is the method of near-similarity of operators introduced by G. V. Rozenblum and developed in works of J. Janas, S. Naboko, and E. A. Yanovich (Tur).</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Behavior of large eigenvalues for the two-photon asymmetric quantum Rabi model\",\"authors\":\"A. Boutet de Monvel, M. Charif, L. Zielinski\",\"doi\":\"10.1090/spmj/1793\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The asymptotic behavior of large eigenvalues is studied for the two-photon quantum Rabi model with a finite bias. It is proved that the spectrum of this Hamiltonian model consists of two eigenvalue sequences <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace upper E Subscript n Superscript plus Baseline right-brace Subscript n equals 0 Superscript normal infinity\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:msubsup> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\lbrace E_n^+\\\\rbrace _{n=0}^{\\\\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace upper E Subscript n Superscript minus Baseline right-brace Subscript n equals 0 Superscript normal infinity\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:msubsup> <mml:msubsup> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\lbrace E_n^-\\\\rbrace _{n=0}^{\\\\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and their large <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> asymptotic behavior with error term <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper O left-parenthesis n Superscript negative 1 slash 2 Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">O</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\operatorname {O}(n^{-1/2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is described. The principal tool is the method of near-similarity of operators introduced by G. V. Rozenblum and developed in works of J. Janas, S. Naboko, and E. A. Yanovich (Tur).</p>\",\"PeriodicalId\":51162,\"journal\":{\"name\":\"St Petersburg Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-04-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"St Petersburg Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/spmj/1793\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1793","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: 0
摘要
研究了具有有限偏置的双光子量子拉比模型的大特征值渐近行为。研究证明，该哈密顿模型的谱由两个特征值序列组成 { E n + } n = 0 ∞ \lbrace E_n^+\rbrace _{n=0}^{\infty } ， { E n - } n = 0 ∞ \lbrace E_n^+\rbrace _{n=0}^{\infty } 。 { E n - } n = 0 ∞ \lbrace E_n^-\rbrace _{n=0}^{\infty } ，以及它们的大 n n 渐近线。 描述了它们的大 n n 渐近行为，误差项为 O ( n - 1 / 2 ) \operatorname {O}(n^{-1/2}) 。主要工具是由 G. V. Rozenblum 引入并在 J. Janas、S. Naboko 和 E. A. Yanovich (Tur) 的著作中发展的算子近似方法。
Behavior of large eigenvalues for the two-photon asymmetric quantum Rabi model
The asymptotic behavior of large eigenvalues is studied for the two-photon quantum Rabi model with a finite bias. It is proved that the spectrum of this Hamiltonian model consists of two eigenvalue sequences {En+}n=0∞\lbrace E_n^+\rbrace _{n=0}^{\infty }, {En−}n=0∞\lbrace E_n^-\rbrace _{n=0}^{\infty }, and their large nn asymptotic behavior with error term O(n−1/2)\operatorname {O}(n^{-1/2}) is described. The principal tool is the method of near-similarity of operators introduced by G. V. Rozenblum and developed in works of J. Janas, S. Naboko, and E. A. Yanovich (Tur).
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.
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