Mathematische Nachrichten最新文献

{"title":"Curvature and Weitzenböck formula for spectral triples","authors":"Bram Mesland,&nbsp;Adam Rennie","doi":"10.1002/mana.202400158","DOIUrl":"https://doi.org/10.1002/mana.202400158","url":null,"abstract":"<p>Using the Levi-Civita connection on the noncommutative differential 1-forms of a spectral triple <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>B</mi>\u0000 <mo>,</mo>\u0000 <mi>H</mi>\u0000 <mo>,</mo>\u0000 <mi>D</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(mathcal {B},mathcal {H},mathcal {D})$</annotation>\u0000 </semantics></math>, we define the full Riemann curvature tensor, the Ricci curvature tensor and scalar curvature. We give a definition of Dirac spectral triples and derive a general Weitzenböck formula for them. We apply these tools to <span></span><math>\u0000 <semantics>\u0000 <mi>θ</mi>\u0000 <annotation>$theta$</annotation>\u0000 </semantics></math>-deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under <span></span><math>\u0000 <semantics>\u0000 <mi>θ</mi>\u0000 <annotation>$theta$</annotation>\u0000 </semantics></math>-deformation, whereas the connection Laplacian, Clifford representation of the curvature, and the scalar curvature are all invariant under deformation.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4582-4604"},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400158","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Riemannian 4-manifolds and their twistor spaces: A moving frame approach","authors":"Giovanni Catino,&nbsp;Davide Dameno,&nbsp;Paolo Mastrolia","doi":"10.1002/mana.202300577","DOIUrl":"https://doi.org/10.1002/mana.202300577","url":null,"abstract":"<p>In this paper, we study the twistor space <span></span><math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>$Z$</annotation>\u0000 </semantics></math> of an oriented Riemannian 4-manifold <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> using the moving frame approach, focusing, in particular, on the Einstein, non-self-dual setting. We prove that any general first-order linear condition on the almost complex structures of <span></span><math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>$Z$</annotation>\u0000 </semantics></math> forces the underlying manifold <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> to be self-dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first-order quadratic conditions, showing that the Atiyah–Hitchin–Singer almost Hermitian twistor space of an Einstein 4-manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4651-4670"},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300577","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Visco-elastic damped wave models with time-dependent coefficient","authors":"Halit Sevki Aslan,&nbsp;Michael Reissig","doi":"10.1002/mana.202300341","DOIUrl":"https://doi.org/10.1002/mana.202300341","url":null,"abstract":"<p>In this paper, we study the following Cauchy problem for linear visco-elastic damped wave models with a general time-dependent coefficient <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mo>=</mo>\u0000 <mi>g</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$g=g(t)$</annotation>\u0000 </semantics></math>:\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4535-4581"},"PeriodicalIF":0.8,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300341","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On temporal regularity and polynomial decay of solutions for a class of nonlinear time-delayed fractional reaction–diffusion equations","authors":"Tran Thi Thu,&nbsp;Tran Van Tuan","doi":"10.1002/mana.202300434","DOIUrl":"https://doi.org/10.1002/mana.202300434","url":null,"abstract":"<p>This paper is devoted to analyzing the regularity in time and polynomial decay of solutions for a class of fractional reaction–diffusion equations (FrRDEs) involving delays and nonlinear perturbations in a bounded domain of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mo>R</mo>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <annotation>$operatorname{mathbf {R}}^{d}$</annotation>\u0000 </semantics></math>. By establishing some regularity estimates in both time and space variables of the resolvent operator, we present results on the Hölder and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$C^{1}$</annotation>\u0000 </semantics></math>-regularity in time of solutions for both time-delayed linear and semilinear FrRDEs. Based on the aforementioned results, we study the existence, uniqueness, and regularity of solutions to an identification problem subjected to the delay FrRDE and the additional observations given at final time. Furthermore, under quite reasonable assumptions on nonlinear perturbations and the technique of measure of noncompactness, the existence of decay solutions with polynomial rates for the problem under consideration is shown.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4510-4534"},"PeriodicalIF":0.8,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Curves on Brill–Noether special K3 surfaces","authors":"Richard Haburcak","doi":"10.1002/mana.202300403","DOIUrl":"https://doi.org/10.1002/mana.202300403","url":null,"abstract":"<p>Mukai showed that projective models of Brill–Noether general polarized K3 surfaces of genus 6–10 and 12 are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill–Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill–Noether generality of any polarized K3 surface <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>,</mo>\u0000 <mi>H</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(S,H)$</annotation>\u0000 </semantics></math> is equivalent to the Brill–Noether generality of smooth curves <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math> in the linear system <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>H</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <annotation>$|H|$</annotation>\u0000 </semantics></math>, is still open. Using Lazarsfeld–Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>≤</mo>\u0000 <mn>19</mn>\u0000 </mrow>\u0000 <annotation>$le 19$</annotation>\u0000 </semantics></math>, which provides a new and unified proof even in the genera where Mukai models exist.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4497-4509"},"PeriodicalIF":0.8,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300403","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Boundary controllability of a Korteweg–de Vries-type Boussinesq system","authors":"Vilmos Komornik,&nbsp;Ademir F. Pazoto,&nbsp;Miguel D. Soto Vieira","doi":"10.1002/mana.202400201","DOIUrl":"https://doi.org/10.1002/mana.202400201","url":null,"abstract":"<p>The two-way propagation of a certain class of long-crested water waves is governed approximately by systems of equations of the Boussinesq type. These equations have been put forward in various forms by many authors and their higher-order generalizations arise when modeling the propagation of waves on large lakes, ocean, and in other contexts. Considered here is a class of such system which couple two higher-order Korteweg–de-Vries type equations. Our aim is to investigate the controllability properties of the linearized model posed on a periodic interval. By using the classical duality approach and some theorems on nonharmonic Fourier series, we prove that the system is exactly controllable in certain well-chosen Sobolev spaces by means of suitable boundary controls.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4478-4496"},"PeriodicalIF":0.8,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ε\u0000 $varepsilon$\u0000 -Regularity criteria and the number of singular points for the 3D simplified Ericksen–Leslie system","authors":"Zhongbao Zuo","doi":"10.1002/mana.202400071","DOIUrl":"https://doi.org/10.1002/mana.202400071","url":null,"abstract":"<p>In this paper, we consider the partial regularity of suitable weak solution to the 3D simplify Ericksen–Leslie system modeling the hydrodynamical motion of nematic liquid crystal flow, which is a coupled system with the Navier–Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. We present a new regularity criteria for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Moreover, under the condition\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4370-4388"},"PeriodicalIF":0.8,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142859984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Heat kernel gradient estimates for the Vicsek set","authors":"Fabrice Baudoin,&nbsp;Li Chen","doi":"10.1002/mana.202400180","DOIUrl":"https://doi.org/10.1002/mana.202400180","url":null,"abstract":"<p>We prove pointwise and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$L^p$</annotation>\u0000 </semantics></math> gradient estimates for the heat kernel on the bounded and unbounded Vicsek set and applications to Sobolev inequalities are given. We also define a Hodge semigroup in that setting and prove estimates for its kernel.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4450-4477"},"PeriodicalIF":0.8,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400180","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142859985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dunkl convolution and elliptic regularity for Dunkl operators","authors":"Dominik Brennecken","doi":"10.1002/mana.202300370","DOIUrl":"https://doi.org/10.1002/mana.202300370","url":null,"abstract":"&lt;p&gt;We discuss in which cases the Dunkl convolution &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$u*_kv$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of distributions &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$u,v$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, possibly both with non-compact support, can be defined and study its analytic properties. We prove results on the (singular-)support of Dunkl convolutions. Based on this, we are able to prove a theorem on elliptic regularity for a certain class of Dunkl operators, called elliptic Dunkl operators. Finally, for the root system &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$A_{n-1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; we consider the Riesz distributions &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$(R_alpha)_{alpha in mathbb {C}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and prove that their Dunkl convolution exists and that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$R_alpha *_kR_beta = R_{alpha +b","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4416-4436"},"PeriodicalIF":0.8,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300370","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142862193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the real spectrum of differential operators with PT-symmetric periodic matrix coefficients","authors":"Oktay A. Veliev","doi":"10.1002/mana.202300558","DOIUrl":"https://doi.org/10.1002/mana.202300558","url":null,"abstract":"&lt;p&gt;We study the spectrum of the operator &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;annotation&gt;$T$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; generated by the differential expression of order &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$n&amp;gt;2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mtimes m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; Parity-Time (PT)-symmetric periodic matrix coefficients. The case when &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are the odd numbers was investigated in [18]. In this paper, we consider the all remained cases: (a) &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an odd number and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an even number, (b) &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an even number and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an arbitrary positive integer. We find conditions on the coefficients under which in the cases (a) and (b) the spectrum of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;annotation&gt;$T$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; contains the sets &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(-infty,-H]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;∪&lt;/mo&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$cup [H,infty)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4437-4449"},"PeriodicalIF":0.8,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142862194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}