A sufficient condition is established under which det(ABA−1B−1)=1det (ABA^{-1}B^{-1})=1 for a pair of bounded, invertible operators A,BA,B on a Hilbert space.
对于希尔伯特空间上的一对有界可逆算子 A , B A,B 来说，det ( A B A - 1 B - 1 ) = 1 det (ABA^{-1}B^{-1})=1 是一个充分条件。
{"title":"On Kitaev’s determinant formula","authors":"A. Elgart, M. Fraas","doi":"10.1090/spmj/1796","DOIUrl":"https://doi.org/10.1090/spmj/1796","url":null,"abstract":"<p>A sufficient condition is established under which <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"det left-parenthesis upper A upper B upper A Superscript negative 1 Baseline upper B Superscript negative 1 Baseline right-parenthesis equals 1\"> <mml:semantics> <mml:mrow> <mml:mo movablelimits=\"true\" form=\"prefix\">det</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">det (ABA^{-1}B^{-1})=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a pair of bounded, invertible operators <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A comma upper B\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A,B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a Hilbert space.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"162 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Gesztesy, L. Littlejohn, R. Nichols, M. Piorkowski, J. Stanfill
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a densely defined, closed, symmetric operator in the complex, separable Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with equal deficiency indices and denote by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i Baseline equals kernel left-parenthesis left-parenthesis ModifyingAbove upper A With dot right-parenthesis Superscript asterisk Baseline minus i upper I Subscript script upper H Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ker</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {N}_i = ker ((dot {A})^* - i I_{mathcal {H}})</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="dimension left-parenthesis script upper N Subscript i Baseline right-parenthesis equals k element-of double-struck upper N union StartSet normal infinity EndSet"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">dim (mathcal {N}_i)=kin mathbb {N} cup {infty }</mml:annotation> </mml:
让 A ˙ dot {A} 是复数可分离希尔伯特空间 H mathcal {H} 中的一个密定义、闭合、对称算子，其缺陷指数相等，并用 N i = ker ( ( A ˙ ) ∗ - i I H ) 表示。 mathcal {N}_i = ker ((dot {A})^* - i I_{mathcal {H}}) , dim ( N i ) = k ∈ N∪ { ∞ } dim (mathcal {N}_i)=kin mathbb {N}cup {infty } ，A˙ dot {A} 的相关缺陷子空间。如果 A A 表示 A ˙ dot {A} 在 H mathcal {H} 中的自交扩展，则多诺霍 m m -operator M A , N i D o ( ⋅ ) M_{A,mathcal {N}_i}^{Do} (,cdot ,) 在 N i mathcal {N}_i 中与一对 ( A . ) 相关联、 N i ) (A,mathcal {N}_i) 由 M A , N i D o ( z ) = z I N i + ( z 2 + 1 ) P N i ( A - z I H ) - 1 P N i | N i M_{A,mathcal {N}_i}^{Do}(z)=zI_{mathcal {N}_i}+ (z^2+1) P_{mathcal {N}_i} (A - z I_{mathcal {H}})^{-1} P_{mathcal {N}_i}vert _{mathcal {N}_i} , z ∈ C ∖ R , zin mathbb {C}setminus mathbb {R}, with I N i I_{mathcal {N}_i} the identity operator in N i mathcal {N}_i 、和 P N i P_{{mathcal {N}_i} 是 H mathcal {H} 中到 N i mathcal {N}_i 的正交投影。假定系数 p , q , r p, q,r 的标准局部可整性假设，我们研究与微分表达式 τ = 1 r ( x ) [ - d d x p ( x ) d d x + q ( x ) ] tau =frac {1}{r(x)}[-frac {d}{dx}p(x)frac {d}{dx} + q(x)] a.e. 时对应的所有自联合实现。 x ∈ ( a , b ) ⊆ R xin (a,b) subseteq mathbb {R}, in L 2 ( ( a , b ) ; r d x ) L^2((a,b)；rdx)，并且，作为我们本文的主要目的，在 τ tau 至少在一个区间端点 a a 或 b b 处处于极限圆情况的所有情况下，系统地构建相关的多诺霍 m m - 函数（分别是 ( 2 × 2 ) (2 times 2) 矩阵）。
{"title":"Donoghue 𝑚-functions for Singular Sturm–Liouville operators","authors":"F. Gesztesy, L. Littlejohn, R. Nichols, M. Piorkowski, J. Stanfill","doi":"10.1090/spmj/1795","DOIUrl":"https://doi.org/10.1090/spmj/1795","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper A With dot\"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a densely defined, closed, symmetric operator in the complex, separable Hilbert space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"script\">H</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with equal deficiency indices and denote by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper N Subscript i Baseline equals kernel left-parenthesis left-parenthesis ModifyingAbove upper A With dot right-parenthesis Superscript asterisk Baseline minus i upper I Subscript script upper H Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"script\">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ker</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"script\">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {N}_i = ker ((dot {A})^* - i I_{mathcal {H}})</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=\"dimension left-parenthesis script upper N Subscript i Baseline right-parenthesis equals k element-of double-struck upper N union StartSet normal infinity EndSet\"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"script\">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dim (mathcal {N}_i)=kin mathbb {N} cup {infty }</mml:annotation> </mml:","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"163 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Exact controllability is proved on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first applies a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated with exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.
{"title":"Shape, velocity, and exact controllability for the wave equation on a graph with cycle","authors":"S. Avdonin, J. Edward, Y. Zhao","doi":"10.1090/spmj/1791","DOIUrl":"https://doi.org/10.1090/spmj/1791","url":null,"abstract":"<p>Exact controllability is proved on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first applies a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated with exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note is devoted to the study of complete nonselfadjointness for all maximally dissipative extensions of a Schrödinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. It is shown that all maximally dissipative extensions that preserve the differential expression are completely nonselfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. A characterization of these extensions and the corresponding subspaces is given, accompanied by a specific example.
{"title":"Complete nonselfadjointness for Schrödinger operators on the semi-axis","authors":"C. Fischbacher, S. Naboko, I. Wood","doi":"10.1090/spmj/1802","DOIUrl":"https://doi.org/10.1090/spmj/1802","url":null,"abstract":"<p>This note is devoted to the study of complete nonselfadjointness for all maximally dissipative extensions of a Schrödinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. It is shown that all maximally dissipative extensions that preserve the differential expression are completely nonselfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. A characterization of these extensions and the corresponding subspaces is given, accompanied by a specific example.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"135 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A series of results and methods is presented, which make it possible to trace the relationship between the number of inner zeros of nontrivial solutions of fourth order selfadjoint boundary problems with separated boundary conditions and the negative inertia index.
{"title":"Oscillatory properties of selfadjoint boundary problems of the fourth order","authors":"A. Vladimirov, A. Shkalikov","doi":"10.1090/spmj/1794","DOIUrl":"https://doi.org/10.1090/spmj/1794","url":null,"abstract":"<p>A series of results and methods is presented, which make it possible to trace the relationship between the number of inner zeros of nontrivial solutions of fourth order selfadjoint boundary problems with separated boundary conditions and the negative inertia index.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"135 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is a contribution to the theory of functional models. In particular, it develops the so-called spectral form of the functional model where the selfadjoint dilation of the operator is represented as the operator of multiplication by an independent variable in some auxiliary vector-valued function space. With the help of a Lagrange identity, in the present version the relationship between this auxiliary space and the original Hilbert space will be explicit. A simple example is provided.
{"title":"The spectral form of the functional model for maximally dissipative operators: A Lagrange identity approach","authors":"M. Brown, M. Marletta, S. Naboko, I. Wood","doi":"10.1090/spmj/1792","DOIUrl":"https://doi.org/10.1090/spmj/1792","url":null,"abstract":"<p>This paper is a contribution to the theory of functional models. In particular, it develops the so-called spectral form of the functional model where the selfadjoint dilation of the operator is represented as the operator of multiplication by an independent variable in some auxiliary vector-valued function space. With the help of a Lagrange identity, in the present version the relationship between this auxiliary space and the original Hilbert space will be explicit. A simple example is provided.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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).
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。
{"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":"https://doi.org/10.1090/spmj/1793","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":"42 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a wide class of block Jacobi matrices, the norms of Green matrix (resolvent) entries are estimated. This estimate depends on the rate of growth of the norms of the off-diagonal entries and on the distance from the spectral parameter to the essential spectrum if the latter is nonempty. The sharpness of this estimate is shown by an example.
{"title":"Estimates of Green matrix entries of selfadjoint unbounded block Jacobi matrices","authors":"S. Naboko, S. Simonov","doi":"10.1090/spmj/1800","DOIUrl":"https://doi.org/10.1090/spmj/1800","url":null,"abstract":"<p>In a wide class of block Jacobi matrices, the norms of Green matrix (resolvent) entries are estimated. This estimate depends on the rate of growth of the norms of the off-diagonal entries and on the distance from the spectral parameter to the essential spectrum if the latter is nonempty. The sharpness of this estimate is shown by an example.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The content of the paper is reflected by its title.
论文的标题反映了论文的内容。
{"title":"Absolutely continuous spectrum of a typical Schrödinger operator with an operator-valued potential","authors":"A. Laptev, O. Safronov","doi":"10.1090/spmj/1799","DOIUrl":"https://doi.org/10.1090/spmj/1799","url":null,"abstract":"<p>The content of the paper is reflected by its title.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Quasiperiodic solutions of the Gross–Pitaevskii equation with a periodic potential in dimension three are studied. It is proved that there is an extensive “nonresonant” set G⊂R3mathcal {G}subset mathbb {R}^3 such that for every vv kin mathcal {G} there is a solution asymptotically close to a plane wave Ae^{ilangle vv {k},vv {x}rangle } as |vv k|to infty , given AA is sufficiently small.
研究了三维中具有周期势的格罗斯-皮塔耶夫斯基方程的准周期解。研究证明，存在一个广泛的 "非共振 "集合 G ⊂ R 3 mathcal {G}subset mathbb {R}^3 ，这样对于 mathcal {G} 中的每一个 vv k 都有一个近似接近于平面波 Ae^{ilangle vv {k},vv {x}rangle } 的解，因为 |vv k|to infty ，给定 A A 足够小。
{"title":"Solutions of Gross–Pitaevskii equation with periodic potential in dimension three","authors":"Yu. Karpeshina, Seonguk Kim, R. Shterenberg","doi":"10.1090/spmj/1798","DOIUrl":"https://doi.org/10.1090/spmj/1798","url":null,"abstract":"<p>Quasiperiodic solutions of the Gross–Pitaevskii equation with a periodic potential in dimension three are studied. It is proved that there is an extensive “nonresonant” set <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper G subset-of double-struck upper R cubed\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"script\">G</mml:mi> </mml:mrow> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {G}subset mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that for every <inline-formula content-type=\"math/tex\"> <tex-math> vv kin mathcal {G}</tex-math></inline-formula> there is a solution asymptotically close to a plane wave <inline-formula content-type=\"math/tex\"> <tex-math> Ae^{ilangle vv {k},vv {x}rangle }</tex-math></inline-formula> as <inline-formula content-type=\"math/tex\"> <tex-math> |vv k|to infty </tex-math></inline-formula>, given <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is sufficiently small.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"13 2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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