St Petersburg Mathematical Journal最新文献

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}